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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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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