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</div><h2>HL Paper 3</h2><div class="specification">
<p>The weights, <em>X</em> kg, of the males of a species of bird may be assumed to be normally distributed with mean 4.8 kg and standard deviation 0.2 kg.</p>
</div>
<div class="specification">
<p>The weights, <em>Y</em> kg, of female birds of the same species may be assumed to be normally distributed with mean 2.7 kg and standard deviation 0.15 kg.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a randomly chosen male bird weighs between 4.75 kg and 4.85 kg.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two randomly chosen male birds and three randomly chosen female birds are placed on a weighing machine that has a weight limit of 18 kg. Find the probability that the total weight of these five birds is greater than the weight limit.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 1, accept answers that round correctly to 2 significant figures.</p>
<p>P(4.75 < <em>X</em> < 4.85) = 0.197 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 1, accept answers that round correctly to 2 significant figures.</p>
<p>consider the random variable <em>X</em> − 2<em>Y</em> <em><strong>(M1)</strong></em></p>
<p>E(<em>X</em> − 2<em>Y</em>) = − 0.6 <em><strong>(A1)</strong></em></p>
<p>Var(<em>X</em> − 2<em>Y</em>) = Var(<em>X</em>) + 4Var(<em>Y</em>) <em><strong>(M1)</strong></em></p>
<p>= 0.13 <em><strong>(A1)</strong></em></p>
<p><em>X</em> − 2<em>Y</em> ∼ N(−0.6, 0.13)</p>
<p>P(<em>X</em> − 2<em>Y</em> > 0) <em><strong>(M1)</strong></em></p>
<p>= 0.0480 <strong> <em>A1</em></strong></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 1, accept answers that round correctly to 2 significant figures.</p>
<p>let <em>W</em> = <em>X</em><sub>1</sub> + <em>X</em><sub>2</sub> + <em>Y</em><sub>1</sub> + <em>Y</em><sub>2</sub> + <em>Y</em><sub>3</sub> be the total weight</p>
<p>E(<em>W</em>) = 17.7 <em><strong>(A1)</strong></em></p>
<p>Var(<em>W</em>) = 2Var(<em>X</em>) + 3Var(<em>Y</em>) = 0.1475 <em><strong> (M1)(A1)</strong></em></p>
<p>W ∼ N(17.7, 0.1475)</p>
<p>P(<em>W</em> > 18) = 0.217 <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables \(U,{\text{ }}V\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \).</p>
</div>
<div class="specification">
<p>A random sample of 12 observations on <em>U</em>, <em>V</em> is obtained to determine whether there is a correlation between <em>U and</em> <em>V</em>. The sample product moment correlation coefficient is denoted by <em>r</em>. A test to determine whether or not <em>U</em>, <em>V</em> are independent is carried out at the 1% level of significance.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to investigate whether or not \(U\), \(V\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of \(|r|\) for which the test concludes that \(\rho \ne 0\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({{\text{H}}_0}:\rho = 0;{\text{ }}{{\text{H}}_1}:\rho \ne 0\) <strong><em>A1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\nu = 10\) <strong><em>(A1)</em></strong></p>
<p>\({t_{0.005}} = 3.16927 \ldots \) <strong><em>(M1)(A1)</em></strong></p>
<p>we reject \({{\text{H}}_0}:\rho = 0\) if \(\left| t \right| > 3.16927 \ldots \) <strong><em>(R1)</em></strong></p>
<p>attempting to solve \(\left| r \right|\sqrt {\frac{{10}}{{1 - {r^2}}}} > 3.16927 \ldots \) for \(\left| r \right|\) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow = instead of >.</p>
<p> </p>
<p>(least value of \(\left| r \right|\) is) 0.708 (3 sf) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1M1A0R1M1A0 </em></strong>to candidates who use a one-tailed test. Award <strong><em>A0M1A0R1M1A0 </em></strong>to candidates who use an incorrect number of degrees of freedom or both a one-tailed test and incorrect degrees of freedom.</p>
<p> </p>
<p><strong>Note:</strong> Possible errors are</p>
<p>10 DF 1-tail, \(t = 2.763 \ldots \), least value \( = \) 0.658</p>
<p>11 DF 2-tail, \(t = 3.105 \ldots \), least value \( = \) 0.684</p>
<p>11 DF 1-tail, \(t = 2.718 \ldots \), least value \( = \) 0.634.</p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A biased cubical die has its faces labelled \(1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5\) and \(6\). The probability of rolling a \(6\) <span class="s1">is \(p\), with equal probabilities for the other scores.</span></p>
<p class="p2">The die is rolled once, and the score \({X_1}\) is noted.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \({\text{E}}({X_1})\)<span class="s1">.</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence obtain an unbiased estimator for \(p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The die is rolled a second time, and the score \({X_2}\) is noted.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(k({X_1} - 3) + \left( {\frac{1}{3} - k} \right)({X_2} - 3)\) is also an unbiased estimator for \(p\) for all values of \(k \in \mathbb{R}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value for \(k\), which maximizes the efficiency of this estimator.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">let \(X\) </span>denote the score on the die</p>
<p class="p2"><span class="s2">(i) <span class="Apple-converted-space"> </span></span>\({\text{P}}(X = x) = \left\{ {\begin{array}{*{20}{c}} {\frac{{1 - p}}{5},}&{x = 1,{\text{ 2}},{\text{ 3}},{\text{ 4}},{\text{ 5}}} \\ {p,}&{x = 6} \end{array}} \right.\) <strong><em>(M1)</em></strong></p>
<p class="p2">\(E({X_1}) = (1 + 2 + 3 + 4 + 5)\frac{{1 - p}}{5} + 6p\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2">\( = 3 + 3p\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>so an unbiased estimator for \(p\) would be \(\frac{{{X_1} - 3}}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(E\left( {k({X_1} - 3) + \left( {\frac{1}{3} - k} \right)({X_2} - 3)} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = kE({X_1} - 3) + \left( {\frac{1}{3} - k} \right)E({X_2} - 3)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = k(3p) + \left( {\frac{1}{3} - k} \right)(3p)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">any correct expression involving just \(k\) and \(p\)</p>
<p class="p1">\( = p\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1">hence \(k({X_1} - 3) + \left( {\frac{1}{3} - k} \right)({X_2} - 3)\) is an unbiased estimator of \(p\)</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}\left( {k({X_1} - 3) + \left( {\frac{1}{3} - k} \right)({X_2} - 3)} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = {k^2}{\text{Var}}({X_1} - 3) + {\left( {\frac{1}{3} - k} \right)^2}{\text{Var}}({X_2} - 3)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\( = \left( {{k^2} + {{\left( {\frac{1}{3} - k} \right)}^2}} \right){\sigma ^2}\) (where \({\sigma ^2}\) <span class="s1">denotes </span>\({\text{Var}}(X)\))</p>
<p class="p1">valid attempt to minimise the variance <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(k = \frac{1}{6}\)<span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept an argument which states that the most efficient estimator is the one having equal coefficients of \({X_1}\) and \({X_2}\).</p>
<p class="p1"><em><strong>[7 marks]</strong></em></p>
<p class="p1"><em><strong>Total [11 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">It is known that the standard deviation of the heights of men in a certain country is \(15.0\) cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">One hundred men from that country, selected at random, had their heights measured.</p>
<p class="p2"><span class="s1">The mean of this sample was \(185\) cm. Calculate a \(95\% \) </span>confidence interval for the mean height of the population.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A second random sample of size \(n\) is taken from the same population. Find the minimum value of \(n\) <span class="s1">needed for the width of a \(95\% \) confidence interval to be less than \(3\) cm.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to use \(\bar x \pm z\frac{\sigma }{{\sqrt n }}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\([182,{\text{ }}188]\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept answers that round to the correct \(3\) sf.</p>
<p class="p3"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(1.96 \times \frac{{15.0}}{{\sqrt n }} < 1.5\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1">\(n > {\left( {\frac{{15.0}}{{1.5}} \times 1.96} \right)^2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1</em></strong> for attempting to solve the inequality.</p>
<p class="p2"> </p>
<p class="p1"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Allow the use of \( = \)<span class="s1">.</span></p>
<p class="p2"> </p>
<p class="p1"><span class="s2">minimum value </span>\(n = 385\) <span class="Apple-converted-space"> </span><span class="s3"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s3"><strong><em>[4 marks]</em></strong></span></p>
<p class="p1"><span class="s3"><strong><em>Total [7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The strength of beams compared against the moisture content of the beam is indicated in the following table. You should assume that strength and moisture content are each normally distributed.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-21_om_17.54.38.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the product moment correlation coefficient for these data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Perform a two-tailed test, at the \(5\% \) level of significance, of the hypothesis that strength is independent of moisture content.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If the moisture content of a beam is found to be \(9.5\), use the appropriate regression line to estimate the strength of the beam.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(r = - 0.762\) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept answers that round to \( - 0.76\).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({H_0}:\) Moisture content and strength are independent or \(\rho = 0\)</p>
<p>\({H_1}:\) Moisture content and strength are not independent or \(\rho \ne 0\) <strong><em>A1</em></strong></p>
<p><strong>EITHER</strong></p>
<p>test statistic is \(-3.33\) <strong><em>A1</em></strong></p>
<p>critical value is \(( \pm ){\text{ }}2.306\) <strong><em>A1</em></strong></p>
<p>since \( - 3.33 < - 2.306\) or \(3.33 > 2.306\), <strong><em>R1</em></strong></p>
<p>reject \({H_0}\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>\(p\)-value is \(0.0104\) <strong><em>A2</em></strong></p>
<p>as \(0.0104 < 0.05\), <strong><em>R1</em></strong></p>
<p>reject \({H_0}\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The <strong><em>R1 </em></strong>and <strong><em>A1 </em></strong>can be awarded as follow through from their test statistic or \(p\)-value.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(x = {\text{strength}}\)</p>
<p>\(y = {\text{moisture content}}\)</p>
<p>\(x = - 0.629y + 28.1\) <strong><em>(M1)(A1)</em></strong></p>
<p>if \(y = 9.5\) so \(x = 22.1\) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Only accept answers that round to \(22.1\).</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A1M0A0 </em></strong>for the other regression line \(y = 30.1 - 0.924x\).</p>
<p><em><strong>[4 marks]</strong></em></p>
<p><em><strong>Total [11 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Anna cycles to her new school. She records the times taken for the first ten days with the following results (in minutes).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">12.4 13.7 12.5 13.4 13.8 12.3 14.0 12.8 12.6 13.5</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that these times are a random sample from the \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\) distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Calculate a 95 % confidence interval for \(\mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Before Anna calculated the confidence interval she thought that the value of \(\mu \) would be 12.5. In order to check this, she sets up the null hypothesis \({{\text{H}}_0}:\mu = 12.5\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use the above data to calculate the value of an appropriate test statistic. Find the corresponding <em>p</em>-value using a two-tailed test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Interpret your <em>p</em>-value at the 1 % level of significance, justifying your conclusion.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) estimate of \(\mu = 13.1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">estimate of \({\sigma ^2} = 0.416\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) using a GDC (or otherwise), the 95% confidence interval is <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">[12.6, 13.6] <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept open or closed intervals.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) \(t = \frac{{13.1 - 12.5}}{{0.6446 \ldots /\sqrt {10} }} = 2.94\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 9\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value \( = 2 \times {\text{P}}(T > 2.9433 \ldots )\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0164\,\,\,\,\,\)(accept 0.0165) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) we accept the null hypothesis (the mean travel time is 12.5 minutes) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because 0.0164 (or 0.0165) > 0.01 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow follow through on their <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered by many candidates. In (a), some candidates chose the wrong standard deviation from their calculator and often failed to square their result to obtain the unbiased variance estimate. Candidates should realise that it is the smaller of the two values (ie the one obtained by dividing by (<em>n</em> – 1)) that is required. The most common error was to use the normal distribution instead of the <em>t</em>-distribution. The signpost towards the <em>t</em>-distribution is the fact that the variance had to be estimated in (a). Accuracy penalties were often given for failure to round the confidence limits, the <em>t</em>-statistic or the <em>p</em>-value to three significant figures.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has a geometric distribution with parameter <em>p </em>.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({\text{P}}(X \leqslant n) = 1 - {(1 - p)^n},{\text{ }}n \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Deduce an expression for \({\text{P}}(m < X \leqslant n)\,,{\text{ }}m\,,{\text{ }}n \in {\mathbb{Z}^ + }\) and <em>m </em>< <em>n </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>p </em>= 0.2, find the least value of <em>n </em>for which \({\text{P}}(1 < X \leqslant n) > 0.5\,,{\text{ }}n \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant n) = \sum\limits_{{\text{i}} = 1}^n {{\text{P}}(X = {\text{i}}) = \sum\limits_{{\text{i}} = 1}^n {p{q^{{\text{i}} - 1}}} } \) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = p\frac{{1 - {q^n}}}{{1 - q}}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - {(1 - p)^n}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 - p)^m} - {(1 - p)^n}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[1 mark]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to solve \(0.8 - {(0.8)^n} > 0.5\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">obtain <em>n</em> = 6 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a) some candidates thought that the geometric distribution was continuous, so attempted to integrate the pdf! Others, less seriously, got the end points of the summation wrong.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) It was very disappointing that may candidates, who got an incorrect answer to part (a), persisted with their incorrect answer into this part.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a) some candidates thought that the geometric distribution was continuous, so attempted to integrate the pdf! Others, less seriously, got the end points of the summation wrong.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) It was very disappointing that may candidates, who got an incorrect answer to part (a), persisted with their incorrect answer into this part.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a) some candidates thought that the geometric distribution was continuous, so attempted to integrate the pdf! Others, less seriously, got the end points of the summation wrong.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) It was very disappointing that may candidates, who got an incorrect answer to part (a), persisted with their incorrect answer into this part.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question you may assume that these data are a random sample from a bivariate normal distribution, with population product moment correlation coefficient \(\rho \).</p>
<p class="p1">Richard wishes to do some research on two types of exams which are taken by a large number of students. He takes a random sample of the results of <span class="s1">10 </span>students, which are shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-02_om_14.09.18.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SP/01"></p>
</div>
<div class="specification">
<p class="p1">Using these data, it is decided to test, at the <span class="s1">1% </span>level, the null hypothesis \({H_0}:\rho = 0\) against the alternative hypothesis \({H_1}:\rho > 0\).</p>
</div>
<div class="specification">
<p class="p1">Richard decides to take the exams himself. He scored <span class="s1">11 </span>on Exam 1 but his result on Exam 2 was lost.</p>
</div>
<div class="specification">
<p class="p1">Caroline believes that the population mean mark on Exam 2 is <span class="s1">6 </span>marks higher than the population mean mark on Exam 1. Using the original data from the <span class="s1">10 </span>students, it is decided to test, at the <span class="s1">5% </span>level, this hypothesis against the alternative hypothesis that the mean of the differences, \({\text{d}} = {\text{exam 2 mark }} - {\text{ exam 1 mark}}\), is less than <span class="s1">6 </span>marks.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For these data find the product moment correlation coefficient, \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of the test statistic (including any parameters).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the \(p\)<span class="s1">-value for the test.</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>State the conclusion, in the context of the question, with the word “correlation” in your answer. Justify your answer.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using a suitable regression line, find an estimate for his score on Exam 2, giving your <span class="s1">answer to the nearest integer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of your test statistic (including any parameters).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the \(p\)<span class="s1">-value.</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>State the conclusion, justifying the answer.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(r = 0.804\) </span><strong><em>A2</em></strong></p>
<p class="p2"> </p>
<p class="p3"><span class="s1"><strong>Note: </strong></span>Accept any number that rounds to 0.80.</p>
<p class="p4"> </p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(t\) <span class="s1">distribution with 8 </span>degrees of freedom <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(p{\text{ - value}} = 0.00254\)</span> <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p2"> </p>
<p class="p3"><span class="s2"><strong>Notes: </strong></span>Accept any number that rounds to 0.0025.</p>
<p class="p3"><span class="s2">Award <strong><em>A1 </em></strong></span>for 2-tail test giving an answer that rounds to 0.0051<span class="s2">.</span></p>
<p class="p2"> </p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(p{\text{ - value}} < 0.01\)</span>, so conclude that there is positive correlation <span class="Apple-converted-space"> </span><strong><em>R1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Notes: </strong>Only award the <strong><em>A1 </em></strong>if the <strong><em>R1 </em></strong>is awarded.</p>
<p class="p1">Do not accept just “reject \({H_0}\)” or “accept \({H_1}\)”.</p>
<p class="p1">The words “positive correlation” must be seen.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">regression line of \(y\) (Exam 2 mark) on \(x\) (Exam 1 mark) is <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(y = 0.59407 \ldots x + 21.387 \ldots \) </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(x = 11\) gives \(y = 28\) (to nearest integer) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>applying the \(t\) test to the differences</p>
<p class="p1"><span class="s1">\(t\) distribution with 9 </span>degrees of freedom <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(p{\text{ - value}} = 0.239\)</span> <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p2"> </p>
<p class="p3"><span class="s2"><strong>Notes: </strong></span>Accept any number that rounds to 0.2<span class="s2">4.</span></p>
<p class="p1">Award <strong><em>A1 </em></strong>if subtraction done the wrong way round giving \(p{\text{ - value}} = 0.109\).</p>
<p class="p2"> </p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(p{\text{ - value}} > 0.05\)</span>, so accept \({H_0}\) or \({u_d} = 6\) <span class="Apple-converted-space"> </span><strong><em>R1A1</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable \(X\) is distributed with mean \(\mu \) and variance \({\sigma ^2}\). Two independent random samples of sizes \({n_1}\) and \({n_2}\) are taken from the distribution of \(X\). The sample means are \({\bar X_1}\) and \({\bar X_2}\) respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(U = a{\bar X_1} + (1 - a){\bar X_2},{\text{ }}a \in \mathbb{R}\), is an unbiased estimator of \(\mu \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{Var}}(U) = {a^2}\frac{{{\sigma ^2}}}{{{n_1}}} + {(1 - a)^2}\frac{{{\sigma ^2}}}{{{n_2}}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of \({n_1}\) and \({n_2}\), an expression for \(a\) which gives the most efficient estimator of this form.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find an expression for the most efficient estimator and interpret the result.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}(U) = E(a{\bar X_1} + (1 - a){\bar X_2}) = a{\text{E}}({\bar X_1}) + (1 - a){\text{E}}({\bar X_2})\) <strong><em>(M1)</em></strong></p>
<p>\({\text{E}}({\bar X_1}) = \mu \) and \({\text{E}}({\bar X_2}) = \mu \)</p>
<p>\({\text{E}}(U) = a\mu + (1 - a)\mu \) (or equivalent) <strong><em>A1</em></strong></p>
<p>\( = \mu \) <strong><em>A1</em></strong></p>
<p>hence \(U\) is an unbiased estimator of \(\mu \) <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{Var}}(U) = {\text{Var}}(a{\bar X_1} + (1 - a){\bar X_2})\)</p>
<p>\( = {a^2}{\text{Var}}({\bar X_1}) + {(1 - a)^2}{\text{Var}}({\bar X_2})\) <strong><em>M1</em></strong></p>
<p>stating that \({\text{Var}}({\bar X_1}) = \frac{{{\sigma ^2}}}{{{n_1}}}\) and \({\text{Var}}({\bar X_2}) = \frac{{{\sigma ^2}}}{{{n_2}}}\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow {\text{Var}}(U) = {a^2}\frac{{{\sigma ^2}}}{{{n_1}}} + {(1 - a)^2}\frac{{{\sigma ^2}}}{{{n_2}}}\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Line 3 or equivalent must be seen somewhere.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let \({\text{Var}}(U) = V\)</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>\(\frac{{{\text{d}}V}}{{{\text{d}}a}} = 2a\frac{{{\sigma ^2}}}{{{n_1}}} - 2(1 - a)\frac{{{\sigma ^2}}}{{{n_2}}}\) <strong><em>M1</em></strong></p>
<p>attempting to solve \(\frac{{{\text{d}}V}}{{{\text{d}}a}} = 0\) for \(a\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for obtaining \(a\) in terms of \({n_1},{\text{ }}{n_2}\) and \(\sigma \).</p>
<p> </p>
<p><strong>OR</strong></p>
<p>forming a quadratic in \(a\)</p>
<p>\(V = \left( {\frac{{{\sigma ^2}}}{{{n_1}}} + \frac{{{\sigma ^2}}}{{{n_2}}}} \right){a^2} - 2\frac{{{\sigma ^2}}}{{{n_2}}}a + \frac{{{\sigma ^2}}}{{{n_2}}}\) <strong><em>M1</em></strong></p>
<p>attempting to find the axis of symmetry of <em>V </em><strong><em>R1</em></strong></p>
<p><strong>THEN</strong></p>
<p>\(a = \frac{{\frac{{2{\sigma ^2}}}{{{n_2}}}}}{{2{\sigma ^2}\left( {\frac{1}{{{n_1}}} + \frac{1}{{{n_2}}}} \right)}}\) <strong><em>(A1)</em></strong></p>
<p>\(a = \frac{{{n_1}}}{{{n_1} + {n_2}}}\) <strong><em>A1</em></strong> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting \(a\) into \(U\) <strong><em>(M1)</em></strong></p>
<p>\(U = \frac{{{n_1}{{\bar X}_1} + {n_2}{{\bar X}_2}}}{{{n_1} + {n_2}}}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not <strong><em>FT </em></strong>an incorrect \(a\) for <strong><em>A1</em></strong>, the <strong><em>M1 </em></strong>may however be awarded.</p>
<p> </p>
<p>this is an expression for the mean of the combined samples</p>
<p><strong>OR</strong> this is a weighted mean of the two sample means <strong><em>R1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A teacher has forgotten his computer password. He knows that it is either six of the letter J followed by two of the letter R (<em>i.e.</em> JJJJJJRR) or three of the letter J followed by four of the letter R (<em>i.e.</em> JJJRRRR). The computer is able to tell him at random just two of the letters in his password.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The teacher decides to use the following rule to attempt to find his password.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the computer gives him a J and a J, he will accept the null hypothesis that his password is JJJJJJRR.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Otherwise he will accept the alternative hypothesis that his password is JJJRRRR.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Define a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the probability that the teacher makes a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Define a Type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the probability that the teacher makes a Type II error.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) a Type I error is when \({{\text{H}}_0}\) is rejected, when \({{\text{H}}_0}\) is actually true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{P(}}{{\text{H}}_0}{\text{ rejected}}|{{\text{H}}_0}{\text{ true)}} = {\text{P(at least one R}}|{\text{6 J and 2 R)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(no R}}|{{\text{H}}_0}{\text{ true)}} = \frac{6}{8} \times \frac{5}{7} = \frac{{15}}{{28}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>X</em> count the number of R’s given by the computer under \({{\text{H}}_0},{\text{ }}X \sim {\text{Hyp(}}2,{\text{ }}2,{\text{ }}8)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) = \frac{{\left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> 0 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 2 <br>\end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 2 <br>\end{array}} \right)}} = \frac{{15}}{{28}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(at least one R}}|{{\text{H}}_0}{\text{ true)}} = 1 - \frac{{15}}{{28}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(Type I error)}} = \frac{{13}}{{28}}\,\,\,\,\,( = 0.464)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) a Type II error is when \({{\text{H}}_0}\) is accepted, when \({{\text{H}}_0}\) is actually false <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \({\text{P(}}{{\text{H}}_0}{\text{ accepted}}|{{\text{H}}_0}{\text{ false)}} = {\text{P(2 J}}|{\text{3 J and 4 R)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(2 J}}|{{\text{H}}_0}{\text{ false)}} = \frac{3}{7} \times \frac{2}{6} = \frac{1}{7}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>Y</em> count the number of R’s given by the computer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}\) false implies \(Y \sim {\text{Hyp(}}2,{\text{ }}4,{\text{ }}7)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y = 0) = \frac{{\left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 0 <br>\end{array}} \right)\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> 2 <br>\end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 2 <br>\end{array}} \right)}} = \frac{1}{7}\) (<strong><em>A1)<br></em>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}({\text{Type II error)}} = \frac{1}{7}( = 0.143)\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [9 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Poorer candidates just gained the 2 marks for saying what a Type I and Type II error were and could not then apply the definitions to obtain the conditional probabilities required. It was clear from some crossings out that even the 2 definition continue to cause confusion. Good, clear-thinking candidates were able to do the question correctly.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has the negative binomial distribution NB(3, <em>p</em>) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x)\) denote the probability that <em>X</em> takes the value <em>x</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down an expression for \(f(x)\) , and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\ln f(x) = 3\ln \left( {\frac{p}{{1 - p}}} \right) + \ln (x - 1) + \ln (x - 2) + x\ln (1 - p) - \ln 2{\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the domain of <em>f</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) The domain of <em>f</em> is extended to \(]2,{\text{ }}\infty [\) . Show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{f'(x)}}{{f(x)}} = \frac{1}{{x - 1}} + \frac{1}{{x - 2}} + \ln (1 - p){\text{ .}}\)</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jo has a biased coin which has a probability of 0.35 of showing heads when tossed. She tosses this coin successively and the \({3^{{\text{rd}}}}\) head occurs on the \({Y^{{\text{th}}}}\) toss. Use the result in part (a)(iii) to find the most likely value of <em>Y</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(f(x) = \left( {\begin{array}{*{20}{c}}<br> {x - 1} \\ <br> 2 <br>\end{array}} \right){p^3}{(1 - p)^{x - 3}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0</em></strong> for \(f(x) = \left( {\begin{array}{*{20}{c}}<br> {x - 1} \\ <br> 2 <br>\end{array}} \right){p^3}{q^{x - 3}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking logs, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\ln f(x) = \left( {\ln \left( {\begin{array}{*{20}{c}}<br> {x - 1} \\ <br> 2 <br>\end{array}} \right){p^3}(1 - p){}^{x - 3}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \left( {\frac{{(x - 1)(x - 2)}}{2} \times {p^3}{{(1 - p)}^{x - 3}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for simplifying binomial coefficient, seen anywhere.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \left( {\frac{{(x - 1)(x - 2)}}{2} \times {p^3}\frac{{{{(1 - p)}^x}}}{{{{(1 - p)}^3}}}} \right)\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for correctly splitting \({{{(1 - p)}^{x - 3}}}\) , seen anywhere.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica; min-height: 29.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 3\ln \left( {\frac{p}{{1 - p}}} \right) + \ln (x - 1) + \ln (x - 2) + x\ln (1 - p) - \ln 2\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the domain is {3, 4, 5, …} <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not accept \(x \geqslant 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) differentiating with respect to </span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;"> , </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{f'(x)}}{{f(x)}} = \frac{1}{{x - 1}} + \frac{1}{{x - 2}} + \ln (1 - p)\) <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">setting \(f'(x) = 0\) and putting <em>p</em> = 0.35 ,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{{x - 1}} + \frac{1}{{x - 2}} + \ln 0.65 = 0\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving, <em>x</em> = 6.195… <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we need to check <em>x</em> = 6 and 7</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f</em> (6) = 0.1177… and <em>f</em> (7) = 0.1148… <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the most likely value of <em>Y</em> is 6 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award the final </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for the correct conclusion even if the previous </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> was not awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In general, candidates were able to start this question, but very few wholly correct answers were seen. Most candidates were able to write down the probability function but the process of taking logs was often unconvincing. The vast majority of candidates gave an incorrect domain for <em>f</em>, the most common error being \(x \geqslant 3\) . Most candidates failed to realise that the solution to (b) was to be found by setting the right-hand side of the given equation equal to zero. Many of the candidates who obtained the correct answer, 6.195…, then rounded this to 6 without realising that both 6 and 7 should be checked to see which gave the larger probability.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In general, candidates were able to start this question, but very few wholly correct answers were seen. Most candidates were able to write down the probability function but the process of taking logs was often unconvincing. The vast majority of candidates gave an incorrect domain for <em>f</em>, the most common error being \(x \geqslant 3\) . Most candidates failed to realise that the solution to (b) was to be found by setting the right-hand side of the given equation equal to zero. Many of the candidates who obtained the correct answer, 6.195…, then rounded this to 6 without realising that both 6 and 7 should be checked to see which gave the larger probability.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Jenny and her Dad frequently play a board game. Before she can start Jenny has to throw a “six” on an ordinary six-sided dice. Let the random variable <em>X </em>denote the number of times Jenny has to throw the dice in total until she obtains her first “six”.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If the dice is fair, write down the distribution of <em>X </em>, including the value of any parameter(s).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down E(<em>X </em>) for the distribution in part (a).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the distribution of </span><em style="font-family: 'times new roman', times; font-size: medium;">Y </em><span style="font-family: 'times new roman', times; font-size: medium;">, including the value of any parameter(s).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of </span><em style="font-family: 'times new roman', times; font-size: medium;">y </em><span style="font-family: 'times new roman', times; font-size: medium;">such that \({\text{P}}(Y = y) = \frac{1}{{36}}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(Y \leqslant 6)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Geo}}\left( {\frac{1}{6}} \right){\text{ or NB}}\left( {1,\frac{1}{6}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 50.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 50.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>Y </em>is \({\text{NB}}\left( {2,\frac{1}{6}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 40.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y = y) = \frac{1}{{36}}{\text{ gives }}y = 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 40.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(as all other probabilities would have a factor of 5 in the numerator)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 40.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \leqslant 6) = {\left( {\frac{1}{6}} \right)^2} + 2\left( {\frac{5}{6}} \right){\left( {\frac{1}{6}} \right)^2} + 3{\left( {\frac{5}{6}} \right)^2}{\left( {\frac{1}{6}} \right)^2} + 4{\left( {\frac{5}{6}} \right)^3}{\left( {\frac{1}{6}} \right)^2} + 5{\left( {\frac{5}{6}} \right)^4}{\left( {\frac{1}{6}} \right)^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.263\) <em><strong>A1</strong></em><span style="font: 35.0px Helvetica;"><br></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered as the last question should be the most difficult. It seemed accessible to many candidates, if they realised what the distributions were. The goodness of fit test was well used in (c) with hardly any candidates mistakenly combining cells. Part (e) was made more complicated than it needed to be with calculator solutions when a bit of pure maths would have sufficed. Part (f) caused some problems but good candidates did not have too much difficulty.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered as the last question should be the most difficult. It seemed accessible to many candidates, if they realised what the distributions were. The goodness of fit test was well used in (c) with hardly any candidates mistakenly combining cells. Part (e) was made more complicated than it needed to be with calculator solutions when a bit of pure maths would have sufficed. Part (f) caused some problems but good candidates did not have too much difficulty.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered as the last question should be the most difficult. It seemed accessible to many candidates, if they realised what the distributions were. The goodness of fit test was well used in (c) with hardly any candidates mistakenly combining cells. Part (e) was made more complicated than it needed to be with calculator solutions when a bit of pure maths would have sufficed. Part (f) caused some problems but good candidates did not have too much difficulty.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered as the last question should be the most difficult. It seemed accessible to many candidates, if they realised what the distributions were. The goodness of fit test was well used in (c) with hardly any candidates mistakenly combining cells. Part (e) was made more complicated than it needed to be with calculator solutions when a bit of pure maths would have sufficed. Part (f) caused some problems but good candidates did not have too much difficulty.</span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was well answered as the last question should be the most difficult. It seemed accessible to many candidates, if they realised what the distributions were. The goodness of fit test was well used in (c) with hardly any candidates mistakenly combining cells. Part (e) was made more complicated than it needed to be with calculator solutions when a bit of pure maths would have sufficed. Part (f) caused some problems but good candidates did not have too much difficulty.</span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A shop sells apples and pears. The weights, in grams, of the apples may be assumed to have a \({\text{N}}(200,{\text{ 1}}{{\text{5}}^2})\) distribution and the weights of the pears, in grams, may be assumed to have a \({\text{N}}(120,{\text{ 1}}{{\text{0}}^2})\) distribution.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that the weight of a randomly chosen apple is more than double the weight of a randomly chosen pear.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) A shopper buys 3 apples and 4 pears. Find the probability that the total weight is greater than 1000 grams.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Let <em>X</em>, <em>Y </em>(grams) denote respectively the weights of a randomly chosen apple, pear.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X - 2Y{\text{ is N}}(200 - 2 \times 120,{\text{ }}{15^2} + 4 \times {10^2}),\) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i</em>.<em>e</em>. \({\text{N}}( - 40,{\text{ }}{25^2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">We require</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 2Y) = {\text{P}}(X - 2Y > 0)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0548\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let \(T = {X_1} + {X_2} + {X_3} + {Y_1} + {Y_2} + {Y_3} + {Y_4}\) (grams) denote the total weight.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(T{\text{ is N}}(3 \times 200 + 4 \times 120,{\text{ }}3 \times {15^2} + 4 \times {10^2}),\) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i</em>.<em>e</em>. \({\text{N(1080, 1075)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T > 1000) = 0.993\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The response to this question was disappointing. Many candidates are unable to differentiate between quantities such as \(3X{\text{ and }}{X_1} + {X_2} + {X_3}\) . While this has no effect on the mean, there is a significant difference between the variances of these two random variables.</span></p>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable \(T\) has a probability density function defined by</p>
<p style="text-align: center;">\(f(t) = \left\{ {\begin{array}{*{20}{c}} {\frac{{t(4 - {t^2})}}{4}}&{0 \leqslant t \leqslant 2} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the cumulative distribution function \(F(t)\), for \(0 \leqslant t \leqslant 2\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(F(t)\) for \(0 \leqslant t \leqslant 2\), clearly indicating the coordinates of the endpoints.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(P(T < a) = 0.75\), find the value of \(a\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(F(t) = \int_0^t {\left( {x - \frac{{{x^3}}}{4}} \right){\text{d}}x{\text{ }}\left( { = \int_0^t {\frac{{x(4 - {x^2})}}{4}{\text{d}}x} } \right)} \) <strong><em>M1</em></strong></p>
<p>\( = \left[ {\frac{{{x^2}}}{2} - \frac{{{x^4}}}{{16}}} \right]_0^t{\text{ }}\left( { = \left[ {\frac{{{x^2}(8 - {x^2})}}{{16}}} \right]_0^t} \right){\text{ }}\left( { = \left[ {\frac{{ - 4 - {x^2}{)^2}}}{{16}}} \right]_0^t} \right)\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{{t^2}}}{2} - \frac{{{t^4}}}{{16}}{\text{ }}\left( { = \frac{{{t^2}(8 - {t^2})}}{{16}}} \right){\text{ }}\left( { = 1 - \frac{{{{(4 - {t^2})}^2}}}{{16}}} \right)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone integration involving \(t\) only.</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1A0A0 </em></strong>for integration without limits <em>eg</em>, \(\int {\frac{{t(4 - {t^2})}}{4}{\text{d}}t = \frac{{{t^2}}}{2} - \frac{{{t^4}}}{{16}}} \) or equivalent.</p>
<p> </p>
<p><strong>Note:</strong> But allow integration \( + \) \(C\) then showing \(C = 0\) or even integration without \(C\) if \(F(0) = 0\) or \(F(2) = 1\) is confirmed.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-10_om_18.03.31.png" alt="N17/5/MATHL/HP3/ENG/TZ0/SP/M/01.b.i"></p>
<p>correct shape including correct concavity <strong><em>A1</em></strong></p>
<p>clearly indicating starts at origin and ends at \((2,{\text{ }}1)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone the absence of \((0,{\text{ }}0)\).</p>
<p> </p>
<p><strong>Note:</strong> Accept 2 on the \(x\)-axis and 1 on the \(y\)-axis correctly placed.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve \(\frac{{{a^2}}}{2} - \frac{{{a^4}}}{{16}} = 0.75\) (or equivalent) for \(a\) <strong><em>(M1)</em></strong></p>
<p>\(a = 1.41{\text{ }}( = \sqrt 2 )\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds to 1.4.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables \({X_1}\) and \({X_2}\) are a random sample from \({\text{N}}(\mu ,{\text{ 2}}{\sigma ^2})\). The random variables \({Y_1}\), \({Y_2}\) and \({Y_3}\) are a random sample from \({\text{N}}(2\mu ,{\text{ }}{\sigma ^2})\).</p>
<p>The estimator \(U\) is used to estimate \(\mu \) where \(U = a({X_1} + {X_2}) + b({Y_1} + {Y_2} + {Y_3})\) and \(a\), \(b\) are constants.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(U\) is unbiased, show that \(2a + 6b = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{Var}}(U) = (39{b^2} - 12b + 1){\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of \(a\) and the value of \(b\) which give the best unbiased estimator of this form, giving your answers as fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the variance of this best unbiased estimator.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}(U) = a\left( {{\text{E}}({X_1}) + {\text{E}}({X_2})} \right) + b\left( {{\text{E}}({Y_1}) + {\text{E}}({Y_2}) + {\text{E}}({Y_3})} \right)\) <strong><em>(M1)</em></strong></p>
<p>\( = 2a\mu + 6b\mu \) <strong><em>A1</em></strong></p>
<p>(for an unbiased estimator,) \({\text{E}}(U) = \mu \) <strong><em>R1</em></strong></p>
<p>giving \(2a + 6b = 1\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone omission of E on LHS.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{Var}}(U) = {a^2}\left( {{\text{Var}}({X_1}) + {\text{Var}}({X_2})} \right) + {b^2}\left( {{\text{Var}}({Y_1}) + {\text{Var}}({Y_2}) + {\text{Var}}({Y_3})} \right)\) <strong><em>(M1)</em></strong></p>
<p>\( = 4{a^2}{\sigma ^2} + 3{b^2}{\sigma ^2}\) <strong><em>A1</em></strong></p>
<p>\( = 4{\left( {\frac{{1 - 6b}}{2}} \right)^2}{\sigma ^2} + 3{b^2}{\sigma ^2}\) <strong><em>A1</em></strong></p>
<p>\( = (39{b^2} - 12b + 1){\sigma ^2}\) <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the best unbiased estimator (of this form) will be found by minimising \({\text{Var}}(U)\) <strong><em>(R1)</em></strong></p>
<p>For example, \(\frac{{\text{d}}}{{{\text{d}}b}}\left( {{\text{Var}}(U)} \right) = (78b - 12){\sigma ^2}\) (<strong><em>A1)</em></strong></p>
<p>for a minimum, \(b = \frac{{12}}{{78}}\,\,\,\left( { = \frac{2}{{13}}} \right)\) so that \(a = \frac{3}{{78}}\,\,\,\left( { = \frac{1}{{26}}} \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{Var}}U = \left( {39{{\left( {\frac{2}{{13}}} \right)}^2} - 12\left( {\frac{2}{{13}}} \right) + 1} \right){\sigma ^2}\)</p>
<p>\( = \frac{{{\sigma ^2}}}{{13}}\,\,\,(0.0769{\sigma ^2})\) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>represents the height of a wave on a particular surf beach.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">It is known that <em>X </em>is normally distributed with unknown mean \(\mu \) (metres) and known variance \({\sigma ^2} = \frac{1}{4}{\text{ (metre}}{{\text{s}}^2}{\text{)}}\) . Sally wishes to test the claim made in a surf guide that \(\mu = 3\) against the alternative that \(\mu < 3\) . She measures the heights of 36 waves and calculates their sample mean \({\bar x}\) . She uses this value to test the claim at the 5 % level.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find a simple inequality, of the form \(\bar x < A\) , where <em>A </em>is a number to be determined to 4 significant figures, so that Sally will reject the null hypothesis, that \(\mu = 3\) , if and only if this inequality is satisfied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Define a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Define a Type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Write down the probability that Sally makes a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) The true value of \(\mu \) is 2.75. Calculate the probability that Sally makes a Type II error.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>Y </em>represents the height of a wave on another surf beach. It is known that <em>Y </em>is normally distributed with unknown mean \(\mu \) (metres) and unknown variance \({\sigma ^2}{\text{ (metre}}{{\text{s}}^2}{\text{)}}\) . David wishes to test the claim made in a surf guide that \(\mu = 3\) against the alternative that \(\mu < 3\) . He is also going to perform this test at the 5 % level. He measures the heights of 36 waves and finds that the sample mean, \(\bar y = 2.860\) and the unbiased estimate of the population variance, \(s_{n - 1}^2 = 0.25\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the name of the test that David should perform.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the conclusion of David’s test, justifying your answer by giving the <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Using David’s results, calculate the 90 % confidence interval for \(\mu \) , giving your answers to 4 significant figures.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({H_0}:\mu = 3,{\text{ }}{H_1}:\mu < 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 tailed <em>z </em>test as \({\sigma ^2}\) is known</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">under \({H_0}{\text{, }}X \sim {\text{N}}\left( {3,\frac{1}{4}} \right){\text{ so }}\bar X \sim {\text{N}}\left( {3,\frac{{\frac{1}{4}}}{{36}}} \right) = N\left( {3,\frac{1}{{144}}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = \frac{{\bar x - 3}}{{\frac{1}{{12}}}}{\text{ is N(0, 1)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(z < - 1.64485...) = 0.05\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so inequality is given by \(\frac{{\bar x - 3}}{{\frac{1}{{12}}}} < - 1.64485...{\text{ giving }}\bar x < 2.8629…\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x < 2.863{\text{ (4sf)}}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Candidates can get directly to the answer from \({\text{N}}\left( {3,\frac{1}{{144}}} \right)\) they do not have to go via </span><em style="font-family: 'times new roman', times; font-size: medium;">z </em><span style="font-family: 'times new roman', times; font-size: medium;">is N(0, 1) . However they must give some explanation of what they have done; they cannot just write the answer down.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) a Type I error is accepting \({H_1}\) when \({H_0}\) is true </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) a Type II error is accepting \({H_0}\) when \({H_1}\) is true <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) 0.05 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Accept anything that rounds to 0.050 if they do the conditional calculation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) \(\bar X \sim {\text{N}}\left( {2.75,\frac{1}{{144}}} \right)\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(\bar x > 2.8629...) = 0.0877{\text{ (3sf)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: Accept any answer between 0.0875 and 0.0877 inclusive.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: Accept anything that rounded is between 0.087and 0.089 if there is evidence that the candidate has used tables.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[11 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <em>t</em>-test <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({{\text{H}}_0}:\mu = 3,{\text{ }}{{\text{H}}_1}:\mu < 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1 tailed <em>t </em>test as \({\sigma ^2}\) is unknown</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{\bar y - 3}}{{\frac{1}{{12}}}}\) has the <em>t</em>-distribution with \(v = 35\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the <em>p</em>-value is 0.0509… <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this is \( > 0.05\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so we accept that the mean wave height is 3 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: Allow “Accept \({{\text{H}}_0}\) ” provided \({{\text{H}}_0}\) has been stated.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: Accept </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>FT </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">on the </span><em style="font-family: 'times new roman', times; font-size: medium;">p</em><span style="font-family: 'times new roman', times; font-size: medium;">-value for the </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">s.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(2.719 < \mu < 3.001{\text{ (4 sf)}}\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: \(2.860 \pm 1.6896... \times \frac{{\frac{1}{2}}}{6}\) would gain </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A0 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">if answer are only given to 3sf.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) There were many reasonable answers. In (i) not all candidates explained their method so that they could gain good partial marks even if they had the wrong final answer. A common mistake was to give an answer above 3. It was pleasing that almost all candidates had (ii) and (iii) correct, as this had caused problems in the past. In (iv) it was amusing to see a few candidates work out 5% using conditional probability rather than just write down the answer as asked.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) It was pleasing that almost all candidates realised that it was a <em>t</em>-test rather than a <em>z</em>-test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">There was good understanding on how to use the calculator in parts (ii) and (iii). The correct confidence interval to the desired accuracy was not always given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The most common mistake in question 3 was forgetting to take into account the variance of the sample mean.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) There were many reasonable answers. In (i) not all candidates explained their method so that they could gain good partial marks even if they had the wrong final answer. A common mistake was to give an answer above 3. It was pleasing that almost all candidates had (ii) and (iii) correct, as this had caused problems in the past. In (iv) it was amusing to see a few candidates work out 5% using conditional probability rather than just write down the answer as asked.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) It was pleasing that almost all candidates realised that it was a <em>t</em>-test rather than a <em>z</em>-test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">There was good understanding on how to use the calculator in parts (ii) and (iii). The correct confidence interval to the desired accuracy was not always given.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">The most common mistake in question 3 was forgetting to take into account the variance of the sample mean.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A discrete random variable \(U\) follows a geometric distribution with \(p = \frac{1}{4}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(F(u)\), the cumulative distribution function of \(U\), for \(u = 1,{\text{ }}2,{\text{ }}3 \ldots \)</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, or otherwise, find the value of \(P(U > 20)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that the probability generating function of \(U\) is given by \({G_u}(t) = \frac{t}{{4 - 3t}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \({U_i} \sim {\text{Geo}}\left( {\frac{1}{4}} \right),{\text{ }}i = 1,{\text{ }}2,{\text{ }}3\), and that \(V = {U_1} + {U_2} + {U_3}\)<span class="s1">, find</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>\({\text{E}}(V)\);</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}(V)\);</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\({G_v}(t)\), the probability generating function of \(V\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A third random variable \(W\), has probability generating function \({G_w}(t) = \frac{1}{{{{(4 - 3t)}^3}}}\).</p>
<p class="p1">By differentiating \({G_w}(t)\), <span class="s1">find \({\text{E}}(W)\).</span></p>
<p class="p1"> </p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A third random variable \(W\), has probability generating function \({G_w}(t) = \frac{1}{{{{(4 - 3t)}^3}}}\).</p>
<p class="p1">Prove that \(V = W + 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2">\({\text{P}}(U = u) = \frac{1}{4}{\left( {\frac{3}{4}} \right)^{u - 1}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2">\(F(u) = {\text{P}}(U \le u) = \sum\limits_{r = 1}^u {\frac{1}{4}{{\left( {\frac{3}{4}} \right)}^{r - 1}}\;\;\;} \)(or equivalent)</p>
<p class="p2">\( = \frac{{\frac{1}{4}\left( {1 - {{\left( {\frac{3}{4}} \right)}^u}} \right)}}{{1 - \frac{3}{4}}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2">\( = 1 - {\left( {\frac{3}{4}} \right)^u}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2">\({\text{P}}(U \le u) = 1 - {\text{P}}(U > u)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2">\({\text{P}}(U > u) = \)</span> probability of \(u\) consecutive failures <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">\({\text{P}}(U \le u) = 1 - {\left( {\frac{3}{4}} \right)^u}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(U > 20) = 1 - {\text{P}}(U \le 20)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = {\left( {\frac{3}{4}} \right)^{20}}\;\;\;( = 0.00317)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({G_U}(t) = \sum\limits_{r = 1}^\infty {\frac{1}{4}{{\left( {\frac{3}{4}} \right)}^{r - 1}}{t^r}\;\;\;} \)(or equivalent) <strong><em>M1A1</em></strong></p>
<p>\( = \sum\limits_{r = 1}^\infty {\frac{1}{3}{{\left( {\frac{3}{4}t} \right)}^r}} \) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{\frac{1}{3}\left( {\frac{3}{4}t} \right)}}{{1 - \frac{3}{4}t}}\;\;\;\left( { = \frac{{\frac{1}{4}t}}{{1 - \frac{3}{4}t}}} \right)\) <strong><em>A1</em></strong></p>
<p>\( = \frac{t}{{4 - 3t}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(E(U) = \frac{1}{{\frac{1}{4}}} = 4\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\(E({U_1} + {U_2} + {U_3}{\text{)}} = 4 + 4 + 4 = 12\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1">(ii) <span class="Apple-converted-space"> </span></span>\({\text{Var}}(U) = \frac{{\frac{3}{4}}}{{{{\left( {\frac{1}{4}} \right)}^2}}}=12\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({\text{Var(}}{U_1} + {U_2} + {U_3}) = 12 + 12 + 12 = 36\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1">(iii) <span class="Apple-converted-space"> </span></span>\({G_v}(t) = {\left( {{G_U}(t)} \right)^3}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = {\left( {\frac{t}{{4 - 3t}}} \right)^3}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({G_W}^\prime (t) = - 3{(4 - 3t)^{ - 4}}( - 3)\;\;\;\left( { = \frac{9}{{{{(4 - 3t)}^4}}}} \right)\) <strong><em>(M1)(A1)</em></strong></p>
<p>\(E(W) = {G_W}^\prime (1) = 9\) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow the use of the calculator to perform the differentiation.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p2">probability generating function of the constant 3 <span class="s1">is \({t^3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">\({G_{W - 3}}(t) = E({t^{W + 3}}) = E({t^W})E({t^3})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1">\(W + 3\) has generating function \({G_{W + 3}} = \frac{1}{{{{(4 - 3t)}^3}}} \times {t^3} = {G_V}(t)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">as the generating functions are the same \(V = W + 3\) <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [22 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable \(X\) has cumulative distribution function \(F\) given by \[F(x) = \left\{ {\begin{array}{*{20}{l}} {0,}&{x < 0} \\ {x{{\text{e}}^{x - 1}},}&{0 \leqslant x \leqslant 1.} \\ {1,}&{x > 2} \end{array}} \right.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine \(P(0.25 \leqslant X \leqslant 0.75)\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the median of \(X\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability density function \(f\) of \(X\) is given, for \(0 \leqslant x \leqslant 1\), by</p>
<p>\[f(x) = (x + 1){{\text{e}}^{x - 1}}.\]</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the mean and the variance of \(X\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the central limit theorem. </p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A random sample of 100 observations is obtained from the distribution of \(X\). If \(\bar X\) denotes the sample mean, use the central limit theorem to find an approximate value of \(P(\bar X > 0.65)\). Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(P(0.25 \leqslant X \leqslant 0.75) = F(0.75) - F(0.25)\) <strong><em>(M1)</em></strong></p>
<p>\( = 0.466\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.466.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the median \(m\) satisfies \(F(m) = 0.5\) <strong><em>(M1)</em></strong></p>
<p>\(m = 0.685\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.685.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(f(x) = F’(x)\) <strong><em>(M1)</em></strong></p>
<p>\( = {{\text{e}}^{x - 1}} + x{{\text{e}}^{x - 1}}\) <strong><em>A1</em></strong></p>
<p>\( = (x + 1){{\text{e}}^{x - 1}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\mu = \int\limits_0^1 {x\left( {x + 1} \right){{\text{e}}^{x - 1}}{\text{d}}x} \) <strong><em>(M1)</em></strong></p>
<p>\( = 0.632\,\,\,\left( {1 - \frac{1}{{\text{e}}}} \right)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.632.</p>
<p> </p>
<p>\({\sigma ^2} = \int\limits_0^1 {x\left( {x + 1} \right){{\text{e}}^{x - 1}}{\text{d}}x} - 0.632{ \ldots ^2}\) <strong><em>(M1)</em></strong></p>
<p>\( = 0.0719\,\,\,\left( {\frac{6}{{\text{e}}} - 2 - \frac{1}{{{{\text{e}}^2}}}} \right)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.072.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the central limit theorem states that the mean of a large sample from any distribution (with a finite variance) is approximately normally distributed <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\bar X\) is approximately \(N(0.632 \ldots ,{\text{ }}0.000719 \ldots )\) <strong><em>(M1)(A1)</em></strong></p>
<p>\(P(\bar X > 0.65) = 0.25\) (2 dps required) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p> </p>
<p> </p>
<p> </p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Adam does the crossword in the local newspaper every day. The time taken by Adam, \(X\) <span class="s1">minutes, to complete the crossword is modelled by the normal distribution \({\text{N}}(22,{\text{ }}{5^2})\).</span></p>
</div>
<div class="specification">
<p class="p1">Beatrice also does the crossword in the local newspaper every day. The time taken by Beatrice, \(Y\) <span class="s1">minutes, to complete the crossword is modelled by the normal distribution \({\text{N}}(40,{\text{ }}{6^2})\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that, on a randomly chosen day, the probability that he completes the crossword in less than \(a\) <span class="s1">minutes is equal to 0.8</span>, find the value of \(a\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that the total time taken for him to complete five randomly chosen crosswords exceeds <span class="s1">120 </span>minutes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that, on a randomly chosen day, the time taken by Beatrice to complete the crossword is more than twice the time taken by Adam to complete the crossword. Assume that these two times are independent.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(z = 0.841 \ldots \) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(a = \mu + z\sigma \) </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = 26.2\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">let \(T\) </span>denote the total time taken to complete 5 <span class="s1">crosswords.</span></p>
<p class="p2">\(T\) is \({\text{N}}(110,{\text{ }}125)\) <span class="Apple-converted-space"> </span><strong><em>(A1)(A1)</em></strong></p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span><em>A1 </em></strong>for the mean and <strong><em>A1 </em></strong><span class="s2">for the variance.</span></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(T > 120) = 0.186\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">consider the random variable \(U = Y - 2X\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(U) = - 4\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({\text{Var}}(U) = {\text{Var}}(Y) + 4{\text{Var}}(X)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 136\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\({\text{P}}(Y > 2X) = {\text{P}}(U > 0)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\( = 0.366\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was very well answered with only a very few weak candidates using 0.8 instead of 0.841...</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was well answered with only a few candidates calculating the variance incorrectly.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) was again well answered. The most common errors, not often seen, were writing the variance of \(Y - 2X\) as either \({\text{Var}}(Y) + 2{\text{Var}}(X)\) or \({\text{Var}}(Y) - 2(or{\text{ }}4){\text{Var}}(X)\).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The random variables \(X\)<span class="s1">, </span>\(Y\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \).</p>
</div>
<div class="specification">
<p class="p1">A random sample of 10 <span class="s1">observations on \(X\)</span>, <span class="s1">\(Y\) was obtained and the value of \(r\)</span>, the sample product moment correlation coefficient, was calculated to be 0.486.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State suitable hypotheses to investigate whether or not \(X\)<span class="s1">, </span>\(Y\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Determine the \(p\)-value.</p>
<p class="p1">(ii) State your conclusion at the <span class="s1">5% </span>significance level.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the equation of the regression line of \(y\) on \(x\) should not be used to predict the value of \(y\) corresponding to \(x = {x_0}\), where \({x_0}\) lies within the range of values of \(x\) in the sample.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({H_0}:{\text{ }}\rho = 0;{\text{ }}{H_1}:{\text{ }}\rho \ne 0\) </span><strong><em>A1A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(t = 0.486 \times \sqrt {\frac{{10 - 2}}{{1 - {{0.486}^2}}}} \)</span> <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = 1.572 \ldots \) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">degrees of freedom \( = 8\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(T > 1.5728 \ldots )\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 0.0772\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(p{\text{ - value }} = {\text{ }}0.154\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not follow through for the final <strong><em>A1 </em></strong>if their \({H_1}\) is one-sided.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>accept \({H_0}\) or equivalent statement involving \({H_0}\) or \({H_1}\) <span class="s2">(at the 5% </span>significance level) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p4"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Follow through the candidate’s \(p\)-value.</p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">because the above analysis suggests that \(X\)<span class="s1">, </span>\(Y\) are independent <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">the value of \(r\) suggests that \(X\) and \(Y\) are weakly correlated <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was well answered with only a few candidates using inappropriate symbols, for example \(r\) or \(\mu \)<em>. </em>Also, only very few candidates failed to realise that the wording of the question indicated that a two-tailed test was required.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The test in (b) was generally well carried out and the \(p\)-value found correctly. The most common errors were using incorrect degrees of freedom and evaluating a one-tailed \(p\)-value instead of a two-tailed \(p\)-value.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), many realised that the earlier work meant that the regression line should not be used because the variables had been found to be independent. Incorrect reasons, however, were not uncommon, for example the suggestions that either the regression line of \(x\) on \(y\) should be used or that there were insufficient data.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">John rings a church bell <span class="s1">120 </span>times. The time interval, \({T_i}\), between two successive rings is a random variable with mean of <span class="s1">2 </span>seconds and variance of \(\frac{1}{9}{\text{ second}}{{\text{s}}^2}\).</p>
<p class="p1">Each time interval, \({T_i}\), is independent of the other time intervals. Let \(X = \sum\limits_{i = 1}^{119} {{T_i}} \) be the total time between the first ring and the last ring.</p>
</div>
<div class="specification">
<p class="p1">The church vicar subsequently becomes suspicious that John has stopped coming to ring the bell and that he is letting his friend Ray do it. When Ray rings the bell the time interval, \({T_i}\) has a mean of <span class="s1">2 </span>seconds and variance of \(\frac{1}{{25}}{\text{ second}}{{\text{s}}^2}\).</p>
<p class="p1">The church vicar makes the following hypotheses:</p>
<p class="p1"><span class="s1">\({H_0}\): </span>Ray is ringing the bell; \({H_1}\)<span class="s1">: </span>John is ringing the bell.</p>
<p class="p1">He records four values of \(X\). He decides on the following decision rule:</p>
<p class="p1">If \(236 \leqslant X \leqslant 240\) for all four values of \(X\) he accepts \({H_0}\), otherwise he accepts \({H_1}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p2">(i) <span class="Apple-converted-space"> \({\text{E}}(X)\)</span>;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> \({\text{Var}}(X)\)</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why a normal distribution can be used to give an approximate model for \(X\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Use this model to find the values of \(A\) </span>and \(B\) such that \({\text{P}}(A < X < B) = 0.9\), where \(A\) and \(B\) are symmetrical about the mean of \(X\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the probability that he makes a Type <span class="s1">II </span><span class="s2">error.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \({\text{mean}} = 119 \times 2 = 238\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \({\text{variance}} = 119 \times \frac{1}{9} = \frac{{119}}{9}{\text{ }}( = 13.2)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p1"><span class="s1"><strong>Note: </strong></span>If 120 is used instead of 119 <span class="s1">award <strong><em>A0(M1)A0 </em></strong></span>for part (a) and apply follow through for parts (b)-(d). (b) is unaffected and in (c) the interval becomes \((234,{\text{ }}246)\)<span class="s1">. In (d) the first 2 <strong><em>A1 </em></strong></span>marks are for \(0.3633 \ldots \) and \(0.0174 \ldots \) so the final answer will round to 0.017<span class="s1">.</span></p>
<p class="p2"> </p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">justified by the Central Limit Theorem <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">since \(n\) is large <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Accept \(n > 30\).</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(X \sim N\left( {238,{\text{ }}\frac{{119}}{9}} \right)\)</p>
<p class="p2"><span class="s1">\(Z = \frac{{X - 238}}{{\frac{{\sqrt {119} }}{3}}} \sim N(0,{\text{ }}1)\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(Z < q) = 0.95 \Rightarrow q = 1.644 \ldots \) </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s2">so \({\text{P}}( - 1.644 \ldots < Z < 1.644 \ldots ) = 0.9\)</span> <span class="Apple-converted-space"> </span><span class="s2"><strong><em>(R1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}( - 1.644 \ldots < \frac{{X - 238}}{{\frac{{\sqrt {119} }}{3}}} < 1.644 \ldots ) = 0.9\) </span><span class="s2"><strong><em>(M1)</em></strong></span></p>
<p class="p1">interval is \(232 < X < 244{\text{ }}({\text{3sf}}){\text{ }}(A = 232,{\text{ }}B = 244)\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p2"><strong>Notes: </strong>Accept the use of inverse normal applied to the distribution of \(X\).</p>
<p class="p1">Alternative is to use the GDC <span class="s2">to find a pretend \(Z\) </span>confidence interval for a mean and then convert by multiplying by 119<span class="s2">.</span></p>
<p class="p2">Either \(A\) or \(B\) correct implies the five implied marks.</p>
<p class="p2">Accept any numbers that round to these 3sf numbers.</p>
<p class="p3"> </p>
<p class="p2"><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">under \({{\text{H}}_1},{\text{ }}X \sim N\left( {238,{\text{ }}\frac{{119}}{9}} \right)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(236 \leqslant X \leqslant 240) = 0.41769 \ldots \) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">probability that all 4 values of \(X\) lie in this interval is</p>
<p class="p1"><span class="s2">\({(0.41769 \ldots )^4} = 0.030439 \ldots \) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1">so probability of a Type <span class="s2">II </span><span class="s3">error is 0.0304 </span>(3sf) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"> </p>
<p class="p4"><span class="s1"><strong>Note: </strong></span>Accept any answer that rounds to 0.030<span class="s1">.</span></p>
<p class="p3"> </p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables <em>X</em> , <em>Y</em> follow a bivariate normal distribution with product moment correlation coefficient <em>ρ</em>.</p>
</div>
<div class="specification">
<p>A random sample of 11 observations on <em>X</em>, <em>Y</em> was obtained and the value of the sample product moment correlation coefficient, <em>r</em>, was calculated to be −0.708.</p>
</div>
<div class="specification">
<p>The covariance of the random variables <em>U</em>, <em>V</em> is defined by</p>
<p style="text-align: center;">Cov(<em>U</em>, <em>V</em>) = E((<em>U</em> − E(<em>U</em>))(<em>V</em> − E(<em>V</em>))).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to investigate whether or not a negative linear association exists between <em>X</em> and <em>Y</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the <em>p</em>-value.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State your conclusion at the 1 % significance level.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Cov(<em>U</em>, <em>V</em>) = E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that if <em>U</em>, <em>V</em> are independent random variables then the population product moment correlation coefficient, <em>ρ</em>, is zero.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>H<sub>0 </sub>: <em>ρ</em> = 0; H<sub>1 </sub>: <em>ρ</em> < 0 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(t = - 0.708\sqrt {\frac{{11 - 2}}{{1 - {{\left( { - 0.708} \right)}^2}}}} \,\, = \,\,\left( { - 3.0075 \ldots } \right)\) <em><strong>(M1)</strong></em></p>
<p>degrees of freedom = 9 <em><strong>(A1)</strong></em></p>
<p>P(<em>T</em> < −3.0075...) = 0.00739 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept any answer that rounds to 0.0074.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>reject H<sub>0</sub> or equivalent statement <em><strong> R1</strong></em></p>
<p><strong>Note:</strong> Apply follow through on the candidate’s <em>p</em>-value.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Cov(<em>U</em>, <em>V</em>) + E((<em>U</em> − E(<em>U</em>))(<em>V</em> − E(<em>V</em>)))</p>
<p>= E(<em>UV </em>− E(<em>U</em>)<em>V </em>− E(<em>V</em>)<em>U </em>+ E(<em>U</em>)E(<em>V</em>)) <strong>M1</strong></p>
<p>= E(<em>UV</em>) − E(E(<em>U</em>)<em>V</em>) − E(E(<em>V</em>)<em>U</em>) + E(E(<em>U</em>)E(<em>V</em>)) <em><strong>(A1)</strong></em></p>
<p>= E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>) − E(<em>V</em>)E(<em>U</em>) + E(<em>U</em>)E(<em>V</em>) <em><strong>A1</strong></em></p>
<p>Cov(<em>U</em>, <em>V</em>) = E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>) <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>E(<em>UV</em>) = E(<em>U</em>)E(<em>V</em>) (independent random variables) <em><strong>R1</strong></em></p>
<p>⇒Cov(<em>U</em>, <em>V</em>) = E(<em>U</em>)E(<em>V</em>) − E(<em>U</em>)E(<em>V</em>) = 0 <em><strong>A1</strong></em></p>
<p>hence, <em>ρ = </em>\(\frac{{{\text{Cov}}\left( {U,\,V} \right)}}{{\sqrt {{\text{Var}}\left( U \right)\,{\text{Var}}\left( V \right)} }} = 0\) <em><strong>A1AG</strong></em></p>
<p><strong>Note:</strong> Accept the statement that Cov(<em>U</em>,<em>V</em>) is the numerator of the formula for <em>ρ</em>.</p>
<p><strong>Note:</strong> Only award the first <em><strong>A1</strong> </em>if the <em><strong>R1</strong> </em>is awarded.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A smartphone’s battery life is defined as the number of hours a fully charged battery can be used before the smartphone stops working. A company claims that the battery life of a model of smartphone is, on average, 9.5 hours. To test this claim, an experiment is conducted on a random sample of 20 smartphones of this model. For each smartphone, the battery life, \(b\) hours, is measured and the sample mean, \({\bar b}\), calculated. It can be assumed the battery lives are normally distributed with standard deviation 0.4 hours.</p>
</div>
<div class="specification">
<p>It is then found that this model of smartphone has an average battery life of 9.8 hours.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses for a two-tailed test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the critical region for testing \({\bar b}\) at the 5 % significance level.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type II error.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Another model of smartphone whose battery life may be assumed to be normally distributed with mean <em>μ</em> hours and standard deviation 1.2 hours is tested. A researcher measures the battery life of six of these smartphones and calculates a confidence interval of [10.2, 11.4] for <em>μ</em>.</p>
<p>Calculate the confidence level of this interval.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 3, accept answers that round correctly to 2 significant figures.</p>
<p>\({{\text{H}}_0}\,{\text{:}}\,\mu = 9.5{\text{;}}\,\,{{\text{H}}_1}\,{\text{:}}\,\mu \ne 9.5\) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 3, accept answers that round correctly to 2 significant figures.</p>
<p>the critical values are \(9.5 \pm 1.95996 \ldots \times \frac{{0.4}}{{\sqrt {20} }}\) <em><strong>(M1)(A1)</strong></em></p>
<p>i.e. 9.3247…, 9.6753…</p>
<p>the critical region is \({\bar b}\) < 9.32, \({\bar b}\) > 9.68 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct inequalities, <em><strong>A1</strong> </em>for correct values.</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if <em>t</em>-distribution used, note that <em>t</em>(19)<sub>97.5</sub> = 2.093 …</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 3, accept answers that round correctly to 2 significant figures.</p>
<p>\(\bar B \sim {\text{N}}\left( {9.8,\,{{\left( {\frac{{0.4}}{{\sqrt {20} }}} \right)}^2}} \right)\) <em><strong> (A1)</strong></em></p>
<p>\({\text{P}}\left( {9.3247 \ldots < \bar B < 9.6753 \ldots } \right)\) <em><strong>(M1)</strong></em></p>
<p>=0.0816 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> FT the critical values from (b). Note that critical values of 9.32 and 9.68 give 0.0899.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In question 3, accept answers that round correctly to 2 significant figures.</p>
<p><strong>METHOD 1</strong></p>
<p>\(X \sim {\text{N}}\left( {{\text{10}}{\text{.8,}}\,\frac{{{{1.2}^2}}}{6}} \right)\) <em><strong>(M1)(A1)</strong></em></p>
<p>P(10.2 < <em>X</em> < 11.4) = 0.7793… <em><strong> (A1)</strong></em></p>
<p>confidence level is 77.9% <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept 78%.</p>
<p><strong>METHOD 2</strong></p>
<p>\(11.4 - 10.2 = 2z \times \frac{{1.2}}{{\sqrt 6 }}\) <em><strong>(M1)</strong></em></p>
<p>\(z = 1.224 \ldots \) <em><strong> (A1)</strong></em></p>
<p>P(−1.224… < <em>Z</em> < 1.224…) = 0.7793… <em><strong>(A1)</strong></em></p>
<p>confidence level is 77.9% <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept 78%.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has the negative binomial distribution NB(5, <em>p</em>), where <em>p</em> < 0.5, and \({\text{P}}(X = 10) = 0.05\). By first finding the value of <em>p</em>, find the value of \({\text{P}}(X = 11)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 10) = \left( {\begin{array}{*{20}{c}}<br> 9 \\ <br> 4 <br>\end{array}} \right){p^5}{(1 - p)^5}\) (= 0.05) <strong><em>(M1)A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> First <strong><em>A1</em></strong> is for the binomial coefficient. Second <strong><em>A1</em></strong> is for the rest.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving by any method, \(p = 0.297 \ldots \) <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Award <strong><em>A2</em></strong> for anything which rounds to 0.703.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Do not apply any <strong><em>AP</em></strong> at this stage.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 10) = \left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> 4 <br>\end{array}} \right) \times {(0.297...)^5} \times {(1 - 0.297...)^6}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.0586 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow follow through for incorrect <em>p</em>-values.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Questions on these discrete distributions have not been generally well answered in the past and it was pleasing to note that many candidates submitted a reasonably good solution to this question. In (b), the determination of the value of <em>p</em> was often successful using a variety of methods including solving the equation \(p(1 - p) = {(0.000396{\text{ }} \ldots )^{1/5}}\), graph plotting or using SOLVER on the GDC or even expanding the equation into a \({10^{{\text{th}}}}\) degree polynomial and solving that. Solutions to this particular question exceeded expectations.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of adult monkeys of a certain species are known to be normally distributed, the males with mean 30 kg and standard deviation 3 kg and the females with mean 20 kg and standard deviation 2.5 kg.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the weight of a randomly selected male is more than twice the weight of a randomly selected female.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two males and five females stand together on a weighing machine. Find the probability that their total weight is less than 175 kg.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we are given that \(M \sim {\text{N(30, 9)}}\) and \(F \sim {\text{N(20, 6.25)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(X = M - 2F;{\text{ }}X \sim {\text{N}}\)(\( - 10\), \(34\)) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we require \({\text{P}}(X > 0)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.0432 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(Y = {M_1} + {M_2} + {F_1} + {F_2} + {F_3} + {F_4} + {F_5};{\text{ }}Y \sim {\text{N(160, 49.25)}}\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we require \({\text{P}}(Y < 175) = 0.984\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A teacher decides to use the marks obtained by a random sample of 12 students in Geography and History examinations to investigate whether or not there is a positive association between marks obtained by students in these two subjects. You may assume that the distribution of marks in the two subjects is bivariate normal.</p>
</div>
<div class="specification">
<p>He gives the marks to Anne, one of his students, and asks her to use a calculator to carry out an appropriate test at the 5% significance level. Anne reports that the \(p\)-value is 0.177.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses for this investigation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in context, what conclusion should be drawn from this \(p\)-value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The teacher then asks Anne for the values of the \(t\)-statistic and the product moment correlation coefficient \(r\) produced by the calculator but she has deleted these. Starting with the \(p\)-value, calculate these values of \(t\) and \(r\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({H_0}:\rho = 0;{\text{ }}{H_1}:\rho > 0\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not accept \(r\) in place of \(\rho \).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>insufficient evidence to conclude that there is a (positive) association between marks in these two subjects (or equivalent statement in context) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>degrees of freedom \( = 10\) <strong><em>(A1)</em></strong></p>
<p>required value of \(t = {\text{inverse }}t(0.823)\) <strong><em>(M1)</em></strong></p>
<p>\( = 0.972\) <strong><em>A1</em></strong></p>
<p>attempt to solve \(t = r\sqrt {\frac{{n - 2}}{{1 - {r^2}}}} \) <strong><em>(M1)</em></strong></p>
<p>\(r = 0.294\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any \(r\) value that rounds to 0.29.</p>
<p> </p>
<p><strong>Note:</strong> Follow through their \(t\) value to determine \(r\).</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The heating in a residential school is to be increased on the third frosty day during the term. If the probability that a day will be frosty is 0.09, what is the probability that the heating is increased on the \({25^{{\text{th}}}}\) day of the term?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) On which day is the heating most likely to be increased?</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) the distribution is NB(3, 0.09) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the probability is \(\left( {\begin{array}{*{20}{c}}<br> {24} \\ <br> 2 <br>\end{array}} \right){0.91^{22}} \times {0.09^3} = 0.0253\) <strong><em>(M1)(A1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) P(Heating increased on \({n^{{\text{th}}}}\) day)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {n - 1} \\ <br> 2 <br>\end{array}} \right){0.91^{n - 3}} \times {0.09^3}\) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">by trial and error <em>n</em> = 23 gives the maximum probability <strong><em>(M1)A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(neighbouring values: 0.02551 (<em>n</em> = 22) ; 0.02554 (<em>n</em> = 23) ; 0.02545 (<em>n</em> = 24) )</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [12 marks]</em></strong></span></p>
<div><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates understood the context of this question, and the negative binomial distribution was usually applied, albeit occasionally with incorrect parameters. Good solutions were seen to part(b), using lists in their GDC or trial and error.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(X\) and \(Y\) are two random variables such that \({\text{E}}(X) = {\mu _X}\) and \({\text{E}}(Y) = {\mu _Y}\) then \({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X - {\mu _X})(Y - {\mu _Y})} \right)\).</p>
<p class="p1">Prove that if \(X\) and \(Y\) are independent then \({\text{Cov}}(X,{\text{ }}Y) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In a particular company, it is claimed that the distance travelled by employees to work is independent of their salary. To test this, 20 randomly selected employees are asked about the distance they travel to work and the size of their salaries. It is found that the product moment correlation coefficient, \(r\), for the sample is \( - 0.35\).</p>
<p class="p1">You may assume that both salary and distance travelled to work follow normal distributions.</p>
<p class="p1">Perform a one-tailed test at the \(5\% \) significance level to test whether or not the distance travelled to work and the salaries of the employees are independent.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">\({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X - {\mu _X})(Y - {\mu _Y})} \right)\)</p>
<p class="p1">\( = {\text{E}}(XY - X{\mu _Y} - Y{\mu _X} + {\mu _X}{\mu _Y})\) <span class="Apple-converted-space"> </span><span class="s1">(<strong><em>M1)</em></strong></span></p>
<p class="p1">\( = {\text{E}}(XY) - {\mu _Y}{\text{E}}(X) - {\mu _X}{\text{E}}(Y) + {\mu _X}{\mu _Y}\)</p>
<p class="p1">\( = {\text{E}}(XY) - {\mu _X}{\mu _Y}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">as \(X\) and \(Y\) are independent \({\text{E}}(XY) = {\mu _X}{\mu _Y}\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">\({\text{Cov}}(X,{\text{ }}Y) = 0\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X - {\mu _x})(Y - {\mu _y})} \right)\)</p>
<p class="p1">\( = {\text{E}}(X - {\mu _x}){\text{E}}(Y - {\mu _y})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">since \(X,Y\) are independent <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">\( = ({\mu _x} - {\mu _x})({\mu _y} - {\mu _y})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\( = 0\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({H_0}:\rho = 0\;\;\;{H_1}:\rho < 0\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The hypotheses must be expressed in terms of \(\rho \).</p>
<p> </p>
<p>test statistic \({t_{test}} = - 0.35\sqrt {\frac{{20 - 2}}{{1 - {{( - 0.35)}^2}}}} \) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = - 1.585 \ldots \) <strong><em>(A1)</em></strong></p>
<p>\({\text{degrees of freedom}} = 18\) <strong><em>(A1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p>\(p{\text{ - value}} = 0.0652\) <strong><em>A1</em></strong></p>
<p>this is greater than \(0.05\) <strong><em>M1</em></strong></p>
<p><strong>OR</strong></p>
<p>\({t_{5\% }}(18) = - 1.73\) <strong><em>A1</em></strong></p>
<p>this is less than \( - {\text{1.59}}\) <strong><em>M1</em></strong></p>
<p><strong>THEN</strong></p>
<p>hence accept \({H_0}\) or reject \({H_1}\) or equivalent or contextual equivalent <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow follow through for the final <strong><em>R1 </em></strong>mark.</p>
<p><em><strong>[8 marks]</strong></em></p>
<p><em><strong>Total [11 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (a) were often disappointing with few candidates gaining full marks, a common error being failure to state that</p>
<p class="p1">\(E(XY) = E(X)E(Y)\) or \({\text{E}}\left( {(X - {\mu _x})(Y - {\mu _y})} \right) = {\text{E}}(X - {\mu _x}){\text{E}}(Y - {\mu _y})\) in the case of independence.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In (b), the hypotheses were sometimes given incorrectly. Some candidates gave \({H_1}\) as \(\rho \ne 0\), not seeing that a one-tailed test was required. A more serious error was giving the hypotheses as \({H_0}:r = 0,{\text{ }}{H_1}:r < 0\) which shows a complete misunderstanding of the situation. Subsequent parts of the question were well answered in general.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the recurrence relation</p>
<p style="text-align: center;">\({u_n} = 5{u_{n - 1}} - 6{u_{n - 2}},{\text{ }}{u_0} = 0\) and \({u_1} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for \({u_n}\) in terms of \(n\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For every prime number \(p > 3\), show that \(p|{u_{p - 1}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the auxiliary equation is \({\lambda ^2} - 5\lambda + 6 = 0\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow \lambda = 2,{\text{ }}3\) <strong><em>(A1)</em></strong></p>
<p>the general solution is \({u_n} = A \times {2^n} + B \times {3^n}\) <strong><em>A1</em></strong></p>
<p>imposing initial conditions (substituting \(n = 0,{\text{ }}1\)) <strong><em>M1</em></strong></p>
<p>\(A + B = 0\) and \(2A + 3B = 1\) <strong><em>A1</em></strong></p>
<p>the solution is \(A = - 1,{\text{ }}B = 1\)</p>
<p>so that \({u_n} = {3^n} - {2^n}\) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({u_{p - 1}} = {3^{p - 1}} - {2^{p - 1}}\)</p>
<p>\(p > 3\), therefore 3 or 2 are not divisible by \(p\) <strong><em>R1</em></strong></p>
<p>hence by FLT, \({3^{p - 1}} \equiv 1 \equiv {2^{p - 1}}(\bmod p)\) for \(p > 3\) <strong><em>M1A1</em></strong></p>
<p>\({u_{p - 1}} \equiv 0(\bmod p)\) <strong><em>A1</em></strong></p>
<p>\(p|{u_{p - 1}}\) for every prime number \(p > 3\) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The students in a class take an examination in Applied Mathematics which consists of two papers. Paper 1 is in Mechanics and Paper 2 is in Statistics. The marks obtained by the students in Paper 1 and Paper 2 are denoted by \((x,{\text{ }}y)\) respectively and you may assume that the values of \((x,{\text{ }}y)\) form a random sample from a bivariate normal distribution with correlation coefficient \(\rho \) . The teacher wishes to determine whether or not there is a positive association between marks in Mechanics and marks in Statistics.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State suitable hypotheses.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The marks obtained by the 12 students who sat both papers are given in the following table.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 24px;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the product moment correlation coefficient for these data and state its <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Interpret your <em>p</em>-value in the context of the problem.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">George obtained a mark of 63 on Paper 1 but was unable to sit Paper 2 because of illness. Predict the mark that he would have obtained on Paper 2.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Another class of 16 students sat examinations in Physics and Chemistry and the product moment correlation coefficient between the marks in these two subjects was calculated to be 0.524. Using a 1 % significance level, determine whether or not this value suggests a positive association between marks in Physics and marks in Chemistry.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}:\rho = 0;{\text{ }}{{\text{H}}_1}:\rho > 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) correlation coefficient = 0.905 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value \( = 2.61 \times {10^{ - 5}}\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) very strong evidence to indicate a positive association between marks in Mechanics and marks in Statistics <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the regression line of <em>y</em> on <em>x</em> is \(y = 8.71 + 0.789x\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">George’s estimated mark on Paper 2 \( = 8.71 + 0.789 \times 63\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 58 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = r\sqrt {\frac{{n - 2}}{{1 - {r^2}}}} = 2.3019 \ldots \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">degrees of freedom = 14 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value \( = 0.0186 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">at the 1 % significance level, this does not indicate a positive association between the marks in Physics and Chemistry <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has a Poisson distribution with unknown mean \(\mu \) . It is required to test the hypotheses</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({H_0}:\mu = 3\) against \({H_1}:\mu \ne 3\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>S</em> denote the sum of 10 randomly chosen values of <em>X</em> . The critical region is defined as \((S \leqslant 22) \cup (S \geqslant 38)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the significance level of the test.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the value of \(\mu \) is actually 2.5, determine the probability of a Type II error.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">under \({H_0}\) , \(S{\text{ is Po}}(30)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S \leqslant 22) = {\text{0.080569}} \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S \geqslant 38) = {\text{0.089012}} \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">significance level = 0.080569… + 0.089012… <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.170 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S \leqslant 22) = {\text{0.080569}} \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S \leqslant 37) = {\text{0.910987}} \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">significance level = 1 – (0.910987…) + 0.089012… <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.170 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept 17 % or 0.17.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award 2 marks out of the final 4 marks for correct use of the Central Limit Theorem, giving 0.144 without a continuity correction and 0.171 with a continuity correction. The first </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> is independent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>S</em> is now Po(25) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P (Type II error) = P (accept \({H_0}|\mu = 2.5\)) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{P}}\left( {23 \leqslant S \leqslant 37|S{\text{ is Po}}(25)} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Only one of the above </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> marks can be implied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.990789… – 0.317533… </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.673 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award 2 marks out of the final 4 marks for correct use of the Central Limit Theorem, giving 0.647 without a continuity correction and 0.685 with a continuity correction. The first </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> is independent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to this question were often disappointing with many candidates not knowing what had to be done. Even those candidates who knew what to do sometimes made errors in evaluating the probabilities, often by misinterpreting the inequality signs. Candidates who used the Central Limit Theorem to evaluate the probabilities were given only partial credit on the grounds that the answers obtained were approximate and not exact.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to this question were often disappointing with many candidates not knowing what had to be done. Even those candidates who knew what to do sometimes made errors in evaluating the probabilities, often by misinterpreting the inequality signs. Candidates who used the Central Limit Theorem to evaluate the probabilities were given only partial credit on the grounds that the answers obtained were approximate and not exact.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table gives the average yield of olives per tree, in kg, and the rainfall, in cm, for nine separate regions of Greece. You may assume that these data are a random sample from a bivariate normal distribution, with correlation coefficient \(\rho \).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-11_om_08.55.24.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A scientist wishes to use these data to determine whether there is a positive correlation between rainfall and yield.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State suitable hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the product moment correlation coefficient for these data.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Determine the associated <em>p</em>-value and comment on this value in the context of the question.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the equation of the regression line of <em>y </em>on <em>x</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Hence, estimate the yield per tree in a tenth region where the rainfall was 19 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Determine the angle between the regression line of <em>y </em>on <em>x </em>and that of <em>x </em>on <em>y </em>. Give your answer to the nearest degree.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({H_0}:\rho = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({H_1}:\rho > 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) 0.853 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer that rounds to 0.85.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <em>p</em>-value = 0.00173 (1-tailed) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer that rounds to 0.0017.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds to 0.0035 obtained from 2-tailed test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">strong evidence to reject the hypothesis that there is no correlation between rainfall and yield or to accept the hypothesis that there is correlation between rainfall and yield <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Follow through the <em>p</em>-value for the conclusion.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \(y = 1.78x + 40.5\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept numerical coefficients that round to 1.8 and 41.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) \(y = 1.77 \ldots (19) + 14.5 \ldots \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">74.3 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer that rounds to 74 or 75.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) the gradient of the regression line <em>y </em>on <em>x </em>is 1.78 or equivalent <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the regression line of <em>x </em>on <em>y </em>is \(x = 0.409y - 12.2\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the gradient of the regression line <em>x </em>on <em>y </em>is \(\frac{1}{{0.409}}{\text{ }}( = 2.44)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">calculate \(\arctan (2.44) - \arctan (1.78)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">angle between regression lines is 7 degrees <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer which rounds to ±7 degrees.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [16 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A farmer sells bags of potatoes which he states have a mean weight of 7 kg . An inspector, however, claims that the mean weight is less than 7 kg . In order to test this claim, the inspector takes a random sample of 12 of these bags and determines the weight, \(x\) kg , of each bag. He finds that \[\sum {x = 83.64;{\text{ }}\sum {{x^2} = 583.05.} } \] You may assume that the weights of the bags of potatoes can be modelled by the normal distribution \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to test the inspector’s claim.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Carry out an appropriate test and state the \(p\)-value obtained.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using a 10% significance level and justifying your answer, state your conclusion in context.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({H_0}:\mu = 7,{\text{ }}{H_1}:\mu < 7\) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\bar x = \frac{{83.64}}{{12}} = 6.97\) <strong><em>A1</em></strong></p>
<p>\(s_{n - 1}^2 = \frac{{583.05}}{{11}} - \frac{{{\text{ }}{{83.64}^2}}}{{132}} = 0.0072\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(t = \frac{{6.97 - 7}}{{\sqrt {\frac{{0.0072}}{{12}}} }} = - 1.22(474 \ldots )\) <strong><em>(M1)(A1)</em></strong></p>
<p>\({\text{degrees of freedom}} = 11\) <strong><em>(A1)</em></strong></p>
<p>\(p{\text{ - value}} = 0.123\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.12.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>because \(p > 0.1\) <strong><em>R1</em></strong></p>
<p>the inspector’s claim is not supported (at the 10% level)</p>
<p>(or equivalent in context) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only award the <strong><em>A1 </em></strong>if the <strong><em>R1 </em></strong>has been awarded</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> represents the lifetime in hours of a battery. The lifetime may be assumed to be a continuous random variable <em>X</em> with a probability density function given by \(f(x) = \lambda {{\text{e}}^{ - \lambda x}}\), where \(x \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the cumulative distribution function, \(F(x)\), of <em>X</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the lifetime of a particular battery is more than twice the mean.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the median of <em>X</em> in terms of \(\lambda \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the lifetime of a particular battery lies between the median and the mean.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\lambda {{\text{e}}^{ - \lambda t}}{\text{d}}t = - {{\text{e}}^{ - \lambda t}}{\text{ }}( + c)} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow F(x) = \left[ { - {{\text{e}}^{ - \lambda t}}} \right]_0^x\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - {{\text{e}}^{ - \lambda t}}{\text{ }}(x \geqslant 0)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - F\left( {\frac{2}{\lambda }} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {{\text{e}}^{ - 2}}\,\,\,\,\,( = 0.135)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(F(m) = \frac{1}{2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {{\text{e}}^{ - \lambda m}} = \frac{1}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow - \lambda m = \ln \frac{1}{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m = \frac{1}{\lambda }\ln 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(F\left( {\frac{1}{\lambda }} \right) - F\left( {\frac{{\ln 2}}{\lambda }} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2} - {{\text{e}}^{ - 1}}\,\,\,\,\,( = 0.132)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates the question started well, but many did not appear to understand how to find the cumulative distribution function in (b). Many were able to integrate \(\lambda {{\text{e}}^{ - \lambda x}}\), but then did not know what to do with the integral. Parts (c), (d) and (e) were relatively well done, but even candidates who successfully found the cumulative distribution function often did not use it. This resulted in a lot of time spent integrating the same function.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates the question started well, but many did not appear to understand how to find the cumulative distribution function in (b). Many were able to integrate \(\lambda {{\text{e}}^{ - \lambda x}}\), but then did not know what to do with the integral. Parts (c), (d) and (e) were relatively well done, but even candidates who successfully found the cumulative distribution function often did not use it. This resulted in a lot of time spent integrating the same function.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates the question started well, but many did not appear to understand how to find the cumulative distribution function in (b). Many were able to integrate \(\lambda {{\text{e}}^{ - \lambda x}}\), but then did not know what to do with the integral. Parts (c), (d) and (e) were relatively well done, but even candidates who successfully found the cumulative distribution function often did not use it. This resulted in a lot of time spent integrating the same function.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates the question started well, but many did not appear to understand how to find the cumulative distribution function in (b). Many were able to integrate \(\lambda {{\text{e}}^{ - \lambda x}}\), but then did not know what to do with the integral. Parts (c), (d) and (e) were relatively well done, but even candidates who successfully found the cumulative distribution function often did not use it. This resulted in a lot of time spent integrating the same function.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Anne is a farmer who grows and sells pumpkins. Interested in the weights of pumpkins produced, she records the weights of eight pumpkins and obtains the following results in kilograms.</p>
<p>\[{\text{7.7}}\quad {\text{7.5}}\quad {\text{8.4}}\quad {\text{8.8}}\quad {\text{7.3}}\quad {\text{9.0}}\quad {\text{7.8}}\quad {\text{7.6}}\]</p>
<p>Assume that these weights form a random sample from a \(N(\mu ,{\text{ }}{\sigma ^2})\) distribution. </p>
<p> </p>
</div>
<div class="specification">
<p>Anne claims that the mean pumpkin weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis \({{\text{H}}_0}:\mu = 7.5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use a two-tailed test to determine the \(p\)-value for the above results.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your \(p\)-value at the 5% level of significance, justifying your conclusion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>UE of \(\mu \) is \(8.01{\text{ }}( = 8.0125)\) <strong><em>A1</em></strong></p>
<p>UE of \({\sigma ^2}\) is 0.404 <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers that round correctly to 2 sf.</p>
<p> </p>
<p><strong>Note:</strong> Condone incorrect notation, <em>ie</em>, \(\mu \) instead of UE of \(\mu \) and \({\sigma ^2}\) instead of UE of \({\sigma ^2}\).</p>
<p> </p>
<p><strong>Note:</strong> <strong><em>M0 </em></strong>for squaring \(0.594 \ldots \) giving 0.354, <strong><em>M1A0 </em></strong>for failing to square \(0.635 \ldots \)</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to use the \(t\)-test <strong><em>(M1)</em></strong></p>
<p>\(p\)-value is 0.0566 <strong><em>A2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 2 sf.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(0.0566 > 0.05\) <strong><em>R1</em></strong></p>
<p>we accept the null hypothesis (mean pumpkin weight is 7.5 kg) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Apply follow through on the candidate’s \(p\)-value.</p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A1 </em></strong>if <strong><em>R1 </em></strong>is not awarded.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has probability distribution Po(8).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(X = 6)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find \({\text{P}}(X = 6|5 \leqslant X \leqslant 8)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar X\) denotes the sample mean of \(n > 1\) independent observations from \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down \({\text{E}}(\bar X)\) and \({\text{Var}}(\bar X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, give a reason why \(\bar X\) is not a Poisson distribution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample of \(40\) observations is taken from the distribution for \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(7.1 < \bar X < 8.5)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({\text{P}}\left( {\left| {\bar X - 8} \right| \leqslant k} \right) = 0.95\), find the value of \(k\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{P}}(X = 6) = 0.122\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(X = 6|5 \leqslant X \leqslant 8) = \frac{{{\text{P}}(X = 6)}}{{{\text{P}}(5 \leqslant X \leqslant 8)}} = \frac{{0.122 \ldots }}{{0.592 \ldots - 0.0996 \ldots }}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.248\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{E}}(\bar X) = 8\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(\bar X) = \frac{8}{n}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{E}}(\bar X) \ne {\text{Var}}(\bar X)\) \({\text{(for }}n > 1)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Only award the <strong><em>R1 </em></strong>if the two expressions in (b)(i) are different.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar X \sim {\text{N(8, 0.2)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>M1 </em></strong>for normality, <strong><em>A1 </em></strong>for parameters.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(7.1 < \bar X < 8.5) = 0.846\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The expression is equivalent to</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(283 \leqslant \sum {X \leqslant 339)} \) where \(\sum X \) is \({\text{Po(320)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.840\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept 284, 340 instead of 283, 339</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds correctly to 0.84 or 0.85.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 1.96\frac{\sigma }{{\sqrt n }}\) or \(1.96{\text{ std}}(\bar X)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 0.877\) or \(1.96\sqrt {0.2} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The expression is equivalent to</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(320 - 40k \leqslant \sum {X \leqslant 320 + 40k) = 0.95} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 0.875\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer that rounds to 0.87 or 0.88.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> Award <strong><em>M1A0 </em></strong>if modulus sign ignored and answer obtained rounds to 0.74 or 0.75</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weight of tea in <em>Supermug</em> tea bags has a normal distribution with mean 4.2 g and standard deviation 0.15 g. The weight of tea in <em>Megamug</em> tea bags has a normal distribution with mean 5.6 g and standard deviation 0.17 g.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen <em>Supermug</em> tea bag contains more than 3.9 g of tea.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that, of two randomly chosen <em>Megamug</em> tea bags, one contains more than 5.4 g of tea and one contains less than 5.4 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that five randomly chosen <em>Supermug</em> tea bags contain a total of less than 20.5 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the total weight of tea in seven randomly chosen <em>Supermug</em> tea bags is more than the total weight in five randomly chosen <em>Megamug</em> tea bags.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>S</em> be the weight of tea in a random <em>Supermug</em> tea bag</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(S \sim {\text{N(4.2, 0.1}}{{\text{5}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S > 3.9) = 0.977\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>M</em> be the weight of tea in a random <em>Megamug</em> tea bag</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(M \sim {\text{N(5.6, 0.1}}{{\text{7}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(M > 5.4) = 0.880 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(M < 5.4) = 1 - 0.880 \ldots = 0.119 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">required probability \( = 2 \times 0.880 \ldots \times 0.119 \ldots = 0.211\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}({S_1} + {S_2} + {S_3} + {S_4} + {S_5} < 20.5)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \({S_1} + {S_2} + {S_3} + {S_4} + {S_5} = A\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(A) = 5{\text{E}}(S)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 21 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(A) = 5{\text{Var}}(S)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.1125 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A \sim {\text{N(21, 0.1125}})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A < 20.5) = 0.0680\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}({S_1} + {S_2} + {S_3} + {S_4} + {S_5} + {S_6} + {S_7} - ({M_1} + {M_2} + {M_3} + {M_4} + {M_5}) > 0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \({S_1} + {S_2} + {S_3} + {S_4} + {S_5} + {S_6} + {S_7} - ({M_1} + {M_2} + {M_3} + {M_4} + {M_5}) = B\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(B) = 7{\text{E}}(S) - 5{\text{E}}(M)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 1.4 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Above </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> is independent of first </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(B) = 7{\text{Var}}(S) + 5{\text{Var}}(M)\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.302 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(B > 0) = 0.995\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates this was a reasonable start to the paper with many candidates gaining close to full marks. The most common error was in (b) where, surprisingly, many candidates did not realise the need to multiply the product of the two probabilities by 2 to gain the final answer. Weaker candidates often found problems in understanding how to correctly find the variance in both (c) and (d).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates this was a reasonable start to the paper with many candidates gaining close to full marks. The most common error was in (b) where, surprisingly, many candidates did not realise the need to multiply the product of the two probabilities by 2 to gain the final answer. Weaker candidates often found problems in understanding how to correctly find the variance in both (c) and (d).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates this was a reasonable start to the paper with many candidates gaining close to full marks. The most common error was in (b) where, surprisingly, many candidates did not realise the need to multiply the product of the two probabilities by 2 to gain the final answer. Weaker candidates often found problems in understanding how to correctly find the variance in both (c) and (d).</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For most candidates this was a reasonable start to the paper with many candidates gaining close to full marks. The most common error was in (b) where, surprisingly, many candidates did not realise the need to multiply the product of the two probabilities by 2 to gain the final answer. Weaker candidates often found problems in understanding how to correctly find the variance in both (c) and (d).</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable \(X\) has the following probability distribution.</p>
<p style="padding-left: 90px;">\({\text{P}}(X = x) = \left\{ {\begin{array}{*{20}{l}}<br> {p{q^{\frac{x}{2}}}}&{{\text{for }}x = 0,{\text{ }}2,{\text{ }}4,{\text{ }}6 \ldots {\text{ where }}p + q = 1,{\text{ }}0 < p < 1.} \\ <br> 0&{{\text{otherwise}}} <br>\end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability generating function for \(X\) is given by \(G(t) = \frac{P}{{1 - q{t^2}}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine \({\text{E}}(X)\) in terms of \(p\) and \(q\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable \(Y\) is given by \(Y = 2X + 1\). Find the probability generating function for \(Y\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(G(t) = \sum {P(X = x){t^x}} \) <strong><em>(M1)</em></strong></p>
<p>\( = p + pq{t^2} + p{q^2}{t^4} + \ldots \)</p>
<p>(summing \(GP\)) \({u_1} = p,{\text{ }}r = q{t^2}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{p}{{1 - q{t^2}}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(G’(t) = - \frac{p}{{{{(1 - q{t^2})}^2}}} \times - 2qt\) <strong><em>M1A1</em></strong></p>
<p>\({\text{E}}(X) = G’(1)\) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{2pq}}{{{{(1 - q)}^2}}}\,\,\,\left( { = \frac{{2q}}{p}} \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\({\text{PGF of }}Y = \sum {P(Y = y){t^y}} \) (<strong><em>M1)</em></strong></p>
<p>\( = pt + pq{t^5} + p{q^2}{t^9} + \ldots \) <strong><em>A1</em></strong></p>
<p>\( = \frac{{pt}}{{1 - q{t^4}}}\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\({\text{PGF of }}Y = {\text{E}}({t^Y})\) (<strong><em>M1)</em></strong></p>
<p>\( = {\text{E}}({t^{2X + 1}})\)</p>
<p>\( = {\text{E}}\left( {{{({t^2})}^X}} \right) \times {\text{E}}(t)\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{pt}}{{1 - q{t^4}}}\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable \(X\) follows a Poisson distribution with mean \(\lambda \). The probability generating function of \(X\) is given by \({G_X}(t) = {{\text{e}}^{\lambda (t - 1)}}\).</p>
</div>
<div class="specification">
<p>The random variable \(Y\), independent of \(X\), follows a Poisson distribution with mean \(\mu \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find expressions for \({G’_X}(t)\) and \({G’’_X}(t)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \({\text{Var}}(X) = \lambda \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\), show that \(X + Y\) follows a Poisson distribution with mean \(\lambda + \mu \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{P}}(X = x|X + Y = n) = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){\left( {\frac{\lambda }{{\lambda + \mu }}} \right)^x}{\left( {1 - \frac{\lambda }{{\lambda + \mu }}} \right)^{n - x}}\), where \(n\), \(x\) are non-negative integers and \(n \geqslant x\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Identify the probability distribution given in part (c)(i) and state its parameters.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({G’_X}(t) = \lambda {{\text{e}}^{\lambda (t - 1)}}\) <strong><em>A1</em></strong></p>
<p>\({G’’_X}(t) = {\lambda ^2}{{\text{e}}^{\lambda (t - 1)}}\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{Var}}(X) = {G''_X}(1) + {G'_X}(1) - {\left( {{{G'}_X}(1)} \right)^2}\) <strong><em>(M1)</em></strong></p>
<p>\({G’_X}(1) = \lambda \) and \({G’’_X}(1) = {\lambda ^2}\) <strong><em>(A1)</em></strong></p>
<p>\({\text{Var}}(X) = {\lambda ^2} + \lambda - {\lambda ^2}\) <strong><em>A1</em></strong></p>
<p>\( = \lambda \) <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({G_{X + Y}}(t) = {{\text{e}}^{\lambda (t - 1)}} \times {{\text{e}}^{\mu (t - 1)}}\) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The <strong><em>M1 </em></strong>is for knowing to multiply pgfs.</p>
<p> </p>
<p>\( = {{\text{e}}^{(\lambda + \mu )(t - 1)}}\) <strong><em>A1</em></strong></p>
<p>which is the pgf for a Poisson distribution with mean \(\lambda + \mu \) <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Line 3 identifying the Poisson pgf must be seen.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X = x|X + Y = n) = \frac{{{\text{P}}(X = x \cap Y = n - x)}}{{{\text{P}}(X + Y = n)}}\) <strong><em>(M1)</em></strong></p>
<p>\( = \left( {\frac{{{{\text{e}}^{ - \lambda }}{\lambda ^x}}}{{x!}}} \right)\left( {\frac{{{{\text{e}}^{ - \mu }}{\mu ^{n - x}}}}{{(n - x)!}}} \right)\left( {\frac{{n!}}{{{{\text{e}}^{ - (\lambda + \mu )}}{{(\lambda + \mu )}^n}}}} \right)\) (or equivalent) <strong><em>M1A1</em></strong></p>
<p>\( = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right)\frac{{{\lambda ^x}{\mu ^{n - x}}}}{{{{(\lambda + \mu )}^n}}}\) <strong><em>A1</em></strong></p>
<p>\( = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){\left( {\frac{\lambda }{{\lambda + \mu }}} \right)^x}{\left( {\frac{\mu }{{\lambda + \mu }}} \right)^{n - x}}\) <strong><em>A1</em></strong></p>
<p>leading to \({\text{P}}(X = x|X + Y = n) = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){\left( {\frac{\lambda }{{\lambda + \mu }}} \right)^x}{\left( {1 - \frac{\lambda }{{\lambda + \mu }}} \right)^{n - x}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{B}}\left( {n,{\text{ }}\frac{\lambda }{{\lambda + \mu }}} \right)\) <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for stating binomial and <strong><em>A1 </em></strong>for stating correct parameters.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider an unbiased tetrahedral (four-sided) die with faces labelled 1, 2, 3 and 4 respectively.</p>
<p>The random variable <em>X</em> represents the number of throws required to obtain a 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the distribution of <em>X</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability generating function, \(G\left( t \right)\), for <em>X</em> is given by \(G\left( t \right) = \frac{t}{{4 - 3t}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(G'\left( t \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean number of throws required to obtain a 1.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>X</em> is geometric (or negative binomial) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(G\left( t \right) = \frac{1}{4}t + \frac{1}{4}\left( {\frac{3}{4}} \right){t^2} + \frac{1}{4}{\left( {\frac{3}{4}} \right)^2}{t^3} + \ldots \) <em><strong>M1A1</strong></em></p>
<p>recognition of GP \(\left( {{u_1} = \frac{1}{4}t,\,\,r = \frac{3}{4}t} \right)\) <em><strong> (M1)</strong></em></p>
<p>\( = \frac{{\frac{1}{4}t}}{{1 - \frac{3}{4}t}}\) <em><strong>A1</strong></em></p>
<p>leading to \(G\left( t \right) = \frac{t}{{4 - 3t}}\) <em><strong> AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use product or quotient rule <em><strong>M1</strong></em></p>
<p>\(G'\left( t \right) = \frac{4}{{{{\left( {4 - 3t} \right)}^2}}}\) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>4 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1FT</strong></em> to a candidate that correctly calculates the value of \(G'\left( 1 \right)\) from their \(G'\left( t \right)\).</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Alun answers mathematics questions and checks his answer after doing each one.</p>
<p class="p1">The probability that he answers any question correctly is always \(\frac{6}{7}\), independently of all other questions. He will stop for coffee immediately following a second incorrect answer. Let \(X\) be the number of questions Alun answers before he stops for coffee.</p>
</div>
<div class="specification">
<p class="p1">Nic answers mathematics questions and checks his answer after doing each one.</p>
<p class="p1">The probability that he answers any question correctly is initially \(\frac{6}{7}\). After his first incorrect answer, Nic loses confidence in his own ability and from this point onwards, the probability that he answers any question correctly is now only \(\frac{4}{7}\).</p>
<p class="p1">Both before and after his first incorrect answer, the result of each question is independent of the result of any other question. Nic will also stop for coffee immediately following a second incorrect answer. Let \(Y\) be the number of questions Nic answers before he stops for coffee.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of \(X\), <span class="s1">including its parameters.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate \({\text{E}}(X)\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Calculate \({\text{P}}(X = 5)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate \({\text{E}}(Y)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate \({\text{P}}(Y = 5)\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \({\text{NB}}\left( {2,{\text{ }}\frac{1}{7}} \right)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p1"><span class="s1"><strong>Note: </strong>The final <strong><em>A1 </em></strong></span>mark can be awarded for knowing that \(p = \frac{1}{7}\) independent of the other two marks.</p>
<p class="p3"> </p>
<p class="p1">(ii) <span class="Apple-converted-space"> \({\text{E}}(X) = \frac{r}{p} = 14\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(\left( {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right){\left( {\frac{6}{7}} \right)^3}{\left( {\frac{1}{7}} \right)^2} = 0.0514\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p4"><strong>Note: </strong>Accept any number that rounds to this 3sf number.</p>
<p class="p2"> </p>
<p class="p4"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(Y = {Y_1} + {Y_2}\)</span> <span class="s1">(number up to1st + </span>number up to 2nd) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({Y_1} \sim Geo\left( {\frac{1}{7}} \right),{\text{ }}{Y_2} \sim Geo\left( {\frac{3}{7}} \right)\) </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p3"> </p>
<p class="p1"><strong>Notes: </strong>The above <strong><em>(A1) </em></strong>is independent of the <strong><em>(M1)</em></strong><span class="s1">.</span></p>
<p class="p1"><span class="s1">Could have \({\text{NB }}(1,{\text{ }}p)\)</span>, instead of \(Geo(p)\)<span class="s1"><em>.</em></span></p>
<p class="p4"> </p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(Y) = \frac{1}{{\left( {\frac{1}{7}} \right)}} + \frac{1}{{\left( {\frac{3}{7}} \right)}} = 7 + \frac{7}{3} = 9\frac{1}{3}{\text{ (9.33)}}\) </span><span class="s2"><strong><em>M1A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(Y = {Y_1} + {Y_2} = 5\)</span> happens when <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({Y_1} = 1,{\text{ }}{Y_2} = 4\) or \({Y_1} = 2,{\text{ }}{Y_2} = 3\) or \({Y_1} = 3,{\text{ }}{Y_2} = 2\) or \({Y_1} = 4,{\text{ }}{Y_2} = 1\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2">so probability is \(\frac{1}{7}\frac{4}{7}\frac{4}{7}\frac{4}{7}\frac{3}{7} + \frac{6}{7}\frac{1}{7}\frac{4}{7}\frac{4}{7}\frac{3}{7} + \frac{6}{7}\frac{6}{7}\frac{1}{7}\frac{4}{7}\frac{3}{7} + \frac{6}{7}\frac{6}{7}\frac{6}{7}\frac{1}{7}\frac{3}{7}\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>(M1)(A1)</em></strong></span></p>
<p class="p5"><span class="Apple-converted-space">\( = 0.0928{\text{ }}\left( {\frac{{1560}}{{16807}}} \right)\) </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p2"><span class="s2"><strong>Note: </strong></span>Accept any answer that rounds to 0.093.</p>
<p class="p4"> </p>
<p class="p1"><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Two independent discrete random variables \(X\) and \(Y\) have probability generating functions \(G(t)\) and \(H(t)\) respectively. Let \(Z = X + Y\) have probability generating function \(J(t)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down an expression for \(J(t)\) <span class="s1">in terms of \(G(t)\) and \(H(t)\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By differentiating \(J(t)\), <span class="s1">prove that</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> \({\text{E}}(Z) = {\text{E}}(X) + {\text{E}}(Y)\)</span>;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> \({\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)\)</span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(J(t) = G(t)H(t)\) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(J'(t) = G'(t)H(t) + G(t)H'(t)\)</span> <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(J'(1) = G'(1)H(1) + G(1)H'(1)\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(J'(1) = G'(1) + H'(1)\) </span><strong><em>A1</em></strong></p>
<p class="p1">so \(E(Z) = E(X) + E(Y)\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(J''(t) = G''(t)H(t) + G'(t)H'(t) + G'(t)H'(t) + G(t)H''(t)\)</span> <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\(J''(1) = G''(1)H(1) + 2G'(1)H'(1) + G(1)H''(1)\)</p>
<p class="p1"><span class="Apple-converted-space">\( = G''(1) + 2G'(1)H'(1) + H''(1)\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{Var}}(Z) = J''(1) + J'(1) - {\left( {J'(1)} \right)^2}\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = G''(1) + 2G'(1)H'(1) + H''(1) + G'(1) + H'(1) - {\left( {G'(1) + H'(1)} \right)^2}\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = G''(1) + G'(1) - {\left( {G'(1)} \right)^2} + H''(1) + H'(1) - {\left( {H'(1)} \right)^2}\) </span><strong><em>A1</em></strong></p>
<p class="p1">so \({\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>If addition is wrongly used instead of multiplication in (a) it is inappropriate to give <strong><em>FT </em></strong>apart from the second <strong><em>M </em></strong>marks in each part, as the working is too simple.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[10 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>n </em>independent random variables \({X_1},{X_2},…,{X_n}\) all have the distribution \({\text{N}}(\mu ,\,{\sigma ^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the mean and the variance of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({X_1} + {X_2}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(3{X_1}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({X_1} + {X_2} - {X_3}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) \(\bar X = \frac{{({X_1} + {X_2} + ... + {X_n})}}{n}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{E}}(X_1^2)\) in terms of \(\mu \) and \(\sigma \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(2\mu ,{\text{ }}2{\sigma ^2}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(3\mu ,{\text{ }}9{\sigma ^2}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(\mu ,{\text{ }}3{\sigma ^2}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) \(\mu ,{\text{ }}\frac{{{\sigma ^2}}}{n}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note</strong><span style="font-family: 'times new roman', times; font-size: medium;">: If candidate clearly and correctly gives the standard deviations rather than the variances, give </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for 2 or 3 standard deviations and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for 4 standard deviations.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}({X_1}) = {\text{E}}(X_1^2) - {\left( {{\text{E}}({X_1})} \right)^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\sigma ^2} = {\text{E}}(X_1^2) - {\mu ^2}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X_1^2) = {\sigma ^2} + {\mu ^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was very well answered indeed with very many candidates gaining full marks including, pleasingly, part (b). Candidates who could not do question 2, struggled on the whole paper.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was very well answered indeed with very many candidates gaining full marks including, pleasingly, part (b). Candidates who could not do question 2, struggled on the whole paper.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When she tosses it, the score is defined as the number on the uppermost face. One day, she decides to toss the die repeatedly until all the possible scores have occurred at least once.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Having thrown the die once, she lets \({X_2}\) denote the number of additional throws required to obtain a different number from the one obtained on the first throw. State the distribution of \({X_2}\) and hence find \({\text{E}}({X_2})\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) She then lets \({X_3}\) denote the number of additional throws required to obtain a different number from the two numbers already obtained. State the distribution of \({X_3}\) and hence find \({\text{E}}({X_3})\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) By continuing the process, show that the expected number of tosses needed to obtain all six possible scores is 14.7.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({X_2}\) is a geometric random variable <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">with \(p = \frac{5}{6}.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore \({\text{E}}({X_2}) = \frac{6}{5}.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({X_3}\) is a geometric random variable with \(p = \frac{4}{6}.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore \({\text{E}}({X_3}) = \frac{6}{4}.\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \({\text{E}}({X_4}) = \frac{6}{3},{\text{ E}}({X_5}) = \frac{6}{2},{\text{ E}}({X_6}) = \frac{6}{1}\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}({X_1}) = 1\,\,\,\,\,{\text{(or }}{X_1} = 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Expected number of tosses \(\sum\limits_{n = 1}^6 {{\text{E}}({X_n})} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 14.7\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates were unable even to start this question although those who did often made substantial progress.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A coin was tossed 200 times and 115 of these tosses resulted in ‘heads’. Use a two-tailed test with significance level 1 % to investigate whether or not the coin is biased.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of’ ‘heads’ <em>X </em>is B(200, <em>p</em>) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}:p = 0.5;{\text{ }}{{\text{H}}_1}:p \ne 0.5\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A0 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">for the statement “ \({{\text{H}}_0}:\) coin is fair; \({{\text{H}}_1}:\) coin is biased”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">EITHER</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(\left. {X \geqslant 115} \right|{{\text{H}}_0}) = 0.0200\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0400 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This is greater than 0.01. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There is insufficient evidence to conclude that the coin is biased (or the coin is not biased). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(Using a proportion test on a GDC) <em>p</em>-value = 0.0339 <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This is greater than 0.01. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There is insufficient evidence to conclude that the coin is biased (or the coin is not biased). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Under \({{\text{H}}_0}X\) is approximately N(100, 50) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = \frac{{115 - 100}}{{\sqrt {50} }} = 2.12\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(Accept 2.05 with continuity correction)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This is less than 2.58 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There is insufficient evidence to conclude that the coin is biased (or the coin is not biased). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">99 % confidence limits for <em>p </em>are \(\frac{{115}}{{200}} \pm 2.576\sqrt {\frac{{115}}{{200}} \times \frac{{85}}{{200}} \times \frac{1}{{200}}} \) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">giving [0.485, 0.665] <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This interval contains 0.5 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There is insufficient evidence to conclude that the coin is biased (or the coin is not biased). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was well answered in general with several correct methods seen. The most popular method was to use a GDC to carry out a proportion test which is equivalent to using a normal approximation. Relatively few candidates calculated an exact <span style="font: 19.0px Times;"><em>p</em></span>-value using the binomial distribution. Candidates who found a 95% confidence interval for <span style="font: 19.0px Times;"><em>p</em></span>, the probability of obtaining a head, and noted that this contained 0.5 were given full credit.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>Y</em> is such that \({\text{E}}(2Y + 3) = 6{\text{ and Var}}(2 - 3Y) = 11\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) E(<em>Y</em>) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{Var}}(Y)\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{E}}({Y^2})\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Independent random variables <em>R</em> and <em>S</em> are such that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[R \sim {\text{N}}(5,{\text{ 1}}){\text{ and }}S \sim {\text{N(8, 2).}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>V</em> is defined by <em>V</em> = 3<em>S</em> – 4<em>R</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate P(<em>V</em> > 5).</span></p>
<div> </div>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{E}}(2Y + 3) = 6\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2{\text{E}}(Y) + 3 = 6\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(Y) = \frac{3}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{Var}}(2 - 3Y) = 11\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}( - 3Y) = 11\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(9{\text{Var}}(Y) = 11\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(Y) = \frac{{11}}{9}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{E}}({Y^2}) = {\text{Var}}(Y) + {\left[ {{\text{E}}(Y)} \right]^2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{11}}{9} + \frac{9}{4}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{125}}{{36}}\) <strong><em>A1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">E(</span><em style="font-family: 'times new roman', times; font-size: medium;">V</em><span style="font-family: 'times new roman', times; font-size: medium;">) = E(3</span><em style="font-family: 'times new roman', times; font-size: medium;">S – </em><span style="font-family: 'times new roman', times; font-size: medium;">4</span><em style="font-family: 'times new roman', times; font-size: medium;">R</em><span style="font-family: 'times new roman', times; font-size: medium;">)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 20.5px;"> </span><span style="font-family: 'times new roman', times; font-size: medium;">= </span><span style="font-family: 'times new roman', times; font-size: medium;">3E(<em>S</em>) – 4E(<em>R</em>) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">= 24 – 20 = 4 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">Var(3</span><em style="font-family: 'times new roman', times; font-size: medium;">S – </em><span style="font-family: 'times new roman', times; font-size: medium;">4</span><em style="font-family: 'times new roman', times; font-size: medium;">R</em><span style="font-family: 'times new roman', times; font-size: medium;">) = 9Var(</span><em style="font-family: 'times new roman', times; font-size: medium;">S</em><span style="font-family: 'times new roman', times; font-size: medium;">) + 16Var(</span><em style="font-family: 'times new roman', times; font-size: medium;">R</em><span style="font-family: 'times new roman', times; font-size: medium;">) , since </span><em style="font-family: 'times new roman', times; font-size: medium;">R </em><span style="font-family: 'times new roman', times; font-size: medium;">and </span><em style="font-family: 'times new roman', times; font-size: medium;">S </em><span style="font-family: 'times new roman', times; font-size: medium;">are independent random variables </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">=18 + 16 = 34 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">\(V \sim {\text{N}}(4,{\text{ 34}})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(V > 5) = 0.432\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A2 N0</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> <strong style="font-family: 'times new roman', times; font-size: medium;"><em>[6 marks]</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px Helvetica;"> </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>E</em>(<em>Y</em>) was calculated correctly but many could not go further to find \(Var(Y){\text{ and }}E({Y^2})Var(2)\) was often taken to be 2. <em>V</em> was often taken to be discrete leading to calculations such as \(P(V > 5) = 1 - P(V \leqslant 5)\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>E</em>(<em>Y</em>) was calculated correctly but many could not go further to find \(Var(Y){\text{ and }}E({Y^2})Var(2)\) was often taken to be 2. <em>V</em> was often taken to be discrete leading to calculations such as \(P(V > 5) = 1 - P(V \leqslant 5)\).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A baker produces loaves of bread that he claims weigh on average 800 g each. Many customers believe the average weight of his loaves is less than this. A food inspector visits the bakery and weighs a random sample of 10 loaves, with the following results, in grams:</span></p>
<p style="font: normal normal normal 12px/normal Times; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">783, 802, 804, 785, 810, 805, 789, 781, 800, 791.</span></p>
<p style="font: normal normal normal 12px/normal Times; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that these results are taken from a normal distribution.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates for the mean and variance of the distribution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">In spite of these results the baker insists that his claim is correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Stating appropriate hypotheses, test the baker’s claim at the 10 % level of significance.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">unbiased estimate of the mean: 795 (grams) <strong><em>A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">unbiased estimate of the variance: 108 \((gram{s^2})\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">null hypothesis \({H_0}:\mu = 800\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">alternative hypothesis \({H_1}:\mu < 800\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">using 1-tailed <em>t</em>-test <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em> = 0.0812... <strong> <em>A3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">with 9 degrees of freedom <strong><em>(A1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({t_{calc}} = \frac{{\sqrt {10} (795 - 800)}}{{\sqrt {108} }} = - 1.521\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({t_{crit}} = - 1.383\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept 2sf intermediate results.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">so the baker’s claim is rejected <strong><em>R1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept “reject \({H_0}\) ” provided \({H_0}\) has been correctly stated.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>FT </em></strong>for the final <strong><em>R1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A successful question for many candidates. A few candidates did not read the question and adopted a 2-tailed test.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A successful question for many candidates. A few candidates did not read the question and adopted a 2-tailed test.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two species of plant, \(A\) and \(B\), are identical in appearance though it is known that the mean length of leaves from a plant of species \(A\) is \(5.2\) cm, whereas the mean length of leaves from a plant of species \(B\) is \(4.6\) cm. Both lengths can be modelled by normal distributions with standard deviation \(1.2\) cm.</p>
<p>In order to test whether a particular plant is from species \(A\) or species \(B\), \(16\) leaves are collected at random from the plant. The length, \(x\), of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, \(\bar X\), is then performed at the \(5\% \) level, with the hypotheses: \({H_0}:\mu = 5.2\) and \({H_1}:\mu < 5.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(X\) and \(Y\) be independent random variables with \(X \sim {P_o}{\text{ (3)}}\) and \(Y \sim {P_o}{\text{ (2)}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(S = 2X + 3Y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the mean and variance of \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence state with a reason whether or not \(S\) follows a Poisson distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(T = X + Y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \({\text{P}}(T = 3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that \({\text{P}}(T = t) = \sum\limits_{r = 0}^t {{\text{P}}(X = r){\text{P}}(Y = t - r)} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Hence show that \(T\) follows a Poisson distribution with mean 5.</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability of a Type II error if the leaves are in fact from a plant of species B.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{E}}(S) = 2{\text{E}}(X) + 3{\text{E}}(Y) = 6 + 6 = 12\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(S) = 4{\text{Var}}(X) + 9{\text{Var}}(Y) = 12 + 18 = 30\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(S\) does not have a Poisson distribution <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because \({\text{Var}}(S) \ne {\text{E}}(S)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Follow through their \({\text{E}}(S)\) and \({\text{Var}}(S)\) if different.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T = 3) = {\text{P}}\left( {(X,{\text{ }}Y) = (3,{\text{ }}0)} \right) + {\text{P}}\left( {(X,{\text{ }}Y) = (2,{\text{ }}1)} \right) + \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( + {\text{P}}\left( {(X,{\text{ }}Y) = (1,{\text{ }}2)} \right) + {\text{P}}\left( {(X,{\text{ }}Y) = (0,{\text{ }}3)} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{P}}(X = 3){\text{P}}(Y = 0) + {\text{P}}(X = 2){\text{P}}(Y = 1) + \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( + {\text{P}}(X = 1){\text{P}}(Y = 2) + {\text{P}}(X = 0){\text{P}}(Y = 3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{125{e^{ - 5}}}}{6}{\text{ }}( = 0.140)\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept answers which round to 0.14.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(T\) is \({{\text{P}}_o}(2 + 3) = {{\text{P}}_o}(5)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T = 3) = \frac{{125{e^{ - 5}}}}{6}{\text{ }}( = 0.140)\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept answers which round to 0.14.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \({\text{P}}(T = t) = {\text{P}}\left( {(X,{\text{ }}Y) = (0,{\text{ }}t)} \right) + {\text{P}}\left( {(X,{\text{ }}Y) = (1,{\text{ }}t - 1)} \right) + \ldots {\text{P}}\left( {(X,{\text{ }}Y) = (t,{\text{ }}0)} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{P}}(X = 0){\text{P}}(Y = t) + {\text{P}}(X = 1){\text{P}}(Y = t - 1) + \ldots + {\text{P}}(X = t){\text{P}}(Y = 0)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sum\limits_{r = 0}^t {{\text{P}}(X = r){\text{P}}(Y = t - r)} \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) \({\text{P}}(T = t) = \sum\limits_{r = 0}^t {{\text{P}}(X = r){\text{P}}(Y = t - r)} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sum\limits_{r = 0}^t {\frac{{{e^{ - 3}}{3^r}}}{{r!}} \times \frac{{{e^{ - 2}}{2^{t - r}}}}{{(t - r)!}}} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{e^{ - 5}}}}{{t!}}\sum\limits_{r = 0}^t {\frac{{t!}}{{r!(t - r)!}} \times {3^r}{2^{t - r}}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{e^{ - 5}}}}{{t!}}{(3 + 2)^t}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( { = \frac{{{e^{ - 5}}{5^t}}}{{t!}}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(T\) follows a Poisson distribution with mean 5 <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">type II error probability \( = {\text{P}}(\bar X > 4.70654 \ldots |\bar X{\text{ is }}N\left( {4.6,{\text{ }}\frac{{{{1.2}^2}}}{{16}}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = 0.361\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well answered by most candidates. The most common error in (a) was to calculate \(E(2X + 3Y)\) correctly as 12 and then state that, because the sum is Poisson, the variance is also 12. Many of these candidates then stated in (b) that the sum is Poisson because the mean and variance are equal, without apparently realising the circularity of their argument. Although (c) was intended as a possible hint for solving (d) and (e), many candidates simply noted that \(X + Y\) is \({{\text{P}}_o}{\text{(5)}}\) which led immediately to the correct answer. Some candidates tended to merge (d) and (e), often unsuccessfully, while very few candidates completed (e) correctly where the need to insert \(t!\) in the numerator and denominator was not usually spotted.</span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Alan and Brian are athletes specializing in the long jump. When Alan jumps, the length of his jump is a normally distributed random variable with mean 5.2 metres and standard deviation 0.1 metres. When Brian jumps, the length of his jump is a normally distributed random variable with mean 5.1 metres and standard deviation 0.12 metres. For both athletes, the length of a jump is independent of the lengths of all other jumps. During a training session, Alan makes four jumps and Brian makes three jumps. Calculate the probability that the mean length of Alan’s four jumps is less than the mean length of Brian’s three jumps.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Colin joins the squad and the coach wants to know the mean length, \(\mu \) metres, of his jumps. Colin makes six jumps resulting in the following lengths in metres.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">5.21, 5.30, 5.22, 5.19, 5.28, 5.18</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate an unbiased estimate of both the mean \(\mu \) and the variance of the lengths of his jumps.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Assuming that the lengths of these jumps are independent and normally distributed, calculate a 90 % confidence interval for \(\mu \) .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(\bar A,{\text{ }}\bar B\) denote the means of Alan’s and Brian’s jumps</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to find the distributions of \(\bar A,{\text{ }}\bar B\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar A{\text{ is N}}\left( {5.2,\frac{{{{0.1}^2}}}{4}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar B{\text{ is N}}\left( {5.1,\frac{{{{0.12}^2}}}{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to find the distribution of \(\bar A - \bar B\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar A - \bar B{\text{ is N}}\left( {5.2 - 5.1,\frac{{{{0.1}^2}}}{4} + \frac{{{{0.12}^2}}}{3}} \right)\) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> \({\text{N}}(0.1,{\text{ }}0.0073)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(\bar A < \bar B) = {\text{P}}(\bar A - \bar B < 0)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.121\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\sum {x = 31.38,{\text{ }}\sum {{x^2} = 164.1294} } \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = \frac{{31.38}}{6} = 5.23\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s_{n - 1}^2 = \frac{{164.1294}}{5} - \frac{{{{31.38}^2}}}{{5 \times 6}} = 0.00240\) <strong><em>(M1)(A1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({s_{n - 1}} = 0.04899 \Rightarrow s_{n - 1}^2 = 0.00240\) <strong><em>(M1)(A1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept the exact answer 0.0024 without an arithmetic penalty.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) using the <em>t</em>-distribution with DF = 5 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">critical value of <em>t</em> = 2.015 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">90 % confidence limits are \(5.23 \pm 2.015\sqrt {\frac{{0.0024}}{6}} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">giving [5.19, 5.27] <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In (a), it was disappointing to note that many candidates failed to realise that the question was concerned with the mean lengths of the jumps and worked instead with the sums of the lengths.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates obtained correct estimates in (b)(i), usually directly from the GDC. In (b)(ii), however, some candidates found a <em>z</em>-interval instead of a <em>t</em>-interval.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When Ben shoots an arrow, he hits the target with probability 0.4. Successive shots are independent.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) he hits the target exactly 4 times in his first 8 shots;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) he hits the target for the \({4^{{\text{th}}}}\) time with his \({8^{{\text{th}}}}\) shot.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ben hits the target for the \({10^{{\text{th}}}}\) time with his \({X^{{\text{th}}}}\) shot.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the expected value of the random variable <em>X</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down an expression for \({\text{P}}(X = x)\) and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{P}}(X = x)}}{{{\text{P}}(X = x - 1)}} = \frac{{3(x - 1)}}{{5(x - 10)}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence, or otherwise, find the most likely value of <em>X</em>.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the number of hits, \(X \sim {\text{B(8, 0.4)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 4) = \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 4 <br>\end{array}} \right) \times {0.4^4} \times {0.6^4}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.232 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept any answer that rounds to 0.23.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) let the \({4^{{\text{th}}}}\) hit occur on the \({Y^{{\text{th}}}}\) shot so that \(Y \sim {\text{NB(4, 0.4)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(Y = 8) = \left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 3 <br>\end{array}} \right) \times {0.4^4} \times {0.6^4}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.116 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept any answer that rounds to 0.12.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(X \sim {\text{NB(10, 0.4)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = \frac{{10}}{{0.4}} = 25\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) let \({{\text{P}}_x}\) denote \({\text{P}}(X = x)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({P_x} = \left( {\begin{array}{*{20}{c}}<br> {x - 1} \\ <br> 9 <br>\end{array}} \right) \times {0.4^{10}} \times {0.6^{x - 10}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{P_x}}}{{{P_{x - 1}}}} = \frac{{\left( {\begin{array}{*{20}{c}}<br> {x - 1} \\ <br> 9 <br>\end{array}} \right) \times {{0.4}^{10}} \times {{0.6}^{x - 10}}}}{{\left( {\begin{array}{*{20}{c}}<br> {x - 2} \\ <br> 9 <br>\end{array}} \right) \times {{0.4}^{10}} \times {{0.6}^{x - 11}}}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{(x - 1)!}}{{9!(x - 10)!}} \times \frac{{9!(x - 11)! \times 0.6}}{{(x - 2)!}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for correct evaluation of combinatorial terms.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{3(x - 1)}}{{5(x - 10)}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({{\text{P}}_x} > {{\text{P}}_{x - 1}}\) as long as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3x - 3 > 5x - 50\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> \(x < 23.5\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the most likely value is 23 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow solutions based on creating a table of values of \({{\text{P}}_x}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was well answered in general although some candidates were unable to distinguish between the binomial and negative binomial distributions.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b)(ii), most candidates knew what to do but algebraic errors were not uncommon. Candidates often used equal instead of inequality signs and this was accepted if it led to \(x = 23.5\). The difficulty for these candidates was whether to choose \(23\) or \(24\) for the final answer and some made the wrong choice. Some candidates failed to see the relevance of the result in (b)(ii) to finding the most likely value of \(X\) and chose an ‘otherwise’ method, usually by creating a table of probabilities and selecting the largest.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The mean weight of a certain breed of bird is believed to be 2.5 kg. In order to test this belief, it is planned to determine the weights \({x_1}{\text{ , }}{x_2}{\text{ , }}{x_3}{\text{ , }} \ldots {\text{, }}{x_{16}}\) (in kg) of sixteen of these birds and then to calculate the sample mean \({\bar x}\) . You may assume that these weights are a random sample from a normal distribution with standard deviation 0.1 kg.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State suitable hypotheses for a two-tailed test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the critical region for \({\bar x}\) having a significance level of 5 %.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that the mean weight of birds of this breed is actually 2.6 kg, find the probability of making a Type II error.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({H_0}:\mu = 2.5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({H_1}:\mu \ne 2.5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the critical values are \(2.5 \pm 1.96 \times \frac{{0.1}}{{\sqrt {16} }}\) , <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> 2.45, 2.55 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the critical region is \(\bar x < 2.45 \cup \bar x > 2.55\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \( \leqslant ,{\text{ }} \geqslant \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \({\bar X}\) is now \({\text{N}}(2.6,{\text{ }}{0.025^2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a Type II error is accepting \({H_0}\) when \({H_1}\) is true <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">thus we require</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(2.45 < \bar X < 2.55)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0228\,\,\,\,\,\)(Accept 0.0227) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> If critical values of 2.451 and 2.549 are used, accept 0.0207.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In (a), some candidates incorrectly gave the hypotheses in terms of \({\bar x}\) instead of \(\mu \). In (b), many candidates found the correct critical values but then some gave the critical region as \(2.45 < \bar x < 2.55\) instead of \(\bar x < 2.45 \cup \bar x > 2.55\) Many candidates gave the critical values correct to four significant figures and therefore were given an arithmetic penalty. In (c), many candidates correctly defined a Type II error but were unable to calculate the corresponding probability.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The apple trees in a large orchard have, for several years, suffered from a disease for which the outward sign is a red discolouration on some leaves.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The fruit grower knows that the mean number of discoloured leaves per tree is 42.3. The fruit grower suspects that the disease is caused by an infection from a nearby group of cedar trees. He cuts down the cedar trees and, the following year, counts the number of discoloured leaves on a random sample of seven apple trees. The results are given in the table below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) From these data calculate an unbiased estimate of the population variance.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Stating null and alternative hypotheses, carry out an appropriate test at the 10 % level to justify the cutting down of the cedar trees.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(n = 7,{\text{ sample mean }} = 35\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s_{n - 1}^2 = \frac{{\sum {{{(x - 35)}^2}} }}{6} = 322\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) null hypothesis \({{\text{H}}_0}:\mu = 42.3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">alternative hypothesis \({{\text{H}}_1}:\mu < 42.3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">using one-sided <em>t</em>-test</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {{t_{{\text{calc}}}}} \right| = \sqrt 7 \frac{{42.3 - 35}}{{\sqrt {322} }} = 1.076\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">with 6 degrees of freedom , \({t_{{\text{crit}}}} = 1.440 > 1.076\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{(or }}p{\text{-value }} = 0.162 > 0.1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we conclude that there is no justification for cutting down the cedar trees <strong><em>R1</em></strong> <strong><em>N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> <strong><em>FT</em></strong> on their <em>t</em> or <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [9 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well attempted as an example of the <em>t</em>-test. Very few used the <em>Z</em> statistic, and many found <em>p</em>-values.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The discrete random variable <em>X</em> has the following probability distribution, where \(0 < \theta < \frac{1}{3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine \({\text{E}}(X)\) and show that \({\text{Var}}(X) = 6\theta - 16{\theta ^2}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In order to estimate \(\theta \), a random sample of <em>n</em> observations is obtained from the distribution of <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \({\bar X}\) denotes the mean of this sample, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\hat \theta }_1} = \frac{{3 - \bar X}}{4}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is an unbiased estimator for \(\theta \) and write down an expression for the variance of \({{\hat \theta }_1}\) in terms of <em>n</em> and \(\theta \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let <em>Y</em> denote the number of observations that are equal to 1 in the sample. Show that <em>Y</em> has the binomial distribution \({\text{B}}(n,{\text{ }}\theta )\) and deduce that \({{\hat \theta }_2} = \frac{Y}{n}\) is another unbiased estimator for \(\theta \). Obtain an expression for the variance of \({{\hat \theta }_2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that \({\text{Var}}({{\hat \theta }_1}) < {\text{Var}}({{\hat \theta }_2})\) and state, with a reason, which is the more </span><span style="font-family: 'times new roman', times; font-size: medium;">efficient estimator, \({{\hat \theta }_1}\) or \({{\hat \theta }_2}\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 1 \times \theta + 2 \times 2\theta + 3(1 - 3\theta ) = 3 - 4\theta \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = 1 \times \theta + 4 \times 2\theta + 9(1 - 3\theta ) - {(3 - 4\theta )^2}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 6\theta - 16{\theta ^2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{E}}({\hat \theta _1}) = \frac{{3 - {\text{E}}(\bar X)}}{4} = \frac{{3 - (3 - 4\theta )}}{4} = \theta \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \({\hat \theta _1}\) is an unbiased estimator of \(\theta \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}({{\hat \theta }_1}) = \frac{{6\theta - 16{\theta ^2}}}{{16n}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) each of the <em>n</em> observed values has a probability \(\theta \) of having the value 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(Y \sim {\text{B}}(n,{\text{ }}\theta )\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}({{\hat \theta }_2}) = \frac{{{\text{E}}(Y)}}{n} = \frac{{n\theta }}{n} = \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}({{\hat \theta }_2}) = \frac{{n\theta (1 - \theta )}}{{{n^2}}} = \frac{{\theta (1 - \theta )}}{n}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{Var}}({{\hat \theta }_1}) - {\text{Var}}({{\hat \theta }_2}) = \frac{{6\theta - 16{\theta ^2} - 16\theta + 16{\theta ^2}}}{{16n}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{ - 10\theta }}{{16n}} < 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\hat \theta }_1}\) is the more efficient estimator since it has the smaller variance <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[10 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Consider the random variable \(X\) for which \({\text{E}}(X) = a\lambda + b\), where \(a\) and \(b\)are constants and \(\lambda \) is a parameter.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{X - b}}{a}\) is an unbiased estimator for \(\lambda \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The continuous random variable <em>Y </em>has probability density function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(y) = \left\{ \begin{array}{r}{\textstyle{2 \over 9}}(3 + y - \lambda ),\\0,\end{array} \right.\begin{array}{*{20}{l}}{{\rm{ for}}\, \lambda - 3 \le y \le \lambda }\\{{\rm{ otherwise}}}\end{array}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(\lambda \) is a parameter.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Verify that \(f(y)\) is a probability density function for all values of \(\lambda \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine \({\text{E}}(Y)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Write down an unbiased estimator for \(\lambda \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{E}}\left( {\frac{{X - b}}{a}} \right) = \frac{{a\lambda + b - b}}{a}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \lambda \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(Therefore \(\frac{{X - b}}{a}\) is an unbiased estimator for \(\lambda \)) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \(f(y) \geqslant 0\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Only award <strong><em>R1 </em></strong>if this statement is made explicitly.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> recognition or showing that integral of <em>f </em>is 1 (seen anywhere) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\int_{\lambda - 3}^\lambda {\frac{2}{9}(3 + y - \lambda ){\text{d}}y} \) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{2}{9}\left[ {(3 - \lambda )y + \frac{1}{2}{y^2}} \right]_{\lambda - 3}^\lambda \) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{2}{9}\left( {\lambda (3 - \lambda ) + \frac{1}{2}{\lambda ^2} - (3 - \lambda )(\lambda - 3) - \frac{1}{2}{{(\lambda - 3)}^2}} \right)\) or equivalent <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> the graph of the probability density is a triangle with base length 3 and height \(\frac{2}{3}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"> its area is therefore \(\frac{1}{2} \times 3 \times \frac{2}{3}\)</span> <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(Y) = \int_{\lambda - 3}^\lambda {\frac{2}{9}y(3 + y - \lambda ){\text{d}}y} \) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{2}{9}\left[ {(3 - \lambda )\frac{1}{2}{y^2} + \frac{1}{3}{y^3}} \right]_{\lambda - 3}^\lambda \) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{2}{9}\left( {(3 - \lambda )\frac{1}{2}\left( {{\lambda ^2} - {{(\lambda - 3)}^2}} \right) + \frac{1}{3}\left( {{\lambda ^3} - {{(\lambda - 3)}^3}} \right)} \right)\) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \lambda - 1\) <em><strong>A1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong> </strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award 3 marks for noting that the mean is \(\frac{2}{3}{\text{rds}}\) the way along the base and then <strong><em>A1A1 </em></strong>for \(\lambda - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for \(\lambda \) and <strong><em>A1 </em></strong>for –1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) unbiased estimator: \(Y + 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept \(\bar Y + 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> Follow through their \({\text{E}}(Y)\) if linear.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Times; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[11 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> is normally distributed with unknown mean \(\mu \) and unknown variance \({\sigma ^2}\). A random sample of 20 observations on <em>X</em> gave the following results.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\sum {x = 280,{\text{ }}\sum {{x^2} = 3977.57} } \]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a 95 % confidence interval for \(\mu \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the hypotheses</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\text{H}}_0}:\mu = 15;{\text{ }}{{\text{H}}_1}:\mu \ne 15,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the <em>p</em>-value of the above results and state your conclusion at the 1 % significance level.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = 14\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s_{n - 1}^2 = \frac{{3977.57}}{{19}} - \frac{{{{280}^2}}}{{380}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 3.03\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any notation for these estimates including \(\mu \) and \({\sigma ^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M0A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for division by 20.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the 95% confidence limits are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x \pm t\sqrt {\frac{{s_{n - 1}^2}}{n}} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for use of </span><em style="font-family: 'times new roman', times; font-size: medium;">z</em><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;"> </em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;">ie</em><span style="font-family: 'times new roman', times; font-size: medium;">, \(14 \pm 2.093\sqrt {\frac{{3.03}}{{20}}} \) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>FT</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> their mean and variance from (a).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">giving [13.2, 14.8] </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answers which round to 13.2 and 14.8.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of t-statistic \(\left( { = \frac{{14 - 15}}{{\sqrt {\frac{{3.03}}{{20}}} }}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>FT</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> their mean and variance from (a).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for use of </span><em style="font-family: 'times new roman', times; font-size: medium;">z</em><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept \(\frac{{15 - 14}}{{\sqrt {\frac{{3.03}}{{20}}} }}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - 2.569 \ldots \) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept \(2.569 \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p{\text{ - value}} = 0.009392 \ldots \times 2 = 0.0188\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds to 0.019.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)(A1)A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for any answer that rounds to 0.0094.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">insufficient evidence to reject \({{\text{H}}_0}\) (or equivalent, </span><em style="font-family: 'times new roman', times; font-size: medium;">eg</em><span style="font-family: 'times new roman', times; font-size: medium;"> accept \({{\text{H}}_0}\) or reject \({{\text{H}}_1}\)) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>FT</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> on their </span><em style="font-family: 'times new roman', times; font-size: medium;">p</em><span style="font-family: 'times new roman', times; font-size: medium;">-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (a), most candidates estimated the mean correctly although many candidates failed to obtain a correct unbiased estimate for the variance. The most common error was to divide \(\sum {{x^2}} \) by \(20\) instead of \(19\). For some candidates, this was not a costly error since we followed through their variance into (b) and (c).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b) and (c), since the variance was estimated, the confidence interval and test should have been carried out using the t-distribution. It was extremely disappointing to note that many candidates found a Z-interval and used a Z-test and no marks were awarded for doing this. Candidates should be aware that having to estimate the variance is a signpost pointing towards the t-distribution.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b) and (c), since the variance was estimated, the confidence interval and test should have been carried out using the t-distribution. It was extremely disappointing to note that many candidates found a Z-interval and used a Z-test and no marks were awarded for doing this. Candidates should be aware that having to estimate the variance is a signpost pointing towards the t-distribution.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the oranges produced by a farm may be assumed to be normally distributed with mean 205 grams and standard deviation 10 grams.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen orange weighs more than 200 grams.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Five of these oranges are selected at random to be put into a bag. Find the probability that the combined weight of the five oranges is less than 1 kilogram.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The farm also produces lemons whose weights may be assumed to be normally distributed with mean 75 grams and standard deviation 3 grams. Find the probability that the weight of a randomly chosen orange is more than three times the weight of a randomly chosen lemon.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = \frac{{200 - 205}}{{10}} = - 0.5\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">probability = 0.691 (accept 0.692) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 0.309 or 0.308</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>X</em> be the total weight of the 5 oranges</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then \({\text{E}}(X) = 5 \times 205 = 1025\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = 5 \times 100 = 500\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 1000) = 0.132\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>Y</em> = <em>B</em> – 3<em>C</em> where <em>B</em> is the weight of a random orange and <em>C</em> the weight of a random lemon <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(Y) = 205 - 3 \times 75 = - 20\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(Y) = 100 + 9 \times 9 = 181\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y > 0) = 0.0686\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 0.0681 obtained from tables</span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">As might be expected, (a) was well answered by many candidates, although those who gave 0.6915 straight from tables were given an arithmetic penalty. Parts (b) and (c), however, were not so well answered with errors in calculating the variances being the most common source of incorrect solutions. In particular, some candidates are still uncertain about the difference between <em>nX</em> and \(\sum\limits_{i = 1}^n {{X_i}} \) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">As might be expected, (a) was well answered by many candidates, although those who gave 0.6915 straight from tables were given an arithmetic penalty. Parts (b) and (c), however, were not so well answered with errors in calculating the variances being the most common source of incorrect solutions. In particular, some candidates are still uncertain about the difference between <em>nX</em> and \(\sum\limits_{i = 1}^n {{X_i}} \) .</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">As might be expected, (a) was well answered by many candidates, although those who gave 0.6915 straight from tables were given an arithmetic penalty. Parts (b) and (c), however, were not so well answered with errors in calculating the variances being the most common source of incorrect solutions. In particular, some candidates are still uncertain about the difference between <em>nX</em> and \(\sum\limits_{i = 1}^n {{X_i}} \) .</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The continuous random variable <em>X</em> has probability density function <em>f</em> given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {\frac{{3{x^2} + 2x}}{{10}},}&{{\text{for }}1 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine an expression for \(F(x)\), valid for \(1 \leqslant x \leqslant 2\), where <em>F</em> denotes the cumulative distribution function of <em>X</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, determine the median of <em>X</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the central limit theorem.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) A random sample of 150 observations is taken from the distribution of <em>X</em> and \(\bar X\) denotes the sample mean. Use the central limit theorem to find, approximately, the probability that \(\bar X\) is greater than 1.6.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(F(x) = \int_1^x {\frac{{3{u^2} + 2u}}{{10}}{\text{d}}u} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left[ {\frac{{{u^3} + {u^2}}}{{10}}} \right]_1^x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not penalise missing or wrong limits at this stage.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Accept the use of <em>x</em> in the integrand.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{x^3} + {x^2} - 2}}{{10}}\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the median <em>m</em> satisfies the equation \(F(m) = \frac{1}{2}\) so <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({m^3} + {m^2} - 7 = 0\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>FT</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> from an incorrect \(F(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 1.63\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds to 1.6.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the mean of a large sample from any distribution is approximately </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">normal <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> This is the minimum acceptable explanation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) we require the mean \(\mu \) and variance \({\sigma ^2}\) of </span><em style="font-family: 'times new roman', times; font-size: medium;">X</em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = \int_1^2 {\left( {\frac{{3{x^3} + 2{x^2}}}{{10}}} \right){\text{d}}x} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{191}}{{120}}{\text{ }}(1.591666 \ldots )\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\sigma ^2} = \int_1^2 {\left( {\frac{{3{x^4} + 2{x^3}}}{{10}}} \right){\text{d}}x - {\mu ^2}} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.07659722 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the central limit theorem states that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar X \approx N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right),\) <em>i.e.</em> \(N(1.591666 \ldots ,{\text{ }}0.0005106481 \ldots )\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(\bar X > 1.6) = 0.356\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds to 0.36.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solutions to (a)(i) were disappointing in general, suggesting that many candidates are unfamiliar with the concept of the cumulative distribution function. Many candidates knew that it was something to do with the integral of the probability density function but some thought it was \(\int\limits_1^2 {f(x){\text{d}}x} \) which they then evaluated as \(1\) while others thought it was just \(\int {f(x){\text{d}}x} = \frac{{\left( {{x^2} + {x^3}} \right)}}{{10}}\) which is not, in general, a valid method. However, most candidates solved (a)(ii) correctly, usually by integrating the probability density function from \(1\) to \(m\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b)(i), the statement of the central limit theorem was often quite dreadful. The term ‘sample mean’ was often not mentioned and a common misconception appears to be that the actual distribution rather than the sample mean tends to normality as the sample size increases. Solutions to (b)(ii) often failed to go beyond finding the mean and variance of \(X\) . In calculating the variance, some candidates rounded the mean from \(1.5916666..\) to \(1.59\) which resulted in an incorrect value for the variance. It is important to note that calculating a variance usually involves a small difference of two large numbers so that full accuracy must be maintained.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) A random variable, <em>X</em> , has probability density function defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{l}}<br> {100,}&{{\text{for }} - 0.005 \leqslant x < 0.005} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine <em>E</em>(<em>X</em>) and Var(<em>X</em>) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) When a real number is rounded to two decimal places, an error is made.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this error can be modelled by the random variable <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) A list contains 20 real numbers, each of which has been given to two decimal places. The numbers are then added together.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down bounds for the resulting error in this sum.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Using the central limit theorem, estimate to two decimal places the probability that the absolute value of the error exceeds 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) State clearly any assumptions you have made in your calculation.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <em>f</em>(<em>x</em>)is even (symmetrical about the origin) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = {\text{E}}({X^2}) = \int_{ - 0.005}^{0.005} {100{x^2}{\text{d}}x} \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 8.33 \times {10^{ - 6}}\left( {{\text{accept }}0.83 \times {{10}^{ - 5}}{\text{ or }}\frac{1}{{120\,000}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) rounding errors to 2 decimal places are uniformly distributed <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and lie within the interval \( - 0.005 \leqslant x < 0.005.\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this defines <em>X</em> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) using the symbol <em>y</em> to denote the error in the sum of 20 real numbers each rounded to 2 decimal places</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 0.1 \leqslant y( = 20 \times x) < 0.1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(Y \approx {\text{N}}(20 \times 0,{\text{ }}20 \times 8.3 \times {10^{ - 6}}) = {\text{N}}(0,{\text{ }}0.00016)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {\left| Y \right| > 0.01} \right) = 2\left( {1 - {\text{P}}(Y < 0.01)} \right)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\left( {1 - {\text{P}}\left( {Z < \frac{{0.01}}{{0.0129}}} \right)} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.44\) to 2 decimal places <strong><em>A1</em></strong> <strong><em>N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) it is assumed that the errors in rounding the 20 numbers are independent <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and, by the central limit theorem, the sum of the errors can be modelled approximately by a normal distribution <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [15 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the only question on the paper with a conceptually ‘hard’ final part. Part(a) was generally well done, either by integration or by use of the standard formulae for a uniform distribution. Many candidates were not able to provide convincing reasoning in parts (b) and (c)(iii). Part(c)(ii), the application of the Central Limit Theorem was only very rarely tackled competently.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When Andrew throws a dart at a target, the probability that he hits it is \(\frac{1}{3}\) ; when Bill throws a dart at the target, the probability that he hits the it is \(\frac{1}{4}\) . Successive throws are independent. One evening, they throw darts at the target alternately, starting with Andrew, and stopping as soon as one of their darts hits the target. Let <em>X</em> denote the total number of darts thrown.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \({\text{P}}(X = 1)\) and show that \({\text{P}}(X = 2) = \frac{1}{6}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the probability generating function for <em>X</em> is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[G(t) = \frac{{2t + {t^2}}}{{6 - 3{t^2}}}.\]</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence determine \({\text{E}}(X)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 1) = \frac{1}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 2) = \frac{2}{3} \times \frac{1}{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(= \frac{1}{6}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(G(t) = \frac{1}{3}t + \frac{2}{3} \times \frac{1}{4}{t^2} + \frac{2}{3} \times \frac{3}{4} \times \frac{1}{3}{t^3} + \frac{2}{3} \times \frac{3}{4} \times \frac{2}{3} \times \frac{1}{4}{t^4} + \ldots \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{3}t\left( {1 + \frac{1}{2}{t^2} + \ldots } \right) + \frac{1}{6}{t^2}\left( {1 + \frac{1}{2}{t^2} + \ldots } \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\frac{t}{3}}}{{1 - \frac{{{t^2}}}{2}}} + \frac{{\frac{{{t^2}}}{6}}}{{1 - \frac{{{t^2}}}{2}}}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2t + {t^2}}}{{6 - 3{t^2}}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(G'(t) = \frac{{(2 + 2t)(6 - 3{t^2}) + 6t(2t + {t^2})}}{{{{(6 - 3{t^2})}^2}}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = G'(1) = \frac{{10}}{3}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>If \(X\) is a random variable that follows a Poisson distribution with mean \(\lambda > 0\) then the probability generating function of \(X\) is \(G(t) = {e^{\lambda (t - 1)}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Prove that \({\text{E}}(X) = \lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Prove that \({\text{Var}}(X) = \lambda \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(Y\) is a random variable, independent of \(X\), that also follows a Poisson distribution with mean \(\lambda \).</p>
<p class="p1">If \(S = 2X - Y\) find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({\text{E}}(S)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}(S)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(T = \frac{Y}{2} + \frac{Y}{2}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(T\) is an unbiased estimator for \(\lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(T\) is a more efficient unbiased estimator of \(\lambda \) than \(S\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Could either \(S\) or \(T\) model a Poisson distribution? Justify your answer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By consideration of the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\), prove that \(X + Y\) follows a Poisson distribution with mean \(2\lambda \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({G_{X + Y}}(1)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({G_{X + Y}}( - 1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the probability that \(X + Y\) is an even number.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) \(G'(t) = \lambda {e^{\lambda (t - 1)}}\) <strong><em>A1</em></strong></p>
<p>\({\text{E}}(X) = G'(1)\) <strong><em>M1</em></strong></p>
<p>\( = \lambda \) <strong><em>AG</em></strong></p>
<p>(ii) \(G''(t) = {\lambda ^2}{e^{\lambda (t - 1)}}\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow G''(1) = {\lambda ^2}\) <strong><em>(A1)</em></strong></p>
<p>\({\text{Var}}(X) = G''(1) + G'(1) - {\left( {G'(1)} \right)^2}\) <strong><em>(M1)</em></strong></p>
<p>\( = {\lambda ^2} + \lambda - {\lambda ^2}\) <strong><em>A1</em></strong></p>
<p>\( = \lambda \) <strong><em>AG</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \({\text{E}}(S) = 2\lambda - \lambda = \lambda \) <strong><em>A1</em></strong></p>
<p>(ii) \({\text{Var}}(S) = 4\lambda + \lambda = 5\lambda \) <strong><em>(A1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>First <strong><em>A1 </em></strong>can be awarded for either \(4\lambda \) or \(\lambda \).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \({\text{E}}(T) = \frac{\lambda }{2} + \frac{\lambda }{2} = \lambda \;\;\;\)(so <em>\(T\) </em>is an unbiased estimator) <strong><em>A1</em></strong></p>
<p>(ii) \({\text{Var}}(T) = \frac{1}{4}\lambda + \frac{1}{4}\lambda = \frac{1}{2}\lambda \) <strong><em>A1</em></strong></p>
<p>this is less than \({\text{Var}}(S)\)<em>, </em>therefore \(T\) is the more efficient estimator <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through their variances from (b)(ii) and (c)(ii).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">no, mean does not equal the variance <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({G_{X + Y}}(t) = {e^{\lambda (t - 1)}} \times {e^{\lambda (t - 1)}} = {e^{2\lambda (t - 1)}}\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">which is the probability generating function for a Poisson with a mean of \(2\lambda \) <span class="Apple-converted-space"> </span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({G_{X + Y}}(1) = 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({G_{X + Y}}( - 1) = {e^{ - 4\lambda }}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({G_{X + Y}}(1) = p(0) + p(1) + p(2) + p(3) \ldots \)</p>
<p class="p1">\({G_{X + Y}}( - 1) = p(0) - p(1) + p(2) - p(3) \ldots \)</p>
<p class="p1">so \({\text{2P(even)}} = {G_{X + Y}}(1) + {G_{X + Y}}( - 1)\) <span class="Apple-converted-space"> </span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1">\({\text{P(even)}} = \frac{1}{2}(1 + {e^{ - 4\lambda }})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [21 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to the different parts of this question proved to be extremely variable in quality with some parts well answered by the majority of the candidates and other parts accessible to only a few candidates. Part (a) was well answered in general although the presentation was sometimes poor with some candidates doing the differentiation of \(G(t)\) and the substitution of \(t = 1\) simultaneously.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was well answered in general, the most common error being to state that \({\text{Var}}(2X - Y) = {\text{Var}}(2X) - {\text{Var}}(Y)\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were well answered by the majority of candidates.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were well answered by the majority of candidates.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (e), however, were extremely disappointing with few candidates giving correct solutions. A common incorrect solution was the following:</p>
<p class="p1">\(\;\;\;{G_{X + Y}}(t) = {G_X}(t){G_Y}(t)\)</p>
<p class="p1">Differentiating,</p>
<p class="p1">\(\;\;\;{G'_{X + Y}}(t) = {G'_X}(t){G_Y}(t) + {G_X}(t){G'_Y}(t)\)</p>
<p class="p1">\(\;\;\;{\text{E}}(X + Y) = {G'_{X + Y}}(1) = {\text{E}}(X) \times 1 + {\text{E}}(Y) \times 1 = 2\lambda \)</p>
<p class="p1">This is correct mathematics but it does not show that \(X + Y\) is Poisson and it was given no credit. Even the majority of candidates who showed that \({G_{X + Y}}(t) = {{\text{e}}^{2\lambda (t - 1)}}\) failed to state that this result proved that \(X + Y\) is Poisson and they usually differentiated this function to show that \({\text{E}}(X + Y) = 2\lambda \).</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (f), most candidates stated that \({G_{X + Y}}(1) = 1\) even if they were unable to determine \({G_{X + Y}}(t)\) but many candidates were unable to evaluate \({G_{X + Y}}( - 1)\). Very few correct solutions were seen to (g) even if the candidates correctly evaluated \({G_{X + Y}}(1)\) and \({G_{X + Y}}( - 1)\).</p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Jenny tosses seven coins simultaneously and counts the number of tails obtained. She repeats the experiment 750 times. The following frequency table shows her results.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.31.35.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Explain what can be done with this data to decrease the probability of making a type I error.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the meaning of a type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down how to proceed if it is required to decrease the probability of making both a type I and type II error.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">reduce the significance level (or equivalent statement) <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) accepting \({{\text{H}}_0}\) (or failing to reject \({{\text{H}}_0}\)) when it is false (or equivalent) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) increase the number of trials <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">It was disappointing to see that some candidates wrote incorrect hypotheses, eg ‘\({{\text{H}}_0}\): Data are binomial; \({{\text{H}}_1}\): Data are not binomial’ without specifying any parameters. Part (b) caused unexpected problems for many candidates who misunderstood the question and gave ‘increase the number of trials’ as their answer.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> has a binomial distribution with parameters \(n\) and \(p\).</p>
</div>
<div class="specification">
<p>Let \(U = nP\left( {1 - P} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(P = \frac{X}{n}\) is an unbiased estimator of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down an unbiased estimator of Var(<em>X</em>).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}\left( P \right) = {\text{E}}\left( {\frac{X}{n}} \right) = \frac{1}{n}{\text{E}}\left( X \right)\) <em><strong>M1</strong></em></p>
<p>\( = \frac{1}{n}\left( {np} \right) = p\) <em><strong>A1</strong></em></p>
<p>so <em>P</em> is an unbiased estimator of \(p\) <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}\left( {nP\left( {1 - P} \right)} \right) = {\text{E}}\left( {n\left( {\frac{X}{n}} \right)\left( {1 - \frac{X}{n}} \right)} \right)\)</p>
<p>\( = {\text{E}}\left( X \right) = \frac{1}{n}{\text{E}}\left( {{X^2}} \right)\) <em><strong>M1A1</strong></em></p>
<p>use of \({\text{E}}\left( {{X^2}} \right) = {\text{Var}}\left( X \right) + {\left( {{\text{E}}\left( X \right)} \right)^2}\) <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Allow candidates to work with <em>P</em> rather than <em>X</em> for the above 3 marks.</p>
<p>\( = np - \frac{1}{n}\left( {np\left( {1 - p} \right) + {{\left( {np} \right)}^2}} \right)\) <em><strong>A1</strong></em></p>
<p>\( = np - p\left( {1 - p} \right) - n{p^2}\)</p>
<p>\( = np\left( {1 - p} \right) - p\left( {1 - p} \right)\) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the factor of \(\left( {1 - p} \right)\).</p>
<p>\( = \left( {n - 1} \right)p\left( {1 - p} \right)\) <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an unbiased estimator is \(\frac{{{n^2}P\left( {1 - P} \right)}}{{n - 1}}\left( { = \frac{{nU}}{{n - 1}}} \right)\) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Two students are selected at random from a large school with equal numbers of boys and girls. The boys’ heights are normally distributed with mean \(178\) cm and standard deviation \(5.2\) cm, and the girls’ heights are normally distributed with mean \(169\) cm and standard deviation \(5.4\) cm<span class="s1">.</span></p>
<p class="p2">Calculate the probability that the taller of the two students selected is a boy.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">let \(X\) denote boys’ height and \(Y\) denote girls’ height</p>
<p class="p1">if \(BB,{\text{ P(taller is boy)}} = 1\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">if \(GG,{\text{ P(taller is boy)}} = 0\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">if \(BG\) or \(GB\):</p>
<p class="p1">consider \(X - Y\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(E(X - Y) = 178 - 169 = 9\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{Var}}(X - Y) = {5.2^2} + {5.4^2}\;\;\;( = 56.2)\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p1">\({\text{P}}(X - Y > 0) = 0.885\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s1">answer is </span>\(\frac{1}{4} \times 1 + \frac{1}{2} \times 0.885 = 0.693\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[9 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A hospital specializes in treating overweight patients. These patients have weights that are independently, normally distributed with mean 200 kg and standard deviation 15 kg. The elevator in the hospital will break if the total weight of people inside it exceeds 1150 kg. Six patients enter the elevator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the elevator breaks.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(W = \sum\limits_{i = 1}^6 {{w_i}} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({w_i}{\text{ is N}}(200,{\text{ 1}}{{\text{5}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(W) = \sum\limits_{i = 1}^6 {{\text{E}}({w_i}) = 6 \times 200 = 1200} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(W) = \sum\limits_{i = 1}^6 {{\text{Var}}({w_i}) = 6 \times {{15}^2} = 1350} \) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(W{\text{ is N}}(1200,{\text{ 1350}})\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(W > 1150) = 0.913\) by GDC <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Using 6 times the mean or a lower bound for the mean are acceptable methods.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates will often be asked to solve these problems that test if they can distinguish between a number of individuals and a number of copies. The wording of the question was designed to make the difference clear. If candidates wrote \({w_1} + \ldots + {w_6}\) in (a) and 12<em>w</em> in (b), they usually went on to gain full marks.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Eleven students who had under-performed in a philosophy practice examination were given extra tuition before their final examination. The differences between their final examination marks and their practice examination marks were</p>
<p class="p1">\[10,{\text{ }} - 1,{\text{ }}6,{\text{ }}7,{\text{ }} - 5,{\text{ }} - 5,{\text{ }}2,{\text{ }} - 3,{\text{ }}8,{\text{ }}9,{\text{ }} - 2.\]</p>
<p class="p1">Assume that these differences form a random sample from a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine unbiased estimates of \(\mu \) and \({\sigma ^2}\)<span class="s1">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State suitable hypotheses to test the claim that extra tuition improves examination marks.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate the \(p\)-value of the sample.</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Determine whether or not the above claim is supported at the \(5\% \) significance level.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">unbiased estimate of \(\mu \) is \(2.36(36 \ldots )\;\;\;(26/11)\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p1">unbiased estimate of \({\sigma ^2}\) is \(33.65(45 \ldots ) = ({5.801^2})\;\;\;(1851/55)\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept any answer that rounds correctly to \(3\) significant figures.</p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>M1A0</em></strong> for any unbiased estimate of \({\sigma ^2}\) that rounds to \(5.80\).</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \({{\text{H}}_0}:\mu = 0;{\text{ }}{{\text{H}}_1}:\mu > 0\) <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0</em></strong> if an inappropriate symbol is used for the mean, <em>eg</em>, \(r\), \({\rm{\bar d}}\).</p>
<p>(ii) attempt to use <em>t</em>-test <strong><em>(M1)</em></strong></p>
<p>\(t = 1.35\) <strong><em>(A1)</em></strong></p>
<p>\({\text{DF}} = 10\) <strong><em>(A1)</em></strong></p>
<p>\(p\)-value \( = 0.103\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to \(3\) significant figures.</p>
<p>(iii) \(0.103 > 0.05\) <strong><em>A1</em></strong></p>
<p>there is insufficient evidence at the \(5\% \) level to support the claim (that extra tuition improves examination marks)</p>
<p><strong>OR</strong></p>
<p>the claim (that extra tuition improves examination marks) is not supported at the \(5\% \) level (or equivalent statement) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through the candidate’s \(p\)-value.</p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>R1</em></strong> for Accept \({{\text{H}}_0}\) or Reject \({{\text{H}}_1}\).</p>
<p><em><strong>[8 marks]</strong></em></p>
<p><em><strong>Total [12 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Almost every candidate gave the correct estimate of the mean but some chose the wrong variance from their calculators to estimate \({\sigma ^2}\). In (b)(i), the hypotheses were sometimes incorrectly written, usually with an incorrect symbol instead of \(\mu \), for example \(d\), \(\bar x\) and ‘mean’ were seen. Many candidates failed to make efficient use of their calculators in (b)(ii). The intention of the question was that candidates should simply input the data into their calculators and use the software to give the <em>p</em>-value. Instead, many candidates found the <em>p</em>-value by first evaluating \(t\) using the appropriate formula. This was a time consuming process and it gave opportunity for error. In (b)(iii), candidates were expected to refer to the claim so that the answers ‘Accept \({H_0}\)’ or ‘Reject \({H_1}\)’ were not accepted.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Almost every candidate gave the correct estimate of the mean but some chose the wrong variance from their calculators to estimate \({\sigma ^2}\). In (b)(i), the hypotheses were sometimes incorrectly written, usually with an incorrect symbol instead of \(\mu \), for example \(d\), \(\bar x\) and ‘mean’ were seen. Many candidates failed to make efficient use of their calculators in (b)(ii). The intention of the question was that candidates should simply input the data into their calculators and use the software to give the \(p\)-value. Instead, many candidates found the \(p\)-value by first evaluating \(t\) using the appropriate formula. This was a time consuming process and it gave opportunity for error. In (b)(iii), candidates were expected to refer to the claim so that the answers ‘Accept \({H_0}\)’ or ‘Reject \({H_1}\)’ were not accepted.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two species of plant, \(A\) and \(B\), are identical in appearance though it is known that the mean length of leaves from a plant of species \(A\) is \(5.2\) cm, whereas the mean length of leaves from a plant of species \(B\) is \(4.6\) cm. Both lengths can be modelled by normal distributions with standard deviation \(1.2\) cm.</p>
<p>In order to test whether a particular plant is from species \(A\) or species \(B\), \(16\) leaves are collected at random from the plant. The length, \(x\), of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, \(\bar X\), is then performed at the \(5\% \) level, with the hypotheses: \({H_0}:\mu = 5.2\) and \({H_1}:\mu < 5.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the critical region for this test.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">It is now known that in the area in which the plant was found \(90\% \) of all the plants are of species \(A\) and \(10\% \) are of species \(B\).</p>
<p class="p1">Find the probability that \(\bar X\) will fall within the critical region of the test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If, having done the test, the sample mean is found to lie within the critical region, find the probability that the leaves came from a plant of species \(A\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\bar X \sim N\left( {5.2,{\text{ }}\frac{{{{1.2}^2}}}{{16}}} \right)\) (<strong><em>M1)</em></strong></p>
<p>critical value is \(5.2 - 1.64485 \ldots \times \frac{{1.2}}{4} = 4.70654 \ldots \) <strong><em>(A1)</em></strong></p>
<p>critical region is \(] - \infty ,{\text{ }}4.71]\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow follow through for the final <strong><em>A1 </em></strong>from their critical value.</p>
<p> </p>
<p><strong>Note: </strong>Follow through previous values in (b), (c) and (d).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(0.9 \times 0.05 + 0.1 \times (1 - 0.361 \ldots ) = 0.108875997 \ldots = 0.109\) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for a weighted average of probabilities with weights \(0.1,0.9\).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use conditional probability formula <strong><em>M1</em></strong></p>
<p>\(\frac{{0.9 \times 0.05}}{{0.108875997 \ldots }}\) <strong><em>(A1)</em></strong></p>
<p>\( = 0.41334 \ldots = 0.413\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [10 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to this question were generally disappointing.</p>
<p class="p1">In (a), the standard error of the mean was often taken to be \(\sigma (1.2)\) instead of \(\frac{\sigma }{{\sqrt n }}(0.3)\) and the solution sometimes ended with the critical value without the critical region being given.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), the question was often misunderstood with candidates finding the weighted mean of the two means, ie \(0.9 \times 5.2 + 0.1 \times 4.6 = 5.14\) instead of the weighted mean of two probabilities.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Without having the solution to (c), part (d) was inaccessible to most of the candidates so that very few correct solutions were seen.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A factory makes wine glasses. The manager claims that on average 2 % of the glasses are imperfect. A random sample of 200 glasses is taken and 8 of these are found to be imperfect.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Test the manager’s claim at a 1 % level of significance using a one-tailed test.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>X </em>denote the number of imperfect glasses in the sample <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For recognising binomial or proportion or Poisson <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(\(X \sim {\text{B}}(200,{\text{ }}p)\) where <em>p</em>-value is the probability of a glass being imperfect)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({{\text{H}}_0}:p{\text{-value}} = 0.02{\text{ and }}{{\text{H}}_1}:p{\text{-value}} > 0.02\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0493 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the binomial distribution \(p{\text{-value}} = 0.0493 > 0.01{\text{ we accept }}{{\text{H}}_0}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0511 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the Poisson approximation to the binomial distribution since \(p{\text{-value}} = 0.0511 > 0.01{\text{ we accept }}{{\text{H}}_0}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0217 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the one proportion <em>z</em>-test since \(p{\text{-value}} = 0.0217 > 0.01{\text{ we accept }}{{\text{H}}_0}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Use of critical values is acceptable.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates used a <em>t</em>-test on this question. This was possibly because the sample was large enough to approximate normality of a proportion. The need to use a one-tailed test was often missed. When using the <em>z</em>-test of proportions <em>p</em> = 0.04 was often used instead of <em>p</em> = 0.02 . Not many candidates used the binomial distribution.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ten friends try a diet which is claimed to reduce weight. They each weigh themselves before starting the diet, and after a month on the diet, with the following results.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img 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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates of the mean and variance of the loss in weight achieved over the month by people using this diet.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State suitable hypotheses for testing whether or not this diet causes a mean loss in weight.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the value of a suitable statistic for testing your hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the 1 % critical value for your statistic and state your conclusion.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the weight losses are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">2.2\(\,\,\,\,\,\)3.5\(\,\,\,\,\,\)4.3\(\,\,\,\,\,\)–0.5\(\,\,\,\,\,\)4.2\(\,\,\,\,\,\)–0.2\(\,\,\,\,\,\)2.5\(\,\,\,\,\,\)2.7\(\,\,\,\,\,\)0.1\(\,\,\,\,\,\)–0.7 <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sum {x = 18.1} \), \(\sum {{x^2} = 67.55} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">UE of mean = 1.81 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">UE of variance \( = \frac{{67.55}}{9} - \frac{{{{18.1}^2}}}{{90}} = 3.87\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept weight losses as positive or negative. Accept unbiased estimate of mean as positive or negative.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 1.97 as UE of variance.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({H_0}:{\mu _d} = 0\) versus \({H_1}:{\mu _d} > 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept any symbol for \({\mu _d}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) using </span><em style="font-family: 'times new roman', times; font-size: medium;">t</em><span style="font-family: 'times new roman', times; font-size: medium;"> test </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{1.81}}{{\sqrt {\frac{{3.87}}{{10}}} }} = 2.91\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) DF = 9 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award this <strong><em>(A1)</em></strong> if the p-value is given as 0.00864</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">1% critical value = 2.82 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">accept \({H_1}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow <strong><em>FT</em></strong> on final <strong><em>R1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In (a), most candidates gave a correct estimate for the mean but the variance estimate was often incorrect. Some candidates who use their GDC seem to be unable to obtain the unbiased variance estimate from the numbers on the screen. The way to proceed, of course, is to realise that the larger of the two ‘standard deviations’ on offer is the square root of the unbiased estimate so that its square gives the required result. In (b), most candidates realised that the t-distribution should be used although many were awarded an arithmetic penalty for giving either <em>t</em> = 2.911 or the critical value = 2.821. Some candidates who used the <em>p</em>-value method to reach a conclusion lost a mark by omitting to give the critical value. Many candidates found part (c) difficult and although they were able to obtain <em>t</em> = 2.49…, they were then unable to continue to obtain the confidence interval.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In (a), most candidates gave a correct estimate for the mean but the variance estimate was often incorrect. Some candidates who use their GDC seem to be unable to obtain the unbiased variance estimate from the numbers on the screen. The way to proceed, of course, is to realise that the larger of the two ‘standard deviations’ on offer is the square root of the unbiased estimate so that its square gives the required result. In (b), most candidates realised that the t-distribution should be used although many were awarded an arithmetic penalty for giving either <em>t</em> = 2.911 or the critical value = 2.821. Some candidates who used the <em>p</em>-value method to reach a conclusion lost a mark by omitting to give the critical value. Many candidates found part (c) difficult and although they were able to obtain <em>t</em> = 2.49…, they were then unable to continue to obtain the confidence interval.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The owner of a factory is asked to produce bricks of weight <span class="s1">2.2 kg</span>. The quality control manager wishes to test whether or not, on a particular day, the mean weight of bricks being produced is <span class="s1">2.2 kg</span>.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">He therefore collects a random sample of 20 </span>of these bricks and determines the weight, \(x\) <span class="s1">kg</span>, of each brick. He produces the following summary statistics.</p>
<p class="p1">\[\sum {x = 42.0,{\text{ }}\sum {{x^2} = 89.2} } \]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State hypotheses to enable the quality control manager to test the mean weight using a two-tailed test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate unbiased estimates of the mean and the variance of the weights of the bricks being produced.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Assuming that the weights of the bricks are normally distributed, determine the \(p\)<span class="s1">-value of the above results and state the conclusion in context using a 5% </span>significance level.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The owner is more familiar with using confidence intervals. Determine a <span class="s1">95% </span>confidence interval for the mean weight of bricks produced on that particular day.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({H_0}:{\text{ }}\mu = 2.2;{\text{ }}{H_1}:{\text{ }}\mu \ne 2.2\) </span><strong><em>A1A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>UE of mean \( = \frac{{42.0}}{{20}}{\text{ = }}2.1\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">UE of variance \( = \frac{{89.2}}{{19}} - \frac{{20 \times {{2.1}^2}}}{{19}} = 0.0526{\text{ }}\left( {\frac{1}{{19}}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M0) </em></strong></span>for division by 20 where there is no subsequent use of \(\frac{{20}}{{19}}\).</p>
<p class="p1">(ii)</p>
<p class="p4"><span class="Apple-converted-space">\(t = - 1.95\) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{DF}} = 19\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(p - value = 0.0662\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Allow follow through from (b)(i). In particular, 0.05 <span class="s1">for the variance gives \(t = - 2\) and \(p\)</span>-value 0.0600<span class="s1">.</span></p>
<p class="p4">accept \({H_0}\), or equivalent statement involving \({H_0}\) or \({H_1}\)<span class="s2">, indicating that the mean weight is 2.2kg <span class="Apple-converted-space"> </span></span><strong><em>R1</em></strong></p>
<p class="p4"><strong>Note: <span class="Apple-converted-space"> </span></strong>Follow through the candidate’s \(p\)-value.</p>
<p class="p4"><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\([1.99,{\text{ }}2.21]\) </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Allow follow through from (b)(i). In particular, 0.05 for the variance gives \([2.00,{\text{ }}2.20]\).</p>
<p class="p3"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates stated the correct hypotheses in (a).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (b)(i), the mean was invariably found correctly, although to find the variance estimate, quite a few candidates divided by 20 instead of 19. Incorrect variances were followed through in the next part of (b)(i). The \(t\)-test was generally well applied and the correct conclusion drawn. It was, however, surprising to note that many candidate used the appropriate formula to find the value of \(t\) and hence the \(p\)-value as opposed to using their GDC software.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) was generally well answered.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The continuous random variable \(X\) <span class="s1">takes values in the interval \([0,{\text{ }}\theta ]\) and</span></p>
<p class="p2" style="text-align: center;">\({\text{E}}(X) = \frac{\theta }{2}\) and \({\text{Var}}(X) = \frac{{{\theta ^2}}}{{24}}\).</p>
<p class="p1">To estimate the unknown parameter \(\theta \), a random sample of size \(n\) is obtained from the distribution of \(X\). The sample mean is denoted by \(\overline X \) and \(U = k\overline X\) is an unbiased estimator for \(\theta \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate an unbiased estimate for \(\theta \)<span class="s1">, using the random sample,</span></p>
<p class="p2">8.3, 4.2, 6.5, 10.3, 2.7, 1.2, 3.3, 4.3<span class="s2">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Explain briefly why this is not a good estimate for \(\theta \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \({\text{Var}}(U) = \frac{{{\theta ^2}}}{{6n}}\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Show that \({U^2}\) is not an unbiased estimator for \({\theta ^2}\).</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>Find an unbiased estimator for \({\theta ^2}\) in terms of \(U\) and \(n\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{E}}(U) = k{\text{E}}(\overline X ) = k{\text{E}}(X)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{k\theta }}{2}\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2">unbiased when \(k = 2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>for the data, \(\Sigma x = 40.8\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow \bar x = 5.1\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2">so that unbiased estimate for \(\theta = 10.2\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>this is impossible because of the sample value 10.3 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \({\text{Var}}(U) = 4 \times {\text{Var}}(\bar X)\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = 4 \times \frac{{{\theta ^2}}}{{24n}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{\theta ^2}}}{{6n}}\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p2">(ii) \({\text{E}}({U^2}) = {\text{Var}}(U) + {\left( {{\text{E}}(U)} \right)^2}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{\theta ^2}}}{{6n}} + {\theta ^2}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\({\text{E}}({U^2}) \ne {\theta ^2}\) </span><span class="s1"><strong><em>R1</em></strong></span></p>
<p class="p2">so not unbiased <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p2">(iii) <span class="Apple-converted-space"> \({\text{E}}({U^2}) = \frac{{{\theta ^2}}}{{6n}}(1 + 6n)\)</span> <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}\left( {\left( {\frac{{6n}}{{1 + 6n}}} \right){U^2}} \right) = {\theta ^2}\) </span><strong><em>(A1)</em></strong></p>
<p class="p2">therefore \(\left( {\left( {\frac{{6n}}{{1 + 6n}}} \right){U^2}} \right)\) is an unbiased estimator for \({\theta ^2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (a) were often disappointing with some candidates seeming to be confused by the notation used.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (b)(i), many candidates evaluated the sample mean as 5.1 but some failed to convert this to the estimate 10.2 even if they had correctly found the value of \(k\).</p>
<p class="p1">In (b)(ii), very few candidates realised that \(\theta = 10.2\) was not a feasible estimate when one of the sample values was 10.3.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (c) were generally poor.</p>
<p class="p1">In (c)(i), many good answers were seen although some candidates failed to take account of the difference between \({\text{Var}}(X)\) and \({\text{Var}}(\bar X)\).</p>
<p class="p1">In (c)(ii), many candidates thought that \({\text{E}}({\bar X^2}) = {\left[ {{\text{E}}(\bar X)} \right]^2}\) although this had the unfortunate consequence of showing that \({U^2}\) is an unbiased estimator for \({\theta ^2}\). Few candidates realized that an expression for \({\text{E}}({U^2})\) could be found by considering the standard result that \({\text{Var}}(U) = {\text{E}}({U^2}) - {\left[ {{\text{E}}(U)} \right]^2}\) or the equivalent expression for \({\text{Var}}(\bar X)\). Part (c)(iii) was inaccessible to candidates who were unable to solve (ii).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) After a chemical spillage at sea, a scientist measures the amount, <em>x</em> units, of the chemical in the water at 15 randomly chosen sites. The results are summarised in the form \(\sum {x = 18} \) and \(\sum {{x^2} = 28.94} \). Before the spillage occurred the mean level of the chemical in the water was 1.1. Test at the 5 % significance level the hypothesis that there has been an increase in the amount of the chemical in the water.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Six months later the scientist returns and finds that the mean amount of the chemical in the water at the 15 randomly chosen sites is 1.18. Assuming that this sample came from a normal population with variance 0.0256, find a 90 % confidence interval for the mean level of the chemical.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\bar x = \frac{{\sum x }}{n} = 1.2\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s_{n - 1}^2 = 0.524 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">it is a one tailed test</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}:\mu = 1.1,{\text{ }}{{\text{H}}_1}:\mu > 1.1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{1.2 - 1.1}}{{\sqrt {\frac{{0.524 \ldots }}{{15}}} }} = 0.535\) <strong><em>(M1) A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 14\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({t_{crit}} = 1.761\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(0.535 < {t_{crit}}\) we accept \({{\text{H}}_0}\) that there is no increase in the amount of the chemical <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p = 0.301\) <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(p > 0.05\) we accept \({{\text{H}}_0}\) that there is no increase in the amount of the chemical <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) 90 % confidence interval \( = 1.18 \pm 1.645\sqrt {\frac{{0.0256}}{{15}}} \) <strong><em>(M1)A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = [1.11,{\text{ }}1.25]\) <strong><em>A1</em></strong> <strong><em>N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question also proved accessible to a majority of candidates with many wholly correct or nearly wholly correct answers seen. A few candidates did not recognise that part (a) was a <em>t</em>-distribution and part (b) was a Normal distribution, but most recognised the difference. Many candidates received an accuracy penalty on this question for not giving the final answer to part (b) to 3 significant figures.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A shopper buys 12 apples from a market stall and weighs them with the following results (in grams).</span></p>
<p style="font: normal normal normal 27px/normal Helvetica; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">117, 124, 129, 118, 124, 116, 121, 126, 118, 121, 122, 129</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">You may assume that this is a random sample from a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates of \(\mu \) and \({\sigma ^2}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a 99 % confidence interval for \(\mu \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The stallholder claims that the mean weight of apples is 125 grams but the shopper claims that the mean is less than this.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State suitable hypotheses for testing these claims.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the <em>p</em>-value of the above sample.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Giving a reason, state which claim is supported by your <em>p</em>-value using a 5 % significance level.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">unbiased estimate of \(\mu = 122\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">unbiased estimate of \({\sigma ^2} = 4.4406{ \ldots ^2} = 19.7\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 4.44.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the 99 % confidence interval for \(\mu \) is [118, 126] <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({{\text{H}}_0}:\mu = 125;{\text{ }}{{\text{H}}_1}:\mu < 125\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <em>p</em>-value = 0.0220 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) the shopper’s claim is supported because \(0.0220 < 0.05\) <strong><em>A1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Engine oil is sold in cans of two capacities, large and small. The amount, in millilitres, in each can, is normally distributed according to Large \( \sim {\text{N}}(5000,{\text{ }}40)\) and Small \( \sim {\text{N}}(1000,{\text{ }}25)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can is selected at random. Find the probability that the can contains at least \(4995\) millilitres of oil.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and a small can are selected at random. Find the probability that the large can contains at least \(30\) milliliters more than five times the amount contained in the small can.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and five small cans are selected at random. Find the probability that the large can contains at least \(30\) milliliters less than the total amount contained in the small cans.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(L \ge 4995) = 0.785\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept any answer that rounds correctly to \(0.79\).</p>
<p class="p3">Award <strong><em>M1A0</em></strong> for \(0.78\).</p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>M1A0</em></strong> for any answer that rounds to \(0.55\) obtained by taking \({\text{SD}} = 40\).</p>
<p class="p3"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">we are given that \(L \sim {\text{N}}(5000,{\text{ }}40)\) and \(S \sim {\text{N}}(1000,{\text{ }}25)\)</p>
<p class="p1">consider \(X = L - 5S\) <span class="s1">(</span>ignore \( \pm 30\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{E}}(X) = 0\) (\( \pm 30\) consistent with line above<span class="s1">) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{Var}}(X) = {\text{Var}}(L) + 25{\text{Var}}(S) = 40 + 625 = 665\) <span class="Apple-converted-space"> </span><strong><em>(M1)A1</em></strong></p>
<p class="p1">require \({\text{P}}(X \ge 30)\;\;\;({\text{or P}}(X \ge 0){\text{ if }} - 30{\text{ above}})\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">obtain \(0.122\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept any answer that rounds correctly to \(2\) significant figures.</p>
<p class="p1"><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider \(Y = L - ({S_1} + {S_2} + {S_3} + {S_4} + {S_5})\) (ignore \( \pm 30\)) <strong><em>(M1)</em></strong></p>
<p>\({\text{E}}(Y) = 0\) (\( \pm 30\) consistent with line above) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}(Y) = 40 + 5 \times 25 = 165\) <strong><em>A1</em></strong></p>
<p>require \({\text{P}}(Y \le - 30){\text{ (or P}}(Y \le 0){\text{ if }} + 30{\text{ above)}}\) <strong><em>(M1)</em></strong></p>
<p>obtain \(0.00976\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to \(2\) significant figures.</p>
<p> </p>
<p><strong>Note:</strong> Condone the notation \(Y = L - 5S\) if the variance is correct.</p>
<p style="text-align: left;"><em><strong>[5 marks]</strong></em></p>
<p style="text-align: left;"><em><strong>Total [13 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates solved (a) correctly. In (b) and (c), however, many candidates made the usual error of confusing \(\sum\limits_{i = 1}^n {{X_i}} \) and \(nX\)<em>. </em>Indeed some candidates even use the second expression to mean the first. This error leads to an incorrect variance and of course an incorrect answer. Some candidates had difficulty in converting the verbal statements into the correct probability statements, particularly in (c).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates solved (a) correctly. In (b) and (c), however, many candidates made the usual error of confusing \(\sum\limits_{i = 1}^n {{X_i}} \) and \(nX\)<em>. </em>Indeed some candidates even use the second expression to mean the first. This error leads to an incorrect variance and of course an incorrect answer. Some candidates had difficulty in converting the verbal statements into the correct probability statements, particularly in (c).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates solved (a) correctly. In (b) and (c), however, many candidates made the usual error of confusing \(\sum\limits_{i = 1}^n {{X_i}} \) and \(nX\)<em>. </em>Indeed some candidates even use the second expression to mean the first. This error leads to an incorrect variance and of course an incorrect answer. Some candidates had difficulty in converting the verbal statements into the correct probability statements, particularly in (c).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of machine breakdowns occurring in a day in a certain factory may be assumed to follow a Poisson distribution with mean \(\mu \). The value of \(\mu \) is known, from past experience, to be 1.2. In an attempt to reduce the value of \(\mu \), all the machines are fitted with new control units. To investigate whether or not this reduces the value of \(\mu \), the total number of breakdowns, <em>x</em>, occurring during a 30-day period following the installation of these new units is recorded.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">State suitable hypotheses for this investigation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is decided to define the critical region by \(x \leqslant 25\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the significance level.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Assuming that the value of \(\mu \) was actually reduced to 0.75, determine the probability of a Type II error.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}:\mu = 1.2\); \({{\text{H}}_1}:\mu < 1.2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept “ \({{\text{H}}_0}:\) (\(30\)-day) mean \( = 36\); \({{\text{H}}_1}:\) (\(30\)-day) mean \( = 36\) ”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) let <em>X</em> denote the number of breakdowns in 30 days</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">then under \({{\text{H}}_0}\) , \(E(X) = 36\) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{sig level}} = {\text{P}}(X \leqslant 25|{\text{mean}} = 36)\) <em><strong>(M1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">= 0.0345 (3.45%) <em><strong>A1</strong></em></span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept any answer that rounds to 0.035 (3.5%) .</span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Do not accept the use of a normal approximation.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">under \({{\text{H}}_1}\), \(E(X) = 22.5\)</span> <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(P{\text{(Type II error)}} = P(X \geqslant 26|{\text{mean}} = 22.5)\) <em><strong>(M1)(A1)</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">= 0.257 <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept any answer that rounds to 0.26.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Do not accept the use of a normal approximation.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well answered by many candidates. The most common error was to attempt to use a normal approximation to find approximate probabilities instead of the Poisson distribution to find the exact probabilities. Some candidates appeared not to be familiar with the term ‘Type II error probability’ which made (b)(ii) inaccessible. Another fairly common error was to believe that the complement of \(x \leqslant 25\) is \(x \geqslant 25\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well answered by many candidates. The most common error was to attempt to use a normal approximation to find approximate probabilities instead of the Poisson distribution to find the exact probabilities. Some candidates appeared not to be familiar with the term ‘Type II error probability’ which made (b)(ii) inaccessible. Another fairly common error was to believe that the complement of \(x \leqslant 25\) is \(x \geqslant 25\).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the probability generating function for \(X \sim {\text{B}}(1,{\text{ }}p)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the probability generating function for \({\text{B}}(n,{\text{ }}p)\) is a polynomial of degree \(n\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Two independent random variables \({X_1}\) and \({X_2}\) are such that \({X_1} \sim {\text{B}}(1,{\text{ }}{p_1})\) <span class="s1">and \({X_2} \sim {\text{B}}(1,{\text{ }}{p_2})\)</span>. Prove that if \({X_1} + {X_2}\) has a binomial distribution then \({p_1} = {p_2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(X = 0) = 1 - p( = q);{\text{ P}}(X = 1) = p\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)(A1)</em></strong></span></p>
<p class="p1">\({{\text{G}}_x}(t) = \sum\limits_r {{\text{P}}(X = r){t^r}\;\;\;} \)(or writing out term by term) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = q + pt\) <strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">\(PGF\) for \(B(n,{\text{ }}p)\) is \({(q + pt)^n}\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">which is a polynomial of degree \(n\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">in \(n\) independent trials, it is not possible to obtain more than \(n\) succes<span class="s1">ses (or equivalent, <em>eg</em>, \({\text{P}}(X > n) = 0\)) <span class="Apple-converted-space"> </span></span><strong><em>R1</em></strong></p>
<p class="p1">so \({a_r} = 0\) for \(r > n\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">let \(Y = {X_1} + {X_2}\)</p>
<p class="p1">\({G_Y}(t) = ({q_1} + {p_1}t)({q_2} + {p_2}t)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({G_Y}(t)\) has degree two, so if \(Y\) is binomial then</p>
<p class="p1">\(Y \sim {\text{B}}(2,{\text{ }}p)\) for some \(p\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">\({(q + pt)^2} = ({q_1} + {p_1}t)({q_2} + {p_2}t)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>The \(LHS\) could be seen as \({q^2} + 2pqt + {p^2}{t^2}\).</p>
<p class="p2"> </p>
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">by considering the roots of both sides, \(\frac{{{q_1}}}{{{p_1}}} = \frac{{{q_2}}}{{{p_2}}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(\frac{{1 - {p_1}}}{{{p_1}}} = \frac{{1 - {p_2}}}{{{p_2}}}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \({p_1} = {p_2}\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">equating coefficients,</p>
<p class="p1">\({p_1}{p_2} = {p^2},{\text{ }}{q_1}{q_2} = {q^2}{\text{ or }}(1 - {p_1})(1 - {p_2}) = {(1 - p)^2}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">expanding,</p>
<p class="p1">\({p_1} + {p_2} = 2p\) so \({p_1},{\text{ }}{p_2}\) are the roots of \({x^2} - 2px + {p^2} = 0\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \({p_1} = {p_2}\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<p class="p1"><strong><em>Total [11 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (a) were often disappointing with some candidates simply writing down the answer. A common error was to forget the possibility of \(X\) being zero so that \(G(t) = pt\) was often seen.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Explanations in (b) were often poor, again indicating a lack of ability to give a verbal explanation.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Very few complete solutions to (c) were seen with few candidates even reaching the result that \(({q_1} + {p_1}t)({q_2} + {p_2}t)\) must equal \({(q + pt)^2}\) for some \(p\).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ahmed and Brian live in the same house. Ahmed always walks to school and Brian always cycles to school. The times taken to travel to school may be assumed to be independent and normally distributed. The mean and the standard deviation for these times are shown in the table below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that on a particular day Ahmed takes more than 35 minutes to walk to school.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Brian cycles to school on five successive mornings. Find the probability that the total time taken is less than 70 minutes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the probability that, on a particular day, the time taken by Ahmed to walk to school is more than twice the time taken by Brian to cycle to school.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(A \sim {\text{N}}(30,{\text{ }}{3^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A > 35) = 0.0478\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let \(X = {B_1} + {B_2} + {B_3} + {B_4} + {B_5}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 5{\text{E}}(B) = 60\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = 5{\text{Var}}(B) = 20\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 70) = 0.987\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) let \(Y = A - 2B\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(Y) = {\text{E}}(A) - 2{\text{E}}(B) = 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(Y) = {\text{Var}}(A) + 4{\text{Var}}(B) = 25\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y > 0) = 0.885\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [11 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to access this question, but weaker candidates did not always realise that parts (b) and (c) were testing different things. Part (b) proved the hardest with a number of candidates not understanding how to find the variance of the sum of variables.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A manufacturer of stopwatches employs a large number of people to time the winner of a \(100\) </span>metre sprint. It is believed that if the true time of the winner is \(\mu \) seconds, the times recorded are normally distributed with mean \(\mu \) <span class="s1">seconds and standard deviation \(0.03\) seconds.</span></p>
<p class="p2">The times, in seconds, recorded by six randomly chosen people are</p>
<p class="p2">\[9.765,{\text{ }}9.811,{\text{ }}9.783,{\text{ }}9.797,{\text{ }}9.804,{\text{ }}9.798.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate a \(99\% \) <span class="s1">confidence interval for \(\mu \)</span>. Give your answer correct to three decimal places.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Interpret the result found in (a).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the confidence level of the interval that corresponds to halving the width of the \(99\% \) confidence interval. Give your answer as a percentage to the nearest whole number.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">the (unbiased) estimate of \(\mu \) is 9.793 <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">the \(99\% \) CI is \(9.793 \pm 2.576\frac{{0.03}}{{\sqrt 6 }}\) <span class="Apple-converted-space"> </span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1">\( = [9.761,{\text{ }}9.825]\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept \(9.762\) and \(9.824\).</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">if this process is carried out a large number of times <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(approximately) \(99\% \) of the intervals will contain \(\mu \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Award <strong><em>A1A1</em></strong> for a consideration of any specific large value of times \((n \ge 100)\).</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">If the interval is halved, \(2.576\) becomes \(1.288\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">normal tail probability corresponding to \(1.288 = 0.0988 \ldots \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">confidence level \( = 80\% \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">half width \( = 0.5 \times 0.063\) or \(0.062\) or \(0.064 = 0.0315\) or \(0.031\) or \(0.032\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(\frac{{2z \times 0.03}}{{\sqrt 6 }} = 0.0315\) or \(0.031\) or \(0.032\)</p>
<p class="p1">giving \(z = 1.285 \ldots \) or \(1.265 \ldots \) or \(1.306 \ldots \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">confidence level \( = 80\% \) <span class="s1">or \(79\% \) or \(81\% \) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1">Note: <span class="Apple-converted-space"> </span>Follow through values from (a).</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<p class="p1"><em><strong>Total [9 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The intention in (a) was that candidates should input the data into their calculators and use the software to give the confidence interval. However, as in Question 2, many candidates calculated the mean and variance by hand and used the appropriate formulae to determine the confidence limits. Again valuable time was used up and opportunity for error introduced.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Answers to (b) were extremely disappointing with the vast majority giving an incorrect interpretation of a confidence interval. The most common answer given was along the lines of ‘There is a 99% probability that the interval [9.761, 9.825] contains \(\mu \)’<em>. </em>This is incorrect since the interval and \(\mu \) are both constants; the statement that the interval [9.761, 9.825] contains \(\mu \) is either true or false, there is no question of probability being involved. Another common response was ‘I am 99% confident that the interval [9.761, 9.825] contains \(\mu \)’. This is unsatisfactory partly because 99% confident is really a euphemism for 99% probability and partly because it answers the question ‘What is a 99% confidence interval for \(\mu \)<em>’ </em>by simply rearranging the words without actually going anywhere. The expected answer was that if the sampling was carried out a large number of times, then approximately 99% of the calculated confidence intervals would contain \(\mu \)<em>. </em>A more rigorous response would be that a 99% confidence interval for \(\mu \) is an observed value of a random interval which contains \(\mu \) with probability 0.99 just as the number \(\bar x\) is an observed value of the random variable \(\bar X\). The concept of a confidence interval is a difficult one at this level but confidence intervals are part of the programme and so therefore is their interpretation. In view of the widespread misunderstanding of confidence intervals, partial credit was given on this occasion for interpretations involving 99% probability or confidence but this will not be the case in future examinations.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates solved (c) correctly, mostly using Method 2 in the mark scheme.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A random variable \(X\) </span>has a population mean \(\mu \)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain briefly the meaning of</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>an estimator of \(\mu \);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>an unbiased estimator of \(\mu \)<span class="s1">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A random sample \({X_1},{\text{ }}{X_2},{\text{ }}{X_3}\) of three independent observations is taken from the distribution of \(X\).</p>
<p>An unbiased estimator of \(\mu ,{\text{ }}\mu \ne 0\), is given by \(U = \alpha {X_1} + \beta {X_2} + (\alpha - \beta ){X_3}\),</p>
<p>where \(\alpha ,{\text{ }}\beta \in \mathbb{R}\).</p>
<p>(i) Find the value of \(\alpha \).</p>
<p>(ii) Show that \({\text{Var}}(U) = {\sigma ^2}\left( {2{\beta ^2} - \beta + \frac{1}{2}} \right)\) where \({\sigma ^2} = {\text{Var}}(X)\).</p>
<p>(iii) Find the value of \(\beta \) which gives the most efficient estimator of \(\mu \) of this form.</p>
<p>(iv) Write down an expression for this estimator and determine its variance.</p>
<p>(v) Write down a more efficient estimator of \(\mu \) than the one found in (iv), justifying your answer.</p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>an estimator \(T\) is a formula (or statistic) that can be applied to the values in any sample, taken from \(X\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">to estimate the value of \(\mu \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>an estimator is unbiased if \({\text{E}}(T) = \mu \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) using linearity and the definition of an unbiased estimator <strong><em>M1</em></strong></p>
<p>\(\mu = \alpha \mu + \beta \mu + (\alpha - \beta )\mu \) <strong><em>A1</em></strong></p>
<p>obtain \(\alpha = \frac{1}{2}\) <strong><em>A1</em></strong></p>
<p>(ii) attempt to compute \({\text{Var}}(U)\) using correct formula <strong><em>M1</em></strong></p>
<p>\({\text{Var}}(U) = \frac{1}{4}{\sigma ^2} + {\beta ^2}{\sigma ^2} + {\left( {\frac{1}{2} - \beta } \right)^2}{\sigma ^2}\) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}(U) = {\sigma ^2}\left( {2{\beta ^2} - \beta + \frac{1}{2}} \right)\) <strong><em>AG</em></strong></p>
<p>(iii) attempt to minimise quadratic in \(\beta \) (or equivalent) <strong><em>(M1)</em></strong></p>
<p>\(\beta = \frac{1}{4}\) <strong><em>A1</em></strong></p>
<p>(iv) \((U) = \frac{1}{2}{X_1} + \frac{1}{4}{X_2} + \frac{1}{4}{X_3}\) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}(U) = \frac{3}{8}{\sigma ^2}\) <strong><em>A1</em></strong></p>
<p>(v) \(\frac{1}{3}{X_1} + \frac{1}{3}{X_2} + \frac{1}{3}{X_3}\) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}\left( {\frac{1}{3}{X_1} + \frac{1}{3}{X_2} + \frac{1}{3}{X_3}} \right) = \frac{3}{9}{\sigma ^2}\) <strong><em>A1</em></strong></p>
<p>\( < {\text{Var}}(U)\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(\sum\limits_{i = 1}^3 {{\lambda _i}{X_i}} \) if \(\sum\limits_{i = 1}^3 {{\lambda _i} = 1} \) and \(\sum\limits_{i = 1}^3 {\lambda _i^2 < \frac{3}{8}} \) and follow through to the variance if this is the case.</p>
<p><em><strong>[12 marks]</strong></em></p>
<p><em><strong>Total [15 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In general, solutions to (a) were extremely disappointing with the vast majority unable to give correct explanations of estimators and unbiased estimators. Solutions to (b) were reasonably good in general, indicating perhaps that the poor explanations in (a) were due to an inability to explain what they know rather than a lack of understanding.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (b) were reasonably good in general, indicating perhaps that the poor explanations in (a) were due to an inability to explain what they know rather than a lack of understanding.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A shop sells apples, pears and peaches. The weights, in grams, of these three types of fruit may be assumed to be normally distributed with means and standard deviations as given in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Alan buys 1 apple and 1 pear while Brian buys 1 peach. Calculate the probability that the combined weight of Alan’s apple and pear is greater than twice the weight of Brian’s peach.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>X</em>, <em>Y</em>, <em>Z</em> denote respectively the weights, in grams, of a randomly chosen apple, pear, peach</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then \(U = X + Y - 2Z{\text{ is N}}(115 + 110 - 2 \times 105,{\text{ }}{5^2} + {4^2} + {2^2} \times {3^2})\) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1</em></strong> for attempted use of <em>U</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> N(15, 77) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we require</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X + Y > 2Z) = {\text{P}}(U > 0)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.956\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M0A0A2</em></strong> for 0.956 only.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to this question again illustrated the fact that many candidates are unable to distinguish between <em>nX</em> and \(\sum\limits_{i = 1}^n {{X_i}} \) so that many candidates obtained an incorrect variance to evaluate the final probability.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A traffic radar records the speed, \(v\) kilometres per hour (\({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\)), of cars on a section of a road.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table shows a summary of the results for a random sample of 1000 cars whose speeds were recorded on a given day.</span></p>
<p style="font: normal normal normal 20.5px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.17.39.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Using the data in the table,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) show that an estimate of the mean speed of the sample is 113.21 \({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find an estimate of the variance of the speed of the cars on this section of the road.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the 95% confidence interval, \(I\), for the mean speed.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \(J\) be the 90% confidence interval for the mean speed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Without calculating \(J\), explain why \(J \subset I\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\bar v = \frac{1}{{1000}}(55 \times 5 + 65 \times 13 + \ldots + 145 \times 31)\) <strong><em>A1M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>A1</em> </strong>for mid-points, <strong><em>M1 </em></strong>for use of the formula.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{113\,210}}{{1000}} = 113.21\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({s^2} = \frac{{{{(55 - 113.21)}^2} \times 5 + {{(65 - 113.21)}^2} \times 13 + \ldots + {{(145 - 113.21)}^2} \times 31}}{{999}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{362\,295.9}}{{999}} = 362.6585 \ldots = 363\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>if answer rounds to 362 or 363.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Condone division by 1000.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar v \pm \frac{{{t_{0.025}} \times s}}{{\sqrt n }}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence the confidence interval \(I = [112.028,{\text{ }}114.392]\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept answers which round to 112 and 114.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Condone the use of \({z_{0.025}}\) for \({t_{0.025}}\) and \(\sigma \) for \(s\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">less confidence implies narrower interval <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept equivalent statements or arguments having a meaningful diagram and/or relevant percentiles.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence the confidence interval \(I\) at the 95% level contains the confidence interval \(J\) at the 90% level <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In (a)(i), the candidates were required to show that the estimate of the mean is 113.21 so that those who stated simply ‘Using my GDC, mean = 113.21’ were given no credit. Candidates were expected to indicate that the interval midpoints were used and to show the appropriate formula. In (a)(ii), division by either 999 or 1000 was accepted, partly because of the large sample size and partly because the question did not ask for an unbiased estimate of variance.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to (c) were often badly written, often quite difficult to understand exactly what was being stated.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">As soon as Sarah misses a total of 4 lessons at her school an email is sent to her parents. The probability that she misses any particular lesson is constant with a value of \(\frac{1}{3}\). Her decision to attend a lesson is independent of her previous decisions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that an email is sent to Sarah’s parents after the \({8^{{\text{th}}}}\) lesson that Sarah was scheduled to attend.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If an email is sent to Sarah’s parents after the \({X^{{\text{th}}}}\) lesson that she was scheduled to attend, find \({\text{E}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) If after 6 of Sarah’s scheduled lessons we are told that she has missed exactly 2 lessons, find the probability that an email is sent to her parents after a total of 12 scheduled lessons.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) If we know that an email was sent to Sarah’s parents immediately after her \({6^{{\text{th}}}}\) scheduled lesson, find the probability that Sarah missed her \({2^{{\text{nd}}}}\) scheduled lesson.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) we are dealing with the Negative Binomial distribution: \({\text{NB}}\left( {4,\frac{1}{3}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>X</em> be the number of scheduled lessons before the email is sent</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 8) = \left( {\begin{array}{*{20}{c}}<br> 7 \\ <br> 3 <br>\end{array}} \right){\left( {\frac{2}{3}} \right)^4}{\left( {\frac{1}{3}} \right)^4} = 0.0854\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{E}}(X) = \frac{r}{p} = \frac{4}{{\frac{1}{3}}} = 12\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) we are asking for 2 missed lessons in the second 6 lessons, with the last lesson missed so this is \({\text{NB}}\left( {2,\frac{1}{3}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 6) = \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 1 <br>\end{array}} \right){\left( {\frac{2}{3}} \right)^4}{\left( {\frac{1}{3}} \right)^2} = 0.110\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept solutions laid out in terms of conditional probabilities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">We know that she missed the \({6^{{\text{th}}}}\) lesson so she must have missed 3 from the first 5 lessons. All are equally likely so the probability that she missed the \({2^{{\text{nd}}}}\) lesson is \(\frac{3}{5}\). <strong><em>R1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">require \({\text{P(missed }}{{\text{2}}^{{\text{nd}}}}|X = 6) = \frac{{{\text{P(missed }}{{\text{2}}^{{\text{nd}}}}{\text{ and }}X = 6)}}{{{\text{P}}(X = 6)}}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(missed }}{{\text{2}}^{{\text{nd}}}}{\text{ and }}X = 6) = {\text{P(missed }}{{\text{2}}^{{\text{nd}}}}{\text{ and }}{{\text{6}}^{{\text{th}}}}{\text{ and 2 of remaining 4)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[ = \frac{1}{3} \cdot \frac{1}{3} \cdot \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 2 <br>\end{array}} \right){\left( {\frac{1}{3}} \right)^2}{\left( {\frac{2}{3}} \right)^2} = \frac{{24}}{{{3^6}}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 6) = \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 3 <br>\end{array}} \right){\left( {\frac{1}{3}} \right)^4}{\left( {\frac{2}{3}} \right)^2} = \frac{{40}}{{{3^6}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so required probability is \(\frac{{24}}{{{3^6}}} \cdot \frac{{{3^6}}}{{40}} = \frac{3}{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Realising that this was a problem about the Negative Binomial distribution was the crucial thing to realise in this question. All parts of the syllabus do need to be covered.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has a Poisson distribution with mean \(\mu \). The value of \(\mu \) is known to be either 1 or 2 so the following hypotheses are set up.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\text{H}}_0}:\mu = 1;{\text{ }}{{\text{H}}_1}:\mu = 2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample \({x_1},{\text{ }}{x_2},{\text{ }} \ldots ,{\text{ }}{x_{10}}\) of 10 observations is taken from the distribution of <em>X</em> and the following critical region is defined.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\sum\limits_{i = 1}^{10} {{x_i} \geqslant 15} \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the probability of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) a Type I error;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) a Type II error.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) let \(T = \sum\limits_{i = 1}^{10} {{X_i}} \) so that <em>T</em> is Po(10) under \({{\text{H}}_0}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(Type I error)}} = {\text{P }}T \geqslant 15|\mu = 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0835\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Candidates who write the first line and only the correct answer award <strong><em>(M1)M0A0A2</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let \(T = \sum\limits_{i = 1}^{10} {{X_i}} \) so that <em>T</em> is Po(20) under \({{\text{H}}_1}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(Type II error)}} = {\text{P }}T \leqslant 14|\mu = 2\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.105\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Candidates who write the first line and only the correct answer award <strong><em>(M1)M0A0A2</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award 5 marks to a candidate who confuses Type I and Type II errors and has both answers correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question caused problems for many candidates and the solutions were often disappointing. Some candidates seemed to be unaware of the meaning of Type I and Type II errors. Others were unable to calculate the probabilities even when they knew what they represented. Candidates who used a normal approximation to obtain the probabilities were not given full credit – there seems little point in using an approximation when the exact value could be found.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In a game there are <em>n</em> players, where \(n > 2\) . Each player has a disc, one side of which is red and one side blue. When thrown, the disc is equally likely to show red or blue. All players throw their discs simultaneously. A player wins if his disc shows a different colour from all the other discs. Players throw repeatedly until one player wins.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>X</em> be the number of throws each player makes, up to and including the one on which the game is won.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State the distribution of <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find \({\text{P}}(X = x)\) in terms of <em>n</em> and <em>x</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \({\text{E}}(X)\) in terms of <em>n</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Given that <em>n</em> = 7 , find the least number, <em>k</em> , such that \({\text{P}}(X \leqslant k) > 0.5\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) geometric distribution <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let <em>R</em> be the event throwing the disc and it landing on red and </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let <em>B</em> be the event throwing the disc and it landing on blue </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 1) = p = {\text{P}}\left( {1B{\text{ and }}(n - 1)R{\text{ or }}1R{\text{ and }}(n - 1)B} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = n \times \frac{1}{2} \times {\left( {\frac{1}{2}} \right)^{n - 1}} + n \times \frac{1}{2} \times {\left( {\frac{1}{2}} \right)^{n - 1}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{n}{{{2^{n - 1}}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence \({\text{P}}(X = x) = \frac{n}{{{2^{n - 1}}}}{\left( {1 - \frac{n}{{{2^{n - 1}}}}} \right)^{x - 1}},{\text{ }}(x \geqslant 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> \(x \geqslant 1\) not required for final <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Allow <strong><em>FT</em></strong> for final <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \({\text{E}}(X) = \frac{1}{p}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{2^{n - 1}}}}{n}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) when \(n = 7\) , \({\text{P}}(X = x) = {\left( {1 - \frac{7}{{64}}} \right)^{x - 1}} \times \frac{7}{{64}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{7}{{64}} \times {\left( {\frac{{57}}{{64}}} \right)^{x - 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant k) = \sum\limits_{x = 1}^k {\frac{7}{{64}} \times {{\left( {\frac{{57}}{{64}}} \right)}^{x - 1}}} \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \frac{7}{{64}} \times \frac{{1 - {{\left( {\frac{{57}}{{64}}} \right)}^k}}}{{1 - \frac{{57}}{{64}}}} > 0.5\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 1 - {\left( {\frac{{57}}{{64}}} \right)^k} > 0.5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {\left( {\frac{{57}}{{64}}} \right)^k} < 0.5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k > \frac{{\log 0.5}}{{\log \frac{{57}}{{64}}}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k > 5.98\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k = 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Tabular and other GDC methods are acceptable.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was found difficult by the majority of candidates and few fully correct answers were seen. Few candidates were able to find \({\text{P}}(X = x)\) in terms of n and x and many did not realise that the last part of the question required them to find the sum of a series. However, better candidates received over 75% of the marks because the answers could be followed through.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The length of time, <em>T</em>, in months, that a football manager stays in his job before he is removed can be approximately modelled by a normal distribution with population mean \(\mu \) and population variance \({\sigma ^2}\). An independent sample of five values of <em>T</em> is given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">6.5, 12.4, 18.2, 3.7, 5.4</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that \({\sigma ^2} = 9\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find the smallest number of values of <em>T</em> that would be required in a sample for the total width of the 90 % confidence interval for \(\mu \) to be less than 2 months.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If the value of \({\sigma ^2}\) is unknown, use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) as \({\sigma ^2}\) is known \({\bar x}\) is \({\text{N}}\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">CI is \(\bar x - {z^ * }\frac{\sigma }{{\sqrt n }} < \mu < \bar x + {z^ * }\frac{\sigma }{{\sqrt n }}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = 9.24,{\text{ }}{z^ * } = 1.960\) for 95 % CI <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">CI is \(6.61 < \mu < 11.87\) by GDC <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) CI is \(\bar x - {z^ * }\frac{\sigma }{{\sqrt n }} < \mu < \bar x + {z^ * }\frac{\sigma }{{\sqrt n }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">require \(2 \times 1.645\frac{3}{{\sqrt n }} < 2\) <strong><em>R1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4.935 < \sqrt n \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(24.35 < n\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so smallest value for <em>n</em> = 25 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept use of table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) as \({\sigma ^2}\) is not known \({\bar x}\) has the <em>t</em> distribution with <em>v</em> = 4 <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">CI is \(\bar x - {t^ * }\frac{{{s_{n - 1}}}}{{\sqrt n }} < \mu < \bar x + {t^ * }\frac{{{s_{n - 1}}}}{{\sqrt n }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = 9.24,{\text{ }}{s_{n - 1}} = 5.984,{\text{ }}{t^ * } = 2.776\) for 95 % CI <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">CI is \(1.81 < \mu < 16.67\) by GDC <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [15 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The 2 confidence intervals were generally done well by using a calculator. Some marks were dropped by not giving the answers to 2 decimal places as required. Weak candidates did not realise that (b) was a <em>t</em> interval. Part (a) (ii) was not as well answered and often it was the first step that was the problem.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>is normally distributed with unknown mean \(\mu \) and unknown variance \({\sigma ^2}\) . A random sample of 10 observations on <em>X </em>was taken and the following 95 % confidence interval for \(\mu \) was correctly calculated as [4.35, 4.53] .</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Calculate an unbiased estimate for</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) \(\mu \) ,</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \({\sigma ^2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The value of \(\mu \) is thought to be 4.5, so the following hypotheses are defined.\[{{\text{H}}_0}:\mu = 4.5;{\text{ }}{{\text{H}}_1}:\mu < 4.5\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Find the <em>p</em>-value of the observed sample mean.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) State your conclusion if the significance level is</span></p>
<p style="margin: 0px 0px 0px 60px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (a) 1 %,</span></p>
<p style="margin: 0px 0px 0px 60px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (b) 10 %.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) \(\bar x = \frac{{4.35 + 4.53}}{2} = 4.44\) (estimate of \(\mu \)) <em><strong>A2</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><br>(ii) Degrees of freedom = 9 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Critical value of </span><em style="font-family: 'times new roman', times; font-size: medium;">t </em><span style="font-family: 'times new roman', times; font-size: medium;">= 2.262 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2.262 \times \frac{s}{{\sqrt {10} }} = 0.09\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = 0.12582…\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({s^2} = 0.0158\) (estimate of \({\sigma ^2}\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Using </span><em style="font-family: 'times new roman', times; font-size: medium;">t </em><span style="font-family: 'times new roman', times; font-size: medium;">test </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{4.44 - 4.5}}{{\sqrt {\frac{{0.0158}}{{10}}} }} = - 1.50800\) (Accept \( - 1.50946\)) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0829 (Accept 0.0827) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><br>(ii) (a) Accept \({{\text{H}}_0}\) / Reject \({{\text{H}}_1}\) . <strong><em>R1</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong>(b) Reject \({{\text{H}}_0}\) / Accept \({{\text{H}}_1}\) . <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [14 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates realised that the unbiased estimate of the mean was simply the central point of the confidence interval. Many candidates, however, failed to realise that, because the variance was unknown, the <em>t</em>-distribution was used to determine the confidence limits. In (b), although the <em>p</em>-value was asked for specifically, some candidates solved the problem correctly by comparing the value of their statistic with the appropriate critical values. This method was given full credit but, of course, marks were lost by their failure to give the <em>p</em>-value.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the random variable \(X \sim {\text{Geo}}(p)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State \({\text{P}}(X < 4)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the probability generating function for <em>X </em>is given by \({G_X}(t) = \frac{{pt}}{{1 - qt}}\), where \(q = 1 - p\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let the random variable \(Y = 2X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Show that the probability generating function for <em>Y </em>is given by \({G_Y}(t) = {G_X}({t^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) By considering \({G'_Y}(1)\), show that \({\text{E}}(Y) = 2{\text{E}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let the random variable \(W = 2X + 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) (i) Find the probability generating function for <em>W </em>in terms of the probability generating function of <em>Y</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Hence, show that \({\text{E}}(W) = 2{\text{E}}(X) + 1\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) use of \({\text{P}}(X = n) = p{q^{n - 1}}{\text{ }}(q = 1 - p)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 4) = p + pq + p{q^2}{\text{ }}\left( { = 1 - {q^3}} \right){\text{ }}\left( { = 1 - {{(1 - p)}^3}} \right){\text{ }}( = 3p - 3{p^2} + {p^3})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({G_X}(t) = {\text{P}}(X = 1)t + {\text{P}}(X = 2){t^2} + \ldots \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = pt + pq{t^2} + p{q^2}{t^3} + \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">summing an infinite geometric series <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{pt}}{{1 - qt}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \({G_Y}(t) = {\text{P}}(Y = 1)t + {\text{P}}(Y = 2){t^2} + \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 0 \times t + {\text{P}}(X = 1){t^2} + 0 \times {t^3} + {\text{P}}(X = 2){t^4} + \ldots \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = {G_X}({t^2})\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \({G_Y}(t) = E({t^Y}) = E({t^{2X}})\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = E\left( {{{({t^2})}^X}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = {G_X}({t^2})\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \({\text{E}}(Y) = {G'_Y}(1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 2t{G'_X}({t^2})\) evaluated at \(t = 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 2{\text{E}}(X)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{p{t^2}}}{{(1 - q{t^2})}}} \right) = \frac{{2pt(1 - q{t^2}) + 2pq{t^3}}}{{{{(1 - q{t^2})}^2}}}\) evaluated at \(t = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 2 \times \frac{{p(1 - qt) + pqt}}{{{{(1 - qt)}^2}}}\) evaluated at \(t = 1{\text{ (or }}\frac{2}{p})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 2{\text{E}}(X)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) (i) \({G_W}(t) = t{G_Y}(t)\) (or equivalent) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) attempt to evaluate \({G'_W}(t)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> obtain \(1 \times {G_Y}(t) + t \times {G'_Y}(t)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> substitute \(t = 1\) to obtain \(1 \times 1 + 1 \times {G'_Y}(1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = \frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{p{t^3}}}{{(1 - q{t^2})}}} \right) = \frac{{3p{t^2}(1 - q{t^2}) + 2pq{t^4}}}{{{{(1 - q{t^2})}^2}}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> substitute \(t = 1\) to obtain \(1 + \frac{2}{p}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 1 + 2{\text{E}}(X)\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [16 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p class="p1">The continuous random variable \(X\) has probability density function</p>
<p class="p1">\[f(x) = \left\{ {\begin{array}{*{20}{c}} {{{\text{e}}^{ - x}}}&{x \geqslant 0} \\ 0&{x < 0} \end{array}.} \right.\]</p>
<p class="p1">The discrete random variable \(Y\) is defined as the integer part of \(X\), that is the largest integer less than or equal to \(X\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the probability distribution of \(Y\) <span class="s1">is given by \({\text{P}}(Y = y) = {{\text{e}}^{ - y}}(1 - {{\text{e}}^{ - 1}}),{\text{ }}y \in \mathbb{N}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(G(t)\), the probability generating function of \(Y\), is given by \(G(t) = \frac{{1 - {{\text{e}}^{ - 1}}}}{{1 - {{\text{e}}^{ - 1}}t}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence determine the value of \({\text{E}}(Y)\) correct to three significant figures.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(Y = y) = \int_y^{y + 1} {{{\text{e}}^{ - x}}{\text{d}}x} \) </span><strong><em>M1A1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = {[ - {{\text{e}}^{ - x}}]^{y + 1}}y\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = - {{\text{e}}^{ - (y + 1)}} + {{\text{e}}^{ - y}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\( = {{\text{e}}^{ - y}}(1 - {{\text{e}}^{ - 1}})\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>attempt to use \(G(t) = \sum {{\text{P}}(Y = y){t^y}} \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = \sum\limits_{y = 0}^\infty {{{\text{e}}^{ - y}}(1 - {{\text{e}}^{ - 1}}){t^y}} \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong></span>Accept a listing of terms without the use of \(\Sigma \).</p>
<p class="p1">this is an infinite geometric series with first term \(1 - {{\text{e}}^{ - 1}}\) and common ratio \({{\text{e}}^{ - 1}}t\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p5"><span class="Apple-converted-space">\(G(t) = \frac{{1 - {{\text{e}}^{ - 1}}}}{{1 - {{\text{e}}^{ - 1}}t}}\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \({\text{E}}(Y) = G'(1)\)</span> <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="s2">\(G'(t) = \frac{{1 - {{\text{e}}^{ - 1}}}}{{{{(1 - {{\text{e}}^{ - 1}}t)}^2}}} \times {{\text{e}}^{ - 1}}\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(Y) = \frac{{{{\text{e}}^{ - 1}}}}{{(1 - {{\text{e}}^{ - 1}})}}\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 0.582\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Allow the use of GDC to determine \(G'(1)\).</p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (a), it was disappointing to find that very few candidates realised that \({\text{P}}(Y = y)\) could be found by integrating \(f(x)\) from \(y\) to \(y + 1\). Candidates who simply integrated \(f(x)\) to find the cumulative distribution function of \(X\) were given no credit unless they attempted to use their result to find the probability distribution of \(Y\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Solutions to (b)(i) were generally good although marks were lost due to not including the \(y = 0\) term.</p>
<p class="p1">Part (b)(ii) was also well answered in general with the majority of candidates using the GDC to evaluate \(G'(1)\).</p>
<p class="p1">Candidates who tried to differentiate \(G(t)\) algebraically often made errors.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Francisco and his friends want to test whether performance in running 400 metres improves if they follow a particular training schedule. The competitors are tested before and after the training schedule.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The times taken to run 400 metres, in seconds, before and after training are shown in the following table.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.37.33.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Apply an appropriate test at the 1% significance level to decide whether the training schedule improves competitors’ times, stating clearly the null and alternative hypotheses. (It may be assumed that the distributions of the times before and after training are normal.)</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}\): the training schedule does not help improve times (or \(\mu = 0\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_1}\): the training schedule does help improve times (or \(\mu > 0\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Subsequent marks can be awarded even if the hypotheses are not stated.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (Assuming difference of times is normally distributed.)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">let \(d{\text{ time before training }}-{\text{ time after training}}\) <strong><em>(M1)</em></strong></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-18_om_07.47.32.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 5,{\text{ }}\sum {d = 13,{\text{ }}} \sum {{d^2} = 79 \Rightarrow s_{n - 1}^2 = \frac{1}{4}\left( {79 - \frac{{169}}{5}} \right) = 11.3} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(small sample) so use a one-sided <em>t-</em>test <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> The “one-sided” <em>t</em>-test may have been seen above when stating \({{\text{H}}_1}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{2.6}}{{\sqrt {\frac{{11.3}}{5}} }} = 1.7 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 4\), <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">at the 1% level the critical value is 3.7 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \({\text{3.7}} > {\text{1.7}} \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}\) is accepted (insufficient evidence to reject \({{\text{H}}_0}\)) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Follow through their <em>t</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(small sample) so use a one-sided <em>t-</em>test <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p = 0.079 \ldots \) <strong><em>A4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \(0.079 \ldots > 0.01\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{H}}_0}\) is accepted (insufficient evidence to reject \({{\text{H}}_0}\)) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Follow through their <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept \(d = {\text{time after training }}-{\text{ time before training throughout}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: 'times new roman', times; font-size: medium;">It was again disappointing to see many candidates giving incorrect hypotheses. A common error was to give the hypotheses the wrong way around. Candidates should be aware that in this type of problem the null hypothesis always represents the status quo. Also, some candidates defined ‘\(d = {\text{time before }}-{\text{ time after}}\)’ and then gave the hypotheses incorrectly as \({{\text{H}}_0}:d = 0\) or \(\bar d = 0;{\text{ }}{{\text{H}}_1}:d > 0\) or \(\bar d > 0\). It is important to note that the parameter being tested here is \(E(d)\) or \({\mu _d}\) although \(\mu \) was accepted.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A population is known to have a normal distribution with a variance of 3 and an unknown mean \(\mu \) . It is proposed to test the hypotheses \({{\text{H}}_0}:\mu = 13,{\text{ }}{{\text{H}}_1}:\mu > 13\) using the mean of a sample of size 2.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the appropriate critical regions corresponding to a significance level of</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) 0.05;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that the true population mean is 15.2, calculate the probability of making a Type II error when the level of significance is</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) 0.05;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) How is the change in the probability of a Type I error related to the change in the probability of a Type II error?</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) With \({{\text{H}}_0},{\text{ }}\bar X \sim {\text{N}}\left( {13,\frac{3}{2}} \right) = {\text{N(13, 1.5)}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) 5 % for N(0,1) is 1.645</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(\frac{{\bar x - 13}}{{\sqrt {1.5} }} = 1.645\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = 13 + 1.645\sqrt {1.5} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 15.0\,\,\,\,\,{\text{(3 s.f.)}}\) <strong><em>A1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{[15.0, }}\infty {\text{[}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) 1% for N(0, 1) is 2.326</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(\frac{{\bar x - 13}}{{\sqrt {1.5} }} = 2.326\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = 13 + 2.326\sqrt {1.5} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 15.8\,\,\,\,\,{\text{(3 s.f., accept 15.9)}}\) <strong><em>A1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{[15.8, }}\infty {\text{[}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \(\beta = {\text{P}}(\bar X < 15.0147)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.440\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\beta = {\text{P}}(\bar X < 15.8488)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.702\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The probability of a Type II error increases when the probability of a Type I error decreases. <strong><em>R2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [16 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question proved to be the most difficult. The range of solutions ranged from very good to very poor. Many students thought that \(P(TypeI) = 1 - P(TypeII)\) when in fact \(1 - P(TypeII)\) is the power of the test.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Eric plays a game at a fairground in which he throws darts at a target. Each time he throws a dart, the probability of hitting the target is \(0.2\). He is allowed to throw as many darts as he likes, but it costs him \($1\) a throw. If he hits the target a total of three times he wins \($10\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability he has his third success of hitting the target on his sixth throw.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the expected number of throws required for Eric to hit the target three times.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down his expected profit or loss if he plays until he wins the \($10\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If he has just \($8\), find the probability he will lose all his money before he hits the target three times.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">let \(X\) be the number of throws until Eric hits the target three times</p>
<p class="p1">\(X \sim {\text{NB(3, 0.2)}}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{P}}(X = 6) = \left( {\begin{array}{*{20}{c}} 5 \\ 2 \end{array}} \right){0.8^3} \times {0.2^3}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\( = 0.04096\;\;\;\left( { = \frac{{128}}{{3125}}} \right)\;\;\;\)(exact) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">let \(X\) be the number of hits in five throws</p>
<p class="p1">\(X\) is \({\text{B}}(5,{\text{ }}0.2)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{P}}(X = 2) = \left( {\begin{array}{*{20}{c}} 5 \\ 2 \end{array}} \right){0.2^2} \times {0.8^3}\;\;\;(0.2048)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(P\)(3rd hit on 6th throw) \( = \left( {\begin{array}{*{20}{c}} 5 \\ 2 \end{array}} \right){0.2^2} \times {0.8^3} \times 0.2 = 0.04096\left( { = \frac{{128}}{{3125}}} \right)\;\;\;\)(exact) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \({\text{expected number of throws}} = \frac{3}{{0.2}} = 15\) <strong><em>(M1)A1</em></strong></p>
<p>(ii) \({\text{profit}} = (10 - 15) = - \$ 5{\text{ or loss}} = \$ 5\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">let \(Y\) be the number of times the target is hit in \(8\) throws</p>
<p class="p1">\(Y \sim {\text{B}}(8,{\text{ }}0.2)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\({\text{P}}(Y \le 2)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = 0.797\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">let the \({3^{{\text{rd}}}}\) hit occur on the \({Y^{{\text{th}}}}\) throw</p>
<p class="p1">\(Y{\text{ is NB}}(3,{\text{ }}0.2)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\({\text{P}}(Y > 8) = 1 - {\text{P}}(Y \le 8)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = 0.797\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [9 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was well answered, using the negative binomial distribution \(NB(3,{\text{ }}0.2)\), by many candidates. Some candidates began by using the binomial distribution \(B(5,{\text{ }}0.2)\) which is a valid method as long as it is followed by multiplying by 0.2 but this final step was not always carried out successfully.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was well answered by the majority of candidates.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), candidates who used the binomial distribution \(B(8,{\text{ }}0.2)\) were generally successful. Candidates who used the negative binomial distribution \(Y \approx NB(3,{\text{ }}0.2)\) to evaluate \(P(Y > 8)\) were usually unsuccessful because of the large amount of computation involved.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A teacher wants to determine whether practice sessions improve the ability to memorize digits.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">He tests a group of 12 children to discover how many digits of a twelve-digit number could be repeated from memory after hearing them once. He gives them test 1, and following a series of practice sessions, he gives them test 2 one week later. The results are shown in the table below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><img 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" alt></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State appropriate null and alternative hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Test at the 5 % significance level whether or not practice sessions improve ability to memorize digits, justifying your choice of test.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({{\text{H}}_0}:d = 0;{\text{ }}{{\text{H}}_1}:d > 0\), where <em>d </em>is the difference in the number of digits remembered <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b)</span><br><img 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" alt><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: </strong>Award <strong><em>A2 </em></strong>for the correct <em>d </em>values.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Award <strong><em>A1 </em></strong>for one error, <strong><em>A0 </em></strong>for two or more errors.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the <em>t</em>-test because the variance is not known <strong><em>M1R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By GDC</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>t</em> = 2.106… <strong><em>(A2)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>p</em>-value = 0.0295 (accept any value that rounds to this number) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Since 0.0295 < 0.05 there is evidence that practice sessions improve ability to memorize digits <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The critical value of <em>t </em>is 1.796 <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Since 2.106... > 1.796 there is evidence that practice sessions improve ability to memorize digits <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>M1R1A1A1R1 </em></strong>for testing equality of means (<em>t</em> = –1.46, <em>p</em>-value = 0.08) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [11 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this question was reasonably well done the hypotheses were often not stated precisely and the fact that the two data sets were dependent escaped many candidates.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The continuous random variable <em>X </em>has probability density function <em>f </em>given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x,}&{0 \leqslant x \leqslant 0.5,} \\ <br> {\frac{4}{3} - \frac{2}{3}x,}&{0.5 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the function <em>f </em>and show that the lower quartile is 0.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine E(<em>X </em>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{E}}({X^2})\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Two independent observations are made from <em>X </em>and the values are added.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The resulting random variable is denoted <em>Y </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine \({\text{E}}(Y - 2X)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{Var}}\,(Y - 2X)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the cumulative distribution function for <em>X </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, find the median of the distribution.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">piecewise linear graph</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><br><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">correct shape <strong><em>A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">with vertices (0, 0), (0.5, 1) and (2, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">LQ: <em>x</em> = 0.5 , because the area of the triangle is 0.25 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{E}}(X) = \int_0^{0.5} {x \times 2x{\text{d}}x + \int_{0.5}^2 {x \times \left( {\frac{4}{3} - \frac{2}{3}x} \right){\text{d}}x = \frac{5}{6}{\text{ }}( = 0.833...)} } \) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{E}}({X^2}) = \int_0^{0.5} {{x^2} \times 2x{\text{d}}x + \int_{0.5}^2 {{x^2} \times \left( {\frac{4}{3} - \frac{2}{3}x} \right){\text{d}}x = \frac{7}{8}{\text{ }}( = 0.875)} } \) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{E}}(Y - 2X) = 2{\text{E}}(X) - 2{\text{E}}(X) = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{Var}}\,(X) = \left( {{\text{E}}({X^2}) - {\text{E}}{{(X)}^2}} \right) = \frac{{13}}{{72}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(Y = {X_1} + {X_2} \Rightarrow {\text{Var}}\,(Y) = 2{\text{Var }}(X)\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}\,(Y - 2X) = 2{\text{Var}}\,(X) + 4{\text{Var}}\,(X) = \frac{{13}}{{12}}\) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) attempt to use \(cf(x) = \int {f(u){\text{d}}u} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">obtain \(cf(x) = \left\{ {\begin{array}{*{20}{c}}<br> {{x^2},}&{0 \leqslant x \leqslant 0.5,} \\ <br> {\frac{{4x}}{3} - \frac{1}{3}{x^2} - \frac{1}{3},}&{0.5 \leqslant x \leqslant 2,} <br>\end{array}} \right.\) \(\begin{array}{*{20}{c}}<br> {{\boldsymbol{A1}}} \\ <br> {{\boldsymbol{A2}}} <br>\end{array}\)<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) attempt to solve \(cf(x) = 0.5\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{4x}}{3} - \frac{1}{3}{x^2} - \frac{1}{3} = 0.5\) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">obtain 0.775 <strong><em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept attempts in the form of an integral with upper limit the unknown median.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept exact answer \(2 - \sqrt {1.5} \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">There was a curious issue about the lower quartile in part (a): The LQ coincides with a quarter of the range of the distribution \(\frac{2}{4} = 0.5\). Sadly this is wrong reasoning – the correct reasoning involves a consideration of areas.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">There was a curious issue about the lower quartile in part (a): The LQ coincides with a quarter of the range of the distribution \(\frac{2}{4} = 0.5\). Sadly this is wrong reasoning – the correct reasoning involves a consideration of areas.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) many candidates used hand calculation rather than their GDC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable Y was not well understood, and that followed into incorrect calculations involving Y – 2X.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">There was a curious issue about the lower quartile in part (a): The LQ coincides with a quarter of the range of the distribution \(\frac{2}{4} = 0.5\). Sadly this is wrong reasoning – the correct reasoning involves a consideration of areas.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) many candidates used hand calculation rather than their GDC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable Y was not well understood, and that followed into incorrect calculations involving Y – 2X.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">There was a curious issue about the lower quartile in part (a): The LQ coincides with a quarter of the range of the distribution \(\frac{2}{4} = 0.5\). Sadly this is wrong reasoning – the correct reasoning involves a consideration of areas.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b) many candidates used hand calculation rather than their GDC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable Y was not well understood, and that followed into incorrect calculations involving Y – 2X.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) has probability density function</p>
<p class="p1">\(f(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{1}{2}}&{0 \le x < 1} \\ {\frac{1}{4}}&{1 \le x < 3} \\ 0&{x \ge 3} \end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the cumulative distribution function for \(X\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the interquartile range for \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space"><img 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" alt> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Ignore open / closed endpoints and vertical lines.</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong>for a correct graph with scales on both axes and a clear indication of the relevant values.</p>
<p class="p1"><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(F(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{x}{2}}&{0 \le x < 1} \\ {\frac{x}{4} + \frac{1}{4}}&{1 \le x < 3} \\ 1&{x \ge 3} \end{array}} \right.\)</p>
<p class="p1">considering the areas in their sketch or using integration <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(F(x) = 0,{\text{ }}x < 0,{\text{ }}F(x) = 1,{\text{ }}x \ge 3\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(F(x) = \frac{x}{2},{\text{ }}0 \le x < 1\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(F(x) = \frac{x}{4} + \frac{1}{4},{\text{ }}1 \le x < 3\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept \( < \) for \( \le \) in all places and also \( > \) for \( \ge \) first <strong><em>A1</em></strong>.</p>
<p class="p1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({Q_3} = 2,{\text{ }}{Q_1} = 0.5\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1">\({\text{IQR is }}2 - 0.5 = 1.5\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [9 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was correctly answered by most candidates. Some graphs were difficult to mark because candidates drew their lines on top of the ruled lines in the answer book. Candidates should be advised not to do this. Candidates should also be aware that the command term ‘sketch’ requires relevant values to be indicated.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (b), most candidates realised that the cumulative distribution function had to be found by integration but the limits were sometimes incorrect.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), candidates who found the upper and lower quartiles correctly sometimes gave the interquartile range as \([0.5,{\text{ }}2]\). It is important for candidates to realise that that the word range has a different meaning in statistics compared with other branches of mathematics.</p>
<div class="question_part_label">c.</div>
</div>
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