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</div><h2>SL Paper 1</h2><div class="specification">
<p>At an early stage in analysing the marks scored by candidates in an examination paper, the examining board takes a random sample of 250 candidates and finds that the marks, \(x\) , of these candidates give \(\sum {x = 10985} \) and \(\sum {{x^2} = 598736} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate a 90% confidence interval for the population mean mark <em>μ</em> for this paper.</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>The null hypothesis<em> μ</em> = 46.5 is tested against the alternative hypothesis <em>μ</em> < 46.5 at the <em>λ</em>% significance level. Determine the set of values of <em>λ</em> for which the null hypothesis is rejected in favour of the alternative hypothesis.</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>\(\bar x = 43.94\) <em><strong>(A1)</strong></em></p>
<p>unbiased variance estimate = 466.0847 <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Accept sample variance = 464.2204.</p>
<p>⇒ 90% confidence interval is (41.7,46.2) <em><strong>A1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Z-value is −1.87489 or −1.87866 <em><strong>(A1)</strong></em></p>
<p>probability is 0.0304 or 0.0301 <em><strong>(A1)</strong></em></p>
<p>⇒<em> λ</em> ≥ 3.01 <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[4 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>A random sample \({X_1},{\text{ }}{X_2},{\text{ }} \ldots ,{\text{ }}{X_n}\) is taken from the normal distribution \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\), where the value of \(\mu \) is unknown but the value of \({\sigma ^2}\) is known. The mean of the sample is denoted by \(\bar X\).</p>
</div>
<div class="specification">
<p>A mathematics teacher, wishing to apply the above result, generates some artificial data, assumes a value for the variance and calculates the following 95% confidence interval for \(\mu \),</p>
<p>\[[3.12,{\text{ }}3.25].\]</p>
<p>The teacher asks Alun to interpret this result. Alun makes the following statement. “The value of \(\mu \) lies in the interval \([3.12,{\text{ }}3.25]\) with probability 0.95.”</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the distribution of \(\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that, with probability 0.95,</p>
<p>\[\bar X - 1.96\frac{\sigma }{{\sqrt n }} \leqslant \mu \leqslant \bar X + 1.96\frac{\sigma }{{\sqrt n }}.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain briefly why this is an incorrect statement.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give a correct interpretation.</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>\(\frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\) is \({\text{N}}(0,{\text{ }}1)\) or it has the Z-distribution <strong><em>A1</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to make a probability statement <strong><em>R1</em></strong></p>
<p>therefore with probability 0.95,</p>
<p>\( - 1.96 \leqslant \frac{{\bar X - \mu }}{{\frac{\sigma }{{\sqrt n }}}} \leqslant 1.96\) <strong><em>A1</em></strong></p>
<p>\( - 1.96\frac{\sigma }{{\sqrt n }} \leqslant \bar X - \mu \leqslant 1.96\frac{\sigma }{{\sqrt n }}\) <strong><em>A1</em></strong></p>
<p>\(1.96\frac{\sigma }{{\sqrt n }} \geqslant \mu - \bar X \geqslant - 1.96\frac{\sigma }{{\sqrt n }}\) <strong><em>A1</em></strong></p>
<p>\(\bar X + 1.96\frac{\sigma }{{\sqrt n }} \geqslant \mu \geqslant \bar X - 1.96\frac{\sigma }{{\sqrt n }}\)</p>
<p> </p>
<p><strong>Note:</strong> Award the final <strong><em>A1 </em></strong>for either of the above two lines.</p>
<p> </p>
<p>\(\bar X - 1.96\frac{\sigma }{{\sqrt n }} \leqslant \mu \leqslant \bar X + 1.96\frac{\sigma }{{\sqrt n }}\) <strong><em>AG</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>you cannot make a probability statement about a constant lying in a constant interval <strong>OR </strong>the mean either lies in the interval or it does not <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the confidence interval is the observed value of a random interval</p>
<p><strong>OR </strong>if the process is carried out a large number of times, \(\mu \) will lie in the interval 95% of the times <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</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.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>
<br><hr><br><div class="specification">
<p class="p1">The lifetime, in years, of a randomly chosen basic vacuum cleaner is assumed to be modelled by the normal distribution \(B \sim {\text{N}}(14,{\text{ }}{3^2})\).</p>
</div>
<div class="specification">
<p class="p1">The lifetime, in years, of a randomly chosen robust vacuum cleaner is assumed to be modelled by the normal distribution \(R \sim {\text{N}}(20,{\text{ }}{4^2})\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}\left( {B > {\text{E}}(B) + \frac{1}{2}\sqrt {{\text{Var}}(B)} } \right)\).</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"><span class="s1">Find the probability that the total lifetime of </span><span class="s2">7 </span>randomly chosen basic vacuum cleaners is less than <span class="s2">100 </span>years.</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"><span class="s1">Find the probability that the total lifetime of </span><span class="s2">5 </span>randomly chosen robust vacuum cleaners is greater than the total lifetime of <span class="s2">7 </span>randomly chosen basic vacuum cleaners.</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"><span class="Apple-converted-space">\({\text{P}}(B > 15.5){\text{ }}\left( { = {\text{P}}(Z > 0.5)} \right)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = (1 - 0.69146) = 0.309\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><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">consider \(V = {B_1} + {B_2} + {B_3} + {B_4} + {B_5} + {B_6} + {B_7}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(V) = 98\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s2">\({\text{Var}}(V) = 63\) </span>or equivalent <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>No need to state \(V\) is normal.</p>
<p class="p3"> </p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(V < 100) = \left( {{\text{P}}\left( {Z < \frac{2}{{\sqrt {63} }} = 0.251976 \ldots } \right)} \right) = 0.599\) </span><strong><em>A1</em></strong></p>
<p class="p1"><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">consider \(W = {R_1} + {R_2} + {R_3} + {R_4} + {R_5} - ({B_1} + {B_2} + {B_3} + {B_4} + {B_5} + {B_6} + {B_7})\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(W) = 2\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({\text{Var}}(W) = 80 + 63 = 143\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(W > 0) = \left( {{\text{P}}\left( {Z < \frac{2}{{\sqrt {143} }}} \right)} \right)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\( = 0.566\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><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;">
<p class="p1">This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was one of the more successful questions on the paper with many wholly correct answers seen. Only a very small number failed to complete part (a) successfully. There were also many fully correct answers to part (b). Part (c) caused a problem for some candidates where in most of those cases they failed to calculate the variance correctly.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights of male students in a college are modelled by a normal distribution with mean 80 kg and standard deviation 7 kg.</p>
<p>The weights of female students in the college are modelled by a normal distribution with mean 54 kg and standard deviation 5 kg.</p>
</div>
<div class="specification">
<p>The college has a lift installed with a recommended maximum load of 550 kg. One morning, the lift contains 3 male students and 6 female students. You may assume that the 9 students are randomly chosen.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the weight of a randomly chosen male student is more than twice the weight of a randomly chosen female student.</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>Determine the probability that their combined weight exceeds the recommended maximum.</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>let \(M\), \(F\) denote the weights of the male, female</p>
<p>consider \(D = M - 2F\) <strong><em>(M1)</em></strong></p>
<p>\({\text{E}}(D) = 80 - 2 \times 54 = - 28\) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}(D) = {7^2} + 4 \times {5^2}\) <strong><em>(M1)</em></strong></p>
<p>\( = 149\) <strong><em>A1</em></strong></p>
<p>\({\text{P}}(M > 2F) = {\text{P}}(D > 0)\) <strong><em>(M1)</em></strong></p>
<p>\( = 0.0109\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.011.</p>
<p> </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>consider \({\text{S}} = \sum\limits_{i = 1}^3 {{M_i} + \sum\limits_{i = 1}^6 {{F_i}} } \) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone the use of the incorrect notation \(3M + 6F\).</p>
<p>\({\text{E}}(S) = 3 \times 80 + 6 \times 54 = 564\) <strong><em>A1</em></strong></p>
<p>\({\text{Var}}(S) = 3 \times {7^2} + 6 \times {5^2}\) <strong><em>(M1)</em></strong></p>
<p>\( = 297\) <strong><em>A1</em></strong></p>
<p>\({\text{P}}(S > 550) = 0.792\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.792.</p>
<p> </p>
<p><strong><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;">
[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">Jim is investigating the relationship between height and foot length in teenage boys.</p>
<p class="p2"><span class="s1">A sample of 13 </span>boys is taken and the height and foot length of each boy are measured.</p>
<p class="p2">The results are shown in the table.</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-15_om_07.44.05.png" alt></p>
<p class="p2">You may assume that this is a random sample from a bivariate normal distribution.</p>
<p class="p2">Jim wishes to determine whether or not there is a positive association between height and foot length.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the product moment correlation coefficient.</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">Find the \(p\)<span class="s1"><em>-</em></span>value.</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">Interpret the \(p\)<span class="s1"><em>-</em></span>value in the context of the question.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the regression line of \(y\) on \(x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Estimate the foot length of a boy of height <span class="s1">170 cm</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 class="p1"><strong>Note: </strong>In all parts accept answers which round to the correct 2sf answer.</p>
<p class="p2"> </p>
<p class="p1">\(r = 0.806\) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(4.38 \times {10^{ - 4}}\) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(p\)-value represents strong evidence to indicate a (positive) association between height and foot length <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>FT the \(p\)-value</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(y = 0.103x + 12.3\) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempted substitution of \(x = 170\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(y = 29.7\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: </strong>Accept \(y = 29.8\)</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 class="p1">Solutions to this question were often disappointing. Candidates were expected to use appropriate software on their calculators to do the whole question. However, some candidates used their calculators just to evaluate sums and sums of squares and then used the appropriate formulae to calculate the correlation coefficient, the <em>p</em>-value (which required the evaluation of the <em>t</em>-value first) and the equation of the regression line. This was a time consuming exercise and introduced the possibility of arithmetic error.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Solutions to this question were often disappointing. Candidates were expected to use appropriate software on their calculators to do the whole question. However, some candidates used their calculators just to evaluate sums and sums of squares and then used the appropriate formulae to calculate the correlation coefficient, the <em>p</em>-value (which required the evaluation of the <em>t</em>-value first) and the equation of the regression line. This was a time consuming exercise and introduced the possibility of arithmetic error.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Solutions to this question were often disappointing. Candidates were expected to use appropriate software on their calculators to do the whole question. However, some candidates used their calculators just to evaluate sums and sums of squares and then used the appropriate formulae to calculate the correlation coefficient, the <em>p</em>-value (which required the evaluation of the <em>t</em>-value first) and the equation of the regression line. This was a time consuming exercise and introduced the possibility of arithmetic error.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Solutions to this question were often disappointing. Candidates were expected to use appropriate software on their calculators to do the whole question. However, some candidates used their calculators just to evaluate sums and sums of squares and then used the appropriate formulae to calculate the correlation coefficient, the <em>p</em>-value (which required the evaluation of the <em>t</em>-value first) and the equation of the regression line. This was a time consuming exercise and introduced the possibility of arithmetic error.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Solutions to this question were often disappointing. Candidates were expected to use appropriate software on their calculators to do the whole question. However, some candidates used their calculators just to evaluate sums and sums of squares and then used the appropriate formulae to calculate the correlation coefficient, the <em>p</em>-value (which required the evaluation of the <em>t</em>-value first) and the equation of the regression line. This was a time consuming exercise and introduced the possibility of arithmetic error.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bill is investigating whether or not there is a positive association between the heights </span><span style="font-family: times new roman,times; font-size: medium;">and weights of boys of a certain age. He defines the hypotheses\[{{\rm{H}}_0}:\rho = 0;{{\rm{H}}_1}:\rho > 0 ,\]where \(\rho \) denotes the population correlation coefficient between heights and weights </span><span style="font-family: times new roman,times; font-size: medium;">of boys of this age. He measures the height, \(h\) cm, and weight, \(w\) kg, of each of a </span><span style="font-family: times new roman,times; font-size: medium;">random sample of \(20\) boys of this age and he calculates the following statistics.\[\sum {w = 340,\sum {h = 2002,\sum {{w^2} = 5830} } } ,\sum {{h^2} = 201124} ,\sum {hw = 34150} \]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Calculate the correlation coefficient for this sample.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Calculate the \(p\)-value of your result and interpret it at the \(1\% \) level of </span><span style="font-family: times new roman,times; font-size: medium;">significance.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Calculate the equation of the least squares regression line of \(w\) on \(h\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) The height of a randomly selected boy of this age of \(90\) cm. Estimate his </span><span style="font-family: times new roman,times; font-size: medium;">weight.</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><span style="font-family: times new roman,times; font-size: medium;">(i) \(r = \frac{{34150 - 340 \times \frac{{2002}}{{20}}}}{{\sqrt {\left( {5830 - \frac{{{{340}^2}}}{{20}}} \right)} \left( {201124 - \frac{{{{2002}^2}}}{{20}}} \right)}}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)(A1)</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept equivalent formula.</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;">\( = 0.610\) <em><strong>A1</strong></em></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) (\(T = R \times \sqrt {\frac{{n - 2}}{{1 - {R^2}}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">has the <strong><em>t</em></strong>-distribution with \(n - 2\) degrees of freedom)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(t = 0.6097666 \ldots \sqrt {\frac{{18}}{{1 - 0.6097666{ \ldots ^2}}}} \) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">M1</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 3.2640 \ldots \) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{DF}} = 18\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p{\rm{ - value}} = 0.00215 \ldots \) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">this is less than \(0.01\), so we conclude that there is a positive association </span><span style="font-family: times new roman,times; font-size: medium;">between heights and weights of boys of this age <strong><em>R1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></em></strong></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) the equation of the regression line of \(w\) on \(h\) is</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(w - \frac{{340}}{{20}} = \left( {\frac{{20 \times 34150 - 340 \times 2002}}{{20 \times 201124 - {{2002}^2}}}} \right)\left( {h - \frac{{2002}}{{20}}} \right)\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">M1</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(w = 0.160h + 0.957\) <em><strong>A1</strong></em></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) putting \(h = 90\) , \(w = 15.4\) (kg) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>M0A0A0</strong></em> for calculation of \(h\) on \(w\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 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;">
[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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The weights of potatoes in a shop are normally distributed with mean \(98\) grams and </span><span style="font-family: times new roman,times; font-size: medium;">standard deviation \(16\) grams.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shopkeeper places \(100\) randomly chosen potatoes on a weighing machine. </span><span style="font-family: times new roman,times; font-size: medium;">Find the probability that their total weight exceeds \(10\) kilograms.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the minimum number of randomly selected potatoes which are needed to </span><span style="font-family: times new roman,times; font-size: medium;">ensure that their total weight exceeds \(10\) kilograms with probability greater </span><span style="font-family: times new roman,times; font-size: medium;">than \(0.95\).</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">let \(T\) denote the total weight, then</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T \sim N(9800,25600)\) <em><strong>(M1)(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(T > 10000) = 0.106\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">let there be \(n\) potatoes, in this case,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T \sim {\rm{N}}(98n,256n)\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">we require</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(T > 10000) > 0.95\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">or equivalently</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(T \le 10000) < 0.05\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">standardizing,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}\left( {Z \le \frac{{10000 - 98n}}{{16\sqrt n }}} \right) < 0.05\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{10000 - 98n}}{{16\sqrt n }} < - 1.6449 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(98n - 26.32\sqrt n - 10000 > 0\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">solving the corresponding equation, \(n = 104.7 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the required minimum value is \(105\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Part (b) could also be solved using SOLVER and normalcdf, or by trial and </span><span style="font-family: times new roman,times; font-size: medium;">improvement.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Allow the use of \( = \) instead of \( < \) and \( > \) throughout.</span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></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>The mean weight of a certain breed of bird is claimed to be 5.5 kg. In order to test this claim, a random sample of 10 birds of the breed was obtained and weighed, with the following results in kg.</p>
<p>\[5.41\quad \quad \quad 5.22\quad \quad \quad 5.54\quad \quad \quad 5.58\quad \quad \quad 5.20\quad \quad \quad 5.57\quad \quad \quad 5.23\quad \quad \quad 5.32\quad \quad \quad 5.46\quad \quad \quad 5.37\]</p>
<p>You may assume that the weights of this breed of bird are normally distributed.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses for testing the above claim using 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>Calculate unbiased estimates of the mean and the variance of the weights of this breed of bird.</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>Determine the \(p\)-value of the above data.</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>State whether or not the claim is supported by the data, using a significance level of 5%.</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>\({H_0}:\mu = 5.5;{\text{ }}{H_1}:\mu \ne 5.5\) <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>\(\sum {x = 53.9,{\text{ }}\hat \mu = 5.39} \) <strong><em>(M1)A1</em></strong></p>
<p>\(\sum {{x^2} = 290.7132,{\text{ }}{{\hat \sigma }^2} = 0.0214} \) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds correctly to 0.021.</p>
<p> </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>attempt to use the \(t\)-test <strong><em>(M1)</em></strong></p>
<p>\(t = - 2.38{\text{ }}({\text{Accept }} + 2.38)\) <strong><em>(A1)</em></strong></p>
<p>\({\text{DF}} = 9\) <strong><em>(A1)</em></strong></p>
<p>\(p{\text{ - value}} = 0.0412\) <strong><em>A1</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the claim is not supported (not accepted, rejected) at the 5% level of significance <strong><em>A1</em></strong></p>
<p><strong><em>[??? 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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The weights, \(X\) kg , of male birds of a certain species are normally distributed with </span><span style="font-family: times new roman,times; font-size: medium;">mean \(4.5\) kg and standard deviation \(0.2\) kg . The weights, \(Y\) kg , of female birds of this </span><span style="font-family: times new roman,times; font-size: medium;">species are normally distributed with mean \(2.5\) kg and standard deviation \(0.15\) kg .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the mean and variance of \(2Y - X\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that the weight of a randomly chosen male bird is </span><span style="font-family: times new roman,times; font-size: medium;">more than twice the weight of a randomly chosen female bird.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two randomly chosen male birds and three randomly chosen female birds are </span><span style="font-family: times new roman,times; font-size: medium;">placed together on a weighing machine for which the recommended maximum </span><span style="font-family: times new roman,times; font-size: medium;">weight is \(16\) kg . Find the probability that this maximum weight is exceeded.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \({\rm{E}}(2Y - X) = 2 \times 2.5 - 4.5 = 0.5\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(Var(2Y - X) = 4 \times 0.1{5^2} + {0.2^2} = 0.13\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) We require \({\rm{P}}(X > 2Y) = {\rm{P}}(2Y - X < 0)\) <strong><em>M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(0.0828\) <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Using tables, answer is \(0.0823\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(S\) denote the total weight of the \(5\) birds.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Then,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(S) = 2 \times 4.5 + 3 \times 2.5 = 16.5\) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(Var(S) = 2 \times 0.{2^2} + 3 \times 0.1{5^2} = 0.1475\) <strong><em>M1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(S > 16) = 0.904\) <strong><em>A2</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Using tables, answer is \(0.903\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 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;">
[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>Sarah is the quality control manager for the Stronger Steel Corporation which makes steel sheets. The steel sheets should have a mean tensile strength of 430 MegaPascals (MPa). If the mean tensile strength drops to 400 MPa, then Sarah must recommend a change in composition. The tensile strength of these steel sheets follows a normal distribution with a standard deviation of 35 MPa. Sarah defines the following hypotheses</p>
<p>\[{H_0}:\mu = 430\]</p>
<p>\[{H_1}:\mu = 400\]</p>
<p>where \(\mu \) denotes the mean tensile strength in MPa. She takes a random sample of \(n\) steel sheets and defines the critical region as \(\bar x \leqslant k\), where \(\bar x\) notes the mean tensile strength of the sample in MPa and \(k\) is a constant.</p>
<p>Given that the \(P{\text{(Type I Error)}} = 0.0851\) and \(P{\text{(Type II Error)}} = 0.115\), both correct to three significant figures, find the value of \(k\) and the value of \(n\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\(\bar X \sim N\left( {430,{\text{ }}\frac{{{{35}^2}}}{n}} \right)\) <em>(</em><strong><em>M1)(A1)</em></strong></p>
<p><strong>Note: </strong>The M1 is for considering the distribution of \(\bar X\)</p>
<p> </p>
<p>type I error gives \({\text{P}}(\bar X \leqslant k/\mu = 430) = 0.0851\)</p>
<p>\(\frac{{k - 430}}{{\frac{{35}}{{\sqrt n }}}} = - 1.37156 \ldots \) <strong><em>M1A1</em></strong></p>
<p>type II error gives \({\text{P}}(\bar X > k/\mu = 400) = 0.115\)</p>
<p>\(\frac{{k - 400}}{{\frac{{35}}{{\sqrt n }}}} = 1,20035 \ldots \) <strong><em>M1A1</em></strong></p>
<p><strong>Note: </strong>The two <strong><em>M1 </em></strong>marks above are for attempting to standardize \({\bar X}\) and obtain the corresponding equations with inverse normal values</p>
<p> </p>
<p>solving simultaneously <strong><em>(M1)</em></strong></p>
<p>\(k = 414\) <strong><em>A1</em></strong></p>
<p>\(n = 9\) <strong><em>A1</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">This proved to be a difficult question for most candidates with only a minority giving a correct solution. Most candidates either made no attempt at the question or just wrote several lines of irrelevant mathematics.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bottles of iced tea are supposed to contain 500 ml. A random sample of 8 bottles </span><span style="font-family: times new roman,times; font-size: medium;">was selected and the volumes measured (in ml) were as follows:</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">497.2, 502.0, 501.0, 498.6, 496.3, 499.1, 500.1, 497.7 .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Calculate unbiased estimates of the mean and variance.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Test at the \(5\%\) significance level the null hypothesis \({{\rm{H}}_0}:\mu = 500\) against </span><span style="font-family: times new roman,times; font-size: medium;">the alternative hypothesis \({{\rm{H}}_1}:\mu < 500\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A random sample of size four is taken from the distribution N(60, 36) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Calculate the probability that the sum of the sample values is less than 250.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 497.2, 502.0, 501.0, 498.6, 496.3, 499.1, 500.1, 497.7</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">using the GDC</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\overline x = 499.0\) , \({\sigma ^2} = 3.8(0)\) <strong><em>A1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept \(499\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p\)-value = 0.0950 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">since \(0.0950 > 0.05\) accept \({H_0}\) <strong><em>R1A1</em></strong></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({t_{calc}} = - 1.45\) , \({t_{critical}} = - 1.895\) for \(v = 7\) at 5 % level <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">since \({t_{calc}} > {t_{critical}}\) accept \({H_0}\) <em><strong>R1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">each \(X \sim {\rm{N}}(60,36)\) so \(\sum\limits_{n = 1}^4 {{X_n} \sim {\rm{N}}(4(60),4(36)) = {\rm{N}}(240,144)} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Pr}}({\rm{Sum}} < 250) = {\rm{Pr}}\left( {z < \frac{{250 - 240}}{{12}} = \frac{5}{6}} \right)\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)(A1)</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.798\) (by GDC) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Accept \(0.797\) (tables).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> Answer only is awarded <em><strong>M0A0A0(M1)(A1)A1</strong></em>.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 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;">(a)(i) Very few mistakes were made in this question, although sometimes variance and standard deviation were confused. Why both variance and standard deviation are needed might be something that teachers could explore. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Again there were no serious problems although some candidates fail to show all the important parameters such as degrees of freedom. </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 was found to be relatively straightforward except for using the correct variance of \(144\). It would be useful here to make clear the distinction between the sum of random variables and a multiple of a random variable. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The discrete random variables \({X_n},{\text{ }}n \in {\mathbb{Z}^ + }\) have probability generating functions given by \({G_n}(t) = \frac{t}{n}\left( {\frac{{{t^n} - 1}}{{t - 1}}} \right)\).</p>
</div>
<div class="specification">
<p class="p1">Let \({X_{n - 1}}\) and \({X_{n + 1}}\) be independent.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the formula for the sum of a finite geometric series to show that</p>
<p class="p1">\[{\text{P}}({X_n} = k) = \left\{ {\begin{array}{*{20}{l}} {\frac{1}{n}}&{{\text{for }}1 \leqslant k \leqslant n} \\ 0&{{\text{otherwise}}} \end{array}.} \right.\]</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>Find \({\text{E}}({X_n})\).</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 set of values of \(n\) <span class="s1">for which \({\text{E}}({X_{n - 1}} \times {X_{n + 1}}) < 2n\).</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 class="p1">using \(\left( {\frac{{{t^n} - 1}}{{t - 1}}} \right) = 1 + t + {t^2} + \ldots {t^{n - 1}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({G_n}(t) = 0 + \frac{t}{n} + \frac{{{t^2}}}{n} + \frac{{{t^3}}}{n} + \ldots \frac{{{t^n}}}{n} + 0 \times {t^{n + 1}} + 0 \times \ldots \) </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span><em>A1 </em></strong>for the non-zero terms, <strong><em>A1 </em></strong>for the observation that all other terms are zero.</p>
<p class="p2"> </p>
<p class="p1">the statement that the coefficient of \({t^k}\) gives \({\text{P}}({X_n} = k)\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p3">hence \({\text{P}}({X_n} = k) = \left\{ {\begin{array}{*{20}{l}} {\frac{1}{n}}&{{\text{for }}1 \leqslant k \leqslant n} \\ 0&{{\text{otherwise}}} \end{array}} \right.\) <strong><em>AG</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"><span class="s1">\({\text{E}}({X_n}) = 0 \times 0 + 1 \times \frac{1}{n} + 2 \times \frac{1}{n} + 3 \times \frac{1}{n} + \ldots n \times \frac{1}{n} + (n + 1) \times 0 + \ldots \times 0\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p2">\( = \frac{1}{n} \times \sum\limits_{k = 1}^{k = n} k \)</p>
<p class="p2"><span class="Apple-converted-space">\( = \frac{1}{n} \times \frac{1}{2}n(n + 1) = \frac{{n + 1}}{2}\) </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept use of \(G'(1)\).</p>
<p class="p3"> </p>
<p class="p1"><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">\({X_{n - 1}}\) and \({X_{n + 1}}\) are independent \( \Rightarrow {\text{E}}({X_{n - 1}} \times {X_{n + 1}}) = {\text{E}}({X_{n - 1}}) \times {\text{E}}({X_{n + 1}})\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = \frac{n}{2} \times \frac{{n + 2}}{2}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">required to solve \({n^2} < 6n{\text{ }}({\text{or }}n + 2 < 8)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2">solution: \((2 \leqslant ){\text{ }}n < 6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><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 class="p1">Again this was a question that tested candidates and although many started only a very limited number were able to make significant progress. Part (a) was rarely done well with most candidates not understanding what was required. There was a little more success with part (b) but a number of candidates attempted methods that were not going to lead to anything meaningful. Most candidates did not understand what was required from part (c) and few correct answers were seen, even taking into account the fact that follow through marks could be awarded from (b).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Again this was a question that tested candidates and although many started only a very limited number were able to make significant progress. Part (a) was rarely done well with most candidates not understanding what was required. There was a little more success with part (b) but a number of candidates attempted methods that were not going to lead to anything meaningful. Most candidates did not understand what was required from part (c) and few correct answers were seen, even taking into account the fact that follow through marks could be awarded from (b).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Again this was a question that tested candidates and although many started only a very limited number were able to make significant progress. Part (a) was rarely done well with most candidates not understanding what was required. There was a little more success with part (b) but a number of candidates attempted methods that were not going to lead to anything meaningful. Most candidates did not understand what was required from part (c) and few correct answers were seen, even taking into account the fact that follow through marks could be awarded from (b).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Bill buys two biased coins from a toy shop.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shopkeeper claims that when one of the coins is tossed, the probability of </span><span style="font-family: times new roman,times; font-size: medium;">obtaining a head is \(0.6\). Bill wishes to test this claim by tossing the coin \(250\) </span><span style="font-family: times new roman,times; font-size: medium;">times and counting the number of heads obtained.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) State suitable hypotheses for this test.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) He obtains \(140\) heads. Find the \(p\)-value of this result and determine whether </span><span style="font-family: times new roman,times; font-size: medium;">or not it supports the shopkeeper’s claim at the \(5\%\) level of significance.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bill tosses the other coin a large number of times and counts the number of </span><span style="font-family: times new roman,times; font-size: medium;">heads obtained. He correctly calculates a \(95\%\) confidence interval for the </span><span style="font-family: times new roman,times; font-size: medium;">probability that when this coin is tossed, a head is obtained. This is calculated as </span><span style="font-family: times new roman,times; font-size: medium;">[\(0.35199\), \(0.44801\)] where the end-points are correct to five significant figures.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Determine</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the number of times the coin was tossed;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the number of heads obtained.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \({{\rm{H}}_0}:p = 0.6\) ; \({{\rm{H}}_1}:p \ne 0.6\) <strong><em>A1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>EITHER</strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">using a normal approximation, \(p\)-value \( = 0.197\) <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \(0.0984\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the shopkeeper’s claim is supported <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">because \(0.197 > 0.05\) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">using binomial distribution, \(p\)-value \( = 0.221\) <strong><em>A2</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \(0.110\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">the shopkeeper’s claim is supported <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">because \(0.221 > 0.05\) <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Follow through the candidate’s \(p\)-value for <strong><em>A1R1</em></strong>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept \(p\)-values correct to two significant figures.</span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></em></strong></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) \(\hat p = \frac{{0.35199 + 0.44801}}{2} = 0.4\) <em><strong> A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">width of CI \( = 3.92\sqrt {\frac{{0.4 \times 0.6}}{n}} \) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(3.92\sqrt {\frac{{0.4 \times 0.6}}{n}} = 0.44801 - 0.35199 = 0.096(02)\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">solving,</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = {\left( {\frac{{3.92}}{{0.096(02)}}} \right)^2} \times 0.24\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 400\) <em><strong>A1</strong></em></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) \(x = n\widehat p = 400 \times 0.4 = 160\) <strong><em>M1A1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> <br></span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></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;">Part (a) was well answered in general, using the calculator either to carry out a significance test on proportions or to find the \(p\)-value directly using the binomial distribution. Some candidates gave their conclusion in the form "Accept H<sub>0</sub>", this was not accepted since the question asked whether or not the shopkeeper’s claim was supported and a direct answer to this question was required. </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;">Part (b) caused problems for some candidates who were unsure how to proceed. Some candidates used a trial and error method which involved showing that \(\hat p = 0.4\) and then using their calculator to find the confidence interval for appropriate pairs of values for \(n\) and \(p\) until reaching \(400,160\). This was accepted as a valid method although it is not recommended as a general method since its success was based upon the value of \(n\) being one that would probably be tested. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">All members of a large athletics club take part in an annual shotput competition.</p>
<p class="p1">The following data give the distances achieved, in metres, by a random selection of <span class="s1">10 </span>members of the club in the 2016 competition</p>
<p class="p2" style="text-align: center;">11.8, 14.3, 13.8, 10.3, 14.9, 14.7, 12.4, 13.9, 14.0, 11.7</p>
<p class="p1">The president of the club wishes to test whether these data provide evidence that distances achieved have increased since the 2015 competition, when the mean result for the club was <span class="s1">12.4 m</span>. You may assume that the distances achieved follow a normal distribution with mean \(\mu \), variance \({\sigma ^2}\), and that the membership of the club has not changed from 2015 to 2016.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State suitable hypotheses.</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"><span class="s1">(i) <span class="Apple-converted-space"> </span>Give a reason why a \(t\) </span>test is appropriate and write down its degrees of freedom.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the critical region for testing at each of the <span class="s2">5% </span>and <span class="s2">10% </span>significance levels.</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>Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of the test statistic.</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">State the conclusions that the president of the club should reach from this test, <span class="s1">giving reasons for your answer.</span></p>
<div class="marks">[2]</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">\({H_0}:{\text{ }}\mu = 12.4;{\text{ }}{H_1}:{\text{ }}\mu > 12.4\) </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"> </span>\(t\) test is appropriate because the variance (standard deviation) is unknown <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(v = 9\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(t \geqslant 1.83{\text{ }}(5\% );{\text{ }}t \geqslant 1.38{\text{ }}(10\% )\)</span> <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept strict inequalities.</p>
<p class="p3"> </p>
<p class="p1"><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"> </span>unbiased estimate of \(\mu \) <span class="s1">is 13.18 <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong><span class="s1">Accept 13.2</span>.</p>
<p class="p2"> </p>
<p class="p1">unbiased estimate of \({\sigma ^2}\) <span class="s1">is 2.34 \(({1.531^2})\) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \({t_{{\text{calc}}}} = \left( {\frac{{13.18 - 12.4}}{{\frac{{1.531}}{{\sqrt {10} }}}}} \right) = 1.61{\text{ or }}1.65\)</span> <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">as \(1.38 < 1.61 < 1.83\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"><span class="s1">evidence to accept \({H_0}\) </span>at the 5% level, but not at the 10% <span class="s1">level <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p3"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept the use of the \(p\)<span class="s2">-</span>value \( = 0.0708\)<span class="s2">.</span></p>
<p class="p4"> </p>
<p class="p1"><strong><em>[2 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">Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates had an understanding of how to start the question, but only a small number were able to gain full marks. It appeared that many candidates were used to finding \(p\)-values, but showed a lack of understanding when asked to find the critical regions and test a \(t\)-value. The conclusions required in part (d) were often too brief and/or poorly expressed.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sami is undertaking market research on packets of soap powder. He considers the brand “Gleam”. The weight of the contents of a randomly chosen packet of “Gleam” follows a normal distribution with mean 750 grams and standard deviation 20 grams.</p>
<p class="p1">The weight of the packaging follows a different normal distribution with mean 40 grams and standard deviation 5 <span class="s1">grams.</span></p>
<p class="p2">Find:</p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the probability that a randomly chosen packet of “Gleam” has a <strong>total </strong><span class="s2">weight exceeding 780 </span>grams.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>the probability that the total weight of the <strong>contents </strong><span class="s2">of five randomly chosen packets of “Gleam” exceeds 3800 grams.</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 class="p1">Sami now considers the brand “Bright”. The weight of the contents of a randomly chosen packet of “Bright” follow a normal distribution with mean <span class="s1">650 </span>grams and standard deviation <span class="s1">16 </span>grams. Find the probability that the <strong>contents </strong>of six randomly chosen packets of “Bright” weigh more than the <strong>contents </strong>of five randomly chosen packets of “Gleam”.</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"><strong>Note:<span class="Apple-converted-space"> </span></strong>In all parts accept answers which round to the correct 2sf answer.</p>
<p class="p2"> </p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>contents: \(X \sim N(750,{\text{ }}400)\)</p>
<p class="p1">packaging: \(Y \sim N(40,{\text{ }}25)\)</p>
<p class="p1">consider \(X + Y\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{E}}(X + Y) = 790\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{Var}}(X + Y) = 425\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{P}}(X + Y > 780) = 0.686\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"> </p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Let \({X_1} + {X_2} + {X_3} + {X_4} + {X_5} = A\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\({\text{E}}(A) = 5{\text{E}}(X) = 3750\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{Var}}(A) = 5{\text{Var}}(X) = 2000\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{P}}(A > 3800) = 0.132\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: </strong><span class="s1">Condone the notation \(A = 5X\) </span>if the variance is correct, <strong><em>M0 </em></strong>if not</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">contents of Bright: \(B \sim N(650,{\text{ }}256)\)</p>
<p class="p1">let \(G = {B_1} + {B_2} + {B_3} + {B_4} + {B_5} + {B_6} - ({X_1} + {X_2} + {X_3} + {X_4} + {X_5})\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\({\text{E}}(G) = 6 \times 650 - 5 \times 750 = 150\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{Var}}(G) = 6 \times 256 + 5 \times 400 = 3536\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\({\text{P}}(G > 0) = 0.994\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: </strong><span class="s1">Condone the notation \(G = 6B - 5X\) </span>if the variance is correct, <strong><em>M0 </em></strong>if not</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">Part (a)(i) was well answered in general. In (a)(ii) and (b), however, many candidates made the fairly common error of confusing \(\sum\limits_{i = 1}^n {{X_i}} \) with \(nX\) which gives an incorrect variance. This is an important distinction which needs to be emphasized.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a)(i) was well answered in general. In (a)(ii) and (b), however, many candidates made the fairly common error of confusing \(\sum\limits_{i = 1}^n {{X_i}} \) with \(nX\) which gives an incorrect variance. This is an important distinction which needs to be emphasized.</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 random variables \(X\), \(Y\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \). The following table gives a random sample from this distribution.</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_16.35.18.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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine the value of \(r\), the product moment correlation coefficient of this sample.</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) Write down hypotheses in terms of \(\rho \) which would enable you to test whether or not \(X\) and \(Y\) are independent.</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 the <em>p</em>-value of the above sample and state your conclusion at the 5% significance level. Justify your answer.</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) Determine the equation of the regression line of \(y\) on \(x\).</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) State whether or not this equation can be used to obtain an accurate prediction of the value of \(y\) for a given value of \(x\). Give a reason for your answer.</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) \(r = - 0.163\) <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;"><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) (i) \({{\text{H}}_0}:\rho = 0:{{\text{H}}_1}:\rho \ne 0\) <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;">(ii) \(t = r\sqrt {\frac{{n - 2}}{{1 - {r^2}}}} = - 0.468 \ldots \) <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;">\({\text{DF}} = 8\) <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;">\(p{\text{-value}} = 2 \times 0.326 \ldots = 0.652\) <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 \(0.652 > 0.05\), we accept \({{\text{H}}_0}\) <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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>(A1)(A1)A0 </em></strong>if the <em>p</em>-value is given as \(0.326\) without prior working.</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Follow through their <em>p</em>-value for the <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 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;">(c) (i) \(y = - 0.257x + 5.22\) <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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept answers which round to \(–0.26\) and \(5.2\).</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) no, because \(X\) and \(Y\) have been shown to be independent (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;"><strong><em>[2 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="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 shows the probability distribution of the discrete random variable \(X\).</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_15.57.15.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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the probability generating function of \(X\) is given by</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;">\[G(t) = \frac{{t{{(1 + t)}^2}}}{4}.\]</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) Given that \(Y = {X_1} + {X_2} + {X_3} + {X_4}\), where \({X_1},{\text{ }}{X_2},{\text{ }}{X_3},{\text{ }}{X_4}\) is a random sample from the distribution of \(X\),</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) state the probability generating function of \(Y\);</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;">(ii) hence find the value of \({\text{P}}(Y = 8)\).</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) \(G(t) = \frac{1}{4}t + \frac{1}{2}{t^2} + \frac{1}{4}{t^3}\) <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;">\( = \frac{{t{{(1 + t)}^2}}}{4}\) <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;"><strong><em> </em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \({\text{PGF of }} Y = {\left( {G(t)} \right)^4}\left( { = {{\left( {\frac{{t{{(1 + t)}^2}}}{4}} \right)}^4}} \right)\) <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) \({\text{P}}(Y = 8) = {\text{coefficient of }}{t^8}\) <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;">\( = \frac{{^8{{\text{C}}_4}}}{{256}}\) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{35}}{{128}} (0.273)\) <em><strong>A1</strong></em></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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(0.27\) or answers that round to \(0.273\).</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \({X_k}\) be independent normal random variables, where \({\rm{E}}({X_k}) = \mu \) and </span><span style="font-family: times new roman,times; font-size: medium;">\(Var({X_k}) = \sqrt k \) , for \(k = 1,2, \ldots \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The random variable \(Y\) is defined by \(Y = \sum\limits_{k = 1}^6 {\frac{{{{( - 1)}^{k + 1}}}}{{\sqrt k }}} {X_k}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({\rm{E}}(Y)\) in the form \(p\mu \) , where \(p \in \mathbb{R}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(k\) if \({\rm{Var}}({X_k}) < {\rm{Var}}(Y) < {\rm{Var}}({X_{k + 1}})\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A random sample of \(n\) values of \(Y\) was found to have a mean of \(8.76\).</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Given that \(n = 10\) , determine a \(95\%\) confidence interval for \(\mu \) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) The width of the confidence interval needs to be halved. Find the </span><span style="font-family: times new roman,times; font-size: medium;">appropriate value of \(n\) .</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><span style="font-family: times new roman,times; font-size: medium;">(i) \({\rm{E}}(Y) = \frac{1}{{\sqrt 1 }}\mu - \frac{1}{{\sqrt 2 }}\mu + \frac{1}{{\sqrt 3 }}\mu - \frac{1}{{\sqrt 4 }}\mu + \frac{1}{{\sqrt 5 }}\mu - \frac{1}{{\sqrt 6 }}\mu \) <em><strong>(M1)</strong></em><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 0.409\) \((209)\mu \) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept answers which round to \(0.41\). </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) \(Var(Y) = \frac{1}{{\sqrt 1 }} + \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 3 }} + \frac{1}{{\sqrt 4 }} + \frac{1}{{\sqrt 5 }} + \frac{1}{{\sqrt 6 }}\) <strong><em>(M1)</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 3.64\) \((3.6399 \ldots )\) <strong><em> A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(({\rm{Var}}({X_{13}}) = 3.61;{\rm{Var}}({X_{14}}) = 3.74) \Rightarrow k = 13\) <em><strong>A1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></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) \(95\%\) CI for is \({\rm{E}}(Y)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(8.76 \pm 1.96\sqrt {\frac{{3.6399 \ldots }}{{10}}} \) <em><strong>(M1)</strong> </em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \left[ {7.58,9.94} \right]\) <strong><em>A1A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept \(\left[ {7.6,9.9} \right]\) . </span><span style="font-family: times new roman,times; font-size: medium;">Do not penalize answers given to more than 3sf. </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;">Since \(\mu = \frac{{{\rm{E}}(Y)}}{{0.409 \ldots }}\) , CI for \(\mu \) is \(\left[ {18.5,24.3} \right]\) <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Do not penalize answers given to more than 3sf. </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) width of a CI is inversely proportional to the square root of \(n\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so \(n = 40\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em> </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></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;">This question proved to be the most difficult on the paper and few fully correct answers were seen. In part (a) (i) many candidates did know how to find the answers in terms of \(\mu\). Very few candidates successfully completed part (a) (ii).</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 proved to be the most difficult on the paper and few fully correct answers were seen. In part b) (i) a number of candidates made some progress, but few realised or knew how to convert the confidence interval for \({\rm{E}}(Y)\</span><span style="font-family: times new roman,times; font-size: medium;"> into a confidence interval for \(\mu\). For those who persevered to the end of the question, there was a reasonable degree of success in part (b) (ii).</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 weights, in grams, of 10 apples were measured with the following results:</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> \(212.2\) \(216.9\) \(209.0\) \(215.5\) \(215.9\) \(213.5\) \(208.9\) \(213.8\) \(216.4\) \(209.9\)</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;">You may assume that this is a random sample from a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\).</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;">(a) Giving all your answers correct to four significant figures,</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) determine unbiased estimates for \(\mu \) and \({\sigma ^2}\);</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) find a \(95\%\) confidence interval for \(\mu \).</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;">Another confidence interval for \(\mu \), \([211.5, 214.9]\), was calculated using the above data.</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) Find the confidence level of this interval.</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) (i) \(\bar x = {\text{213.2}}\) <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;">\(s = 3.0728 \ldots \) <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;">\({s^2} = 9.442\) <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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \([211.0, 215.4]\) <strong><em>A1A1</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept \(211\) in place of \(211.0\).</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Apart from the above note, accept any answers which round to the correct 4 significant figure answers.</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 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;">(b) use of the fact that the width of the interval is \(2t \times \frac{s}{{\sqrt n }}\) <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;">so that \(3.4 = 2t \times \frac{{3.0728 \ldots }}{{\sqrt {10} }}\) <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;">\(t = 1.749\) <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;">degrees of freedom \( = 9\) <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;">\({\text{P}}(T > 1.749) = 0.0571\) <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;">confidence level \( = 1 - 2 \times 0.0571 = 0.886{\text{ }}(88.6\% )\) <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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award the \({\text{DF}} = 9\) <strong><em>(A1) </em></strong>mark if the following line has \(0.00337\) on the RHS.</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept any answer which rounds to \(88.6\%\).</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 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 sample of size 100 is taken from a normal population with unknown mean <em>μ</em> and known variance 36.</p>
</div>
<div class="specification">
<p>Another investigator decides to use the same data to test the hypotheses <em>H</em><sub>0</sub> : <em>μ</em> = 65 , <em>H<span style="font-size: 11.6667px;">1</span></em> : <em>μ</em> = 67.9.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>An investigator wishes to test the hypotheses <em>H</em><sub>0</sub> : <em>μ</em> = 65, <em>H</em><sub>1</sub> : <em>μ</em> > 65.</p>
<p>He decides on the following acceptance criteria:</p>
<p>Accept <em>H</em><sub>0</sub> if the sample mean \(\bar x\) ≤ 66.5</p>
<p>Accept <em>H</em><sub>1</sub> if \(\bar x\) > 66.5</p>
<p>Find the probability of a Type I error.</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>She decides to use the same acceptance criteria as the previous investigator. Find the probability of a Type II error.</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>Find the critical value for \({\bar x}\) if she wants the probabilities of a Type I error and a Type II error to be equal.</p>
<div class="marks">[3]</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>\(\bar X \sim {\text{N}}\left( {\mu ,\,\frac{{{\sigma ^2}}}{n}} \right)\)</p>
<p>\(\bar X \sim {\text{N}}\left( {65,\,\frac{{36}}{{100}}} \right)\) <em><strong>(A1)</strong></em></p>
<p>P(Type I Error) \( = {\text{P}}\left( {\bar X > 66.5} \right)\) <em><strong> (M1)</strong></em></p>
<p>= 0.00621 <em><strong>A1</strong></em></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>P(Type II Error) = P(accept <em>H</em><sub>0 </sub>| <em>H<sub>1</sub> </em>is true)</p>
<p>\( = {\text{P}}\left( {\bar X \leqslant 66.5\left| {\mu = 67.9} \right.} \right)\) <em><strong>(M1)</strong></em></p>
<p>\( = {\text{P}}\left( {\bar X \leqslant 66.5} \right)\) when \(\bar X \sim {\text{N}}\left( {67.9,\,\frac{{36}}{{100}}} \right)\) <em><strong>(M1)</strong></em></p>
<p>= 0.00982 <em><strong>A1</strong></em></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>the variances of the distributions given by <em>H</em><sub>0</sub> and <em>H<sub>1</sub></em> are equal, <strong> (R1)</strong></p>
<p>by symmetry the value of \({\bar x}\) lies midway between 65 and 67.9 <em><strong>(M1)</strong></em></p>
<p>\( \Rightarrow \bar x = \frac{1}{2}\left( {65 + 67.9} \right) = 66.45\) <em><strong>A1</strong></em></p>
<p><img 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"></p>
<p><em><strong>[3 marks]</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>