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</div><h2>HL Paper 3</h2><div class="specification">
<p>The weights, <em>X</em> kg, of the males of a species of bird may be assumed to be normally distributed with mean 4.8 kg and standard deviation 0.2 kg.</p>
</div>
<div class="specification">
<p>The weights, <em>Y</em> kg, of female birds of the same species may be assumed to be normally distributed with mean 2.7 kg and standard deviation 0.15 kg.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a randomly chosen male bird weighs between 4.75 kg and 4.85 kg.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two randomly chosen male birds and three randomly chosen female birds are placed on a weighing machine that has a weight limit of 18 kg. Find the probability that the total weight of these five birds is greater than the weight limit.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables \(U,{\text{ }}V\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \).</p>
</div>
<div class="specification">
<p>A random sample of 12 observations on <em>U</em>, <em>V</em> is obtained to determine whether there is a correlation between <em>U and</em> <em>V</em>. The sample product moment correlation coefficient is denoted by <em>r</em>. A test to determine whether or not <em>U</em>, <em>V</em> are independent is carried out at the 1% level of significance.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to investigate whether or not \(U\), \(V\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of \(|r|\) for which the test concludes that \(\rho \ne 0\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A biased cubical die has its faces labelled \(1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5\) and \(6\). The probability of rolling a \(6\) <span class="s1">is \(p\), with equal probabilities for the other scores.</span></p>
<p class="p2">The die is rolled once, and the score \({X_1}\) is noted.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \({\text{E}}({X_1})\)<span class="s1">.</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence obtain an unbiased estimator for \(p\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The die is rolled a second time, and the score \({X_2}\) is noted.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(k({X_1} - 3) + \left( {\frac{1}{3} - k} \right)({X_2} - 3)\) is also an unbiased estimator for \(p\) for all values of \(k \in \mathbb{R}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value for \(k\), which maximizes the efficiency of this estimator.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">It is known that the standard deviation of the heights of men in a certain country is \(15.0\) cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">One hundred men from that country, selected at random, had their heights measured.</p>
<p class="p2"><span class="s1">The mean of this sample was \(185\) cm. Calculate a \(95\% \) </span>confidence interval for the mean height of the population.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A second random sample of size \(n\) is taken from the same population. Find the minimum value of \(n\) <span class="s1">needed for the width of a \(95\% \) confidence interval to be less than \(3\) cm.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The strength of beams compared against the moisture content of the beam is indicated in the following table. You should assume that strength and moisture content are each normally distributed.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-21_om_17.54.38.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the product moment correlation coefficient for these data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Perform a two-tailed test, at the \(5\% \) level of significance, of the hypothesis that strength is independent of moisture content.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If the moisture content of a beam is found to be \(9.5\), use the appropriate regression line to estimate the strength of the beam.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Anna cycles to her new school. She records the times taken for the first ten days with the following results (in minutes).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">12.4 13.7 12.5 13.4 13.8 12.3 14.0 12.8 12.6 13.5</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that these times are a random sample from the \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\) distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Calculate a 95 % confidence interval for \(\mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Before Anna calculated the confidence interval she thought that the value of \(\mu \) would be 12.5. In order to check this, she sets up the null hypothesis \({{\text{H}}_0}:\mu = 12.5\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use the above data to calculate the value of an appropriate test statistic. Find the corresponding <em>p</em>-value using a two-tailed test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Interpret your <em>p</em>-value at the 1 % level of significance, justifying your conclusion.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has a geometric distribution with parameter <em>p </em>.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({\text{P}}(X \leqslant n) = 1 - {(1 - p)^n},{\text{ }}n \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Deduce an expression for \({\text{P}}(m < X \leqslant n)\,,{\text{ }}m\,,{\text{ }}n \in {\mathbb{Z}^ + }\) and <em>m </em>< <em>n </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>p </em>= 0.2, find the least value of <em>n </em>for which \({\text{P}}(1 < X \leqslant n) > 0.5\,,{\text{ }}n \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question you may assume that these data are a random sample from a bivariate normal distribution, with population product moment correlation coefficient \(\rho \).</p>
<p class="p1">Richard wishes to do some research on two types of exams which are taken by a large number of students. He takes a random sample of the results of <span class="s1">10 </span>students, which are shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-02_om_14.09.18.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SP/01"></p>
</div>
<div class="specification">
<p class="p1">Using these data, it is decided to test, at the <span class="s1">1% </span>level, the null hypothesis \({H_0}:\rho = 0\) against the alternative hypothesis \({H_1}:\rho > 0\).</p>
</div>
<div class="specification">
<p class="p1">Richard decides to take the exams himself. He scored <span class="s1">11 </span>on Exam 1 but his result on Exam 2 was lost.</p>
</div>
<div class="specification">
<p class="p1">Caroline believes that the population mean mark on Exam 2 is <span class="s1">6 </span>marks higher than the population mean mark on Exam 1. Using the original data from the <span class="s1">10 </span>students, it is decided to test, at the <span class="s1">5% </span>level, this hypothesis against the alternative hypothesis that the mean of the differences, \({\text{d}} = {\text{exam 2 mark }} - {\text{ exam 1 mark}}\), is less than <span class="s1">6 </span>marks.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For these data find the product moment correlation coefficient, \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of the test statistic (including any parameters).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the \(p\)<span class="s1">-value for the test.</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>State the conclusion, in the context of the question, with the word “correlation” in your answer. Justify your answer.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using a suitable regression line, find an estimate for his score on Exam 2, giving your <span class="s1">answer to the nearest integer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of your test statistic (including any parameters).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the \(p\)<span class="s1">-value.</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>State the conclusion, justifying the answer.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable \(X\) is distributed with mean \(\mu \) and variance \({\sigma ^2}\). Two independent random samples of sizes \({n_1}\) and \({n_2}\) are taken from the distribution of \(X\). The sample means are \({\bar X_1}\) and \({\bar X_2}\) respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(U = a{\bar X_1} + (1 - a){\bar X_2},{\text{ }}a \in \mathbb{R}\), is an unbiased estimator of \(\mu \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{Var}}(U) = {a^2}\frac{{{\sigma ^2}}}{{{n_1}}} + {(1 - a)^2}\frac{{{\sigma ^2}}}{{{n_2}}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of \({n_1}\) and \({n_2}\), an expression for \(a\) which gives the most efficient estimator of this form.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find an expression for the most efficient estimator and interpret the result.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A teacher has forgotten his computer password. He knows that it is either six of the letter J followed by two of the letter R (<em>i.e.</em> JJJJJJRR) or three of the letter J followed by four of the letter R (<em>i.e.</em> JJJRRRR). The computer is able to tell him at random just two of the letters in his password.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The teacher decides to use the following rule to attempt to find his password.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the computer gives him a J and a J, he will accept the null hypothesis that his password is JJJJJJRR.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Otherwise he will accept the alternative hypothesis that his password is JJJRRRR.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Define a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the probability that the teacher makes a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Define a Type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the probability that the teacher makes a Type II error.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has the negative binomial distribution NB(3, <em>p</em>) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x)\) denote the probability that <em>X</em> takes the value <em>x</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down an expression for \(f(x)\) , and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\ln f(x) = 3\ln \left( {\frac{p}{{1 - p}}} \right) + \ln (x - 1) + \ln (x - 2) + x\ln (1 - p) - \ln 2{\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the domain of <em>f</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) The domain of <em>f</em> is extended to \(]2,{\text{ }}\infty [\) . Show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{f'(x)}}{{f(x)}} = \frac{1}{{x - 1}} + \frac{1}{{x - 2}} + \ln (1 - p){\text{ .}}\)</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jo has a biased coin which has a probability of 0.35 of showing heads when tossed. She tosses this coin successively and the \({3^{{\text{rd}}}}\) head occurs on the \({Y^{{\text{th}}}}\) toss. Use the result in part (a)(iii) to find the most likely value of <em>Y</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Jenny and her Dad frequently play a board game. Before she can start Jenny has to throw a “six” on an ordinary six-sided dice. Let the random variable <em>X </em>denote the number of times Jenny has to throw the dice in total until she obtains her first “six”.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If the dice is fair, write down the distribution of <em>X </em>, including the value of any parameter(s).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down E(<em>X </em>) for the distribution in part (a).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the distribution of </span><em style="font-family: 'times new roman', times; font-size: medium;">Y </em><span style="font-family: 'times new roman', times; font-size: medium;">, including the value of any parameter(s).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of </span><em style="font-family: 'times new roman', times; font-size: medium;">y </em><span style="font-family: 'times new roman', times; font-size: medium;">such that \({\text{P}}(Y = y) = \frac{1}{{36}}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Before Jenny’s Dad can start, he has to throw two “sixes” using a fair, ordinary six-sided dice. Let the random variable <em>Y </em>denote the total number of times Jenny’s Dad has to throw the dice until he obtains his second “six”.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(Y \leqslant 6)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A shop sells apples and pears. The weights, in grams, of the apples may be assumed to have a \({\text{N}}(200,{\text{ 1}}{{\text{5}}^2})\) distribution and the weights of the pears, in grams, may be assumed to have a \({\text{N}}(120,{\text{ 1}}{{\text{0}}^2})\) distribution.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that the weight of a randomly chosen apple is more than double the weight of a randomly chosen pear.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) A shopper buys 3 apples and 4 pears. Find the probability that the total weight is greater than 1000 grams.</span></p>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable \(T\) has a probability density function defined by</p>
<p style="text-align: center;">\(f(t) = \left\{ {\begin{array}{*{20}{c}} {\frac{{t(4 - {t^2})}}{4}}&{0 \leqslant t \leqslant 2} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the cumulative distribution function \(F(t)\), for \(0 \leqslant t \leqslant 2\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(F(t)\) for \(0 \leqslant t \leqslant 2\), clearly indicating the coordinates of the endpoints.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(P(T < a) = 0.75\), find the value of \(a\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables \({X_1}\) and \({X_2}\) are a random sample from \({\text{N}}(\mu ,{\text{ 2}}{\sigma ^2})\). The random variables \({Y_1}\), \({Y_2}\) and \({Y_3}\) are a random sample from \({\text{N}}(2\mu ,{\text{ }}{\sigma ^2})\).</p>
<p>The estimator \(U\) is used to estimate \(\mu \) where \(U = a({X_1} + {X_2}) + b({Y_1} + {Y_2} + {Y_3})\) and \(a\), \(b\) are constants.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(U\) is unbiased, show that \(2a + 6b = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{Var}}(U) = (39{b^2} - 12b + 1){\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of \(a\) and the value of \(b\) which give the best unbiased estimator of this form, giving your answers as fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the variance of this best unbiased estimator.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>represents the height of a wave on a particular surf beach.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">It is known that <em>X </em>is normally distributed with unknown mean \(\mu \) (metres) and known variance \({\sigma ^2} = \frac{1}{4}{\text{ (metre}}{{\text{s}}^2}{\text{)}}\) . Sally wishes to test the claim made in a surf guide that \(\mu = 3\) against the alternative that \(\mu < 3\) . She measures the heights of 36 waves and calculates their sample mean \({\bar x}\) . She uses this value to test the claim at the 5 % level.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find a simple inequality, of the form \(\bar x < A\) , where <em>A </em>is a number to be determined to 4 significant figures, so that Sally will reject the null hypothesis, that \(\mu = 3\) , if and only if this inequality is satisfied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Define a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Define a Type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Write down the probability that Sally makes a Type I error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) The true value of \(\mu \) is 2.75. Calculate the probability that Sally makes a Type II error.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>Y </em>represents the height of a wave on another surf beach. It is known that <em>Y </em>is normally distributed with unknown mean \(\mu \) (metres) and unknown variance \({\sigma ^2}{\text{ (metre}}{{\text{s}}^2}{\text{)}}\) . David wishes to test the claim made in a surf guide that \(\mu = 3\) against the alternative that \(\mu < 3\) . He is also going to perform this test at the 5 % level. He measures the heights of 36 waves and finds that the sample mean, \(\bar y = 2.860\) and the unbiased estimate of the population variance, \(s_{n - 1}^2 = 0.25\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the name of the test that David should perform.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the conclusion of David’s test, justifying your answer by giving the <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Using David’s results, calculate the 90 % confidence interval for \(\mu \) , giving your answers to 4 significant figures.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A discrete random variable \(U\) follows a geometric distribution with \(p = \frac{1}{4}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(F(u)\), the cumulative distribution function of \(U\), for \(u = 1,{\text{ }}2,{\text{ }}3 \ldots \)</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, or otherwise, find the value of \(P(U > 20)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that the probability generating function of \(U\) is given by \({G_u}(t) = \frac{t}{{4 - 3t}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \({U_i} \sim {\text{Geo}}\left( {\frac{1}{4}} \right),{\text{ }}i = 1,{\text{ }}2,{\text{ }}3\), and that \(V = {U_1} + {U_2} + {U_3}\)<span class="s1">, find</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>\({\text{E}}(V)\);</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}(V)\);</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\({G_v}(t)\), the probability generating function of \(V\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A third random variable \(W\), has probability generating function \({G_w}(t) = \frac{1}{{{{(4 - 3t)}^3}}}\).</p>
<p class="p1">By differentiating \({G_w}(t)\), <span class="s1">find \({\text{E}}(W)\).</span></p>
<p class="p1"> </p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A third random variable \(W\), has probability generating function \({G_w}(t) = \frac{1}{{{{(4 - 3t)}^3}}}\).</p>
<p class="p1">Prove that \(V = W + 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable \(X\) has cumulative distribution function \(F\) given by \[F(x) = \left\{ {\begin{array}{*{20}{l}} {0,}&{x < 0} \\ {x{{\text{e}}^{x - 1}},}&{0 \leqslant x \leqslant 1.} \\ {1,}&{x > 2} \end{array}} \right.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine \(P(0.25 \leqslant X \leqslant 0.75)\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the median of \(X\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability density function \(f\) of \(X\) is given, for \(0 \leqslant x \leqslant 1\), by</p>
<p>\[f(x) = (x + 1){{\text{e}}^{x - 1}}.\]</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the mean and the variance of \(X\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the central limit theorem. </p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A random sample of 100 observations is obtained from the distribution of \(X\). If \(\bar X\) denotes the sample mean, use the central limit theorem to find an approximate value of \(P(\bar X > 0.65)\). Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Adam does the crossword in the local newspaper every day. The time taken by Adam, \(X\) <span class="s1">minutes, to complete the crossword is modelled by the normal distribution \({\text{N}}(22,{\text{ }}{5^2})\).</span></p>
</div>
<div class="specification">
<p class="p1">Beatrice also does the crossword in the local newspaper every day. The time taken by Beatrice, \(Y\) <span class="s1">minutes, to complete the crossword is modelled by the normal distribution \({\text{N}}(40,{\text{ }}{6^2})\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that, on a randomly chosen day, the probability that he completes the crossword in less than \(a\) <span class="s1">minutes is equal to 0.8</span>, find the value of \(a\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that the total time taken for him to complete five randomly chosen crosswords exceeds <span class="s1">120 </span>minutes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that, on a randomly chosen day, the time taken by Beatrice to complete the crossword is more than twice the time taken by Adam to complete the crossword. Assume that these two times are independent.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The random variables \(X\)<span class="s1">, </span>\(Y\) follow a bivariate normal distribution with product moment correlation coefficient \(\rho \).</p>
</div>
<div class="specification">
<p class="p1">A random sample of 10 <span class="s1">observations on \(X\)</span>, <span class="s1">\(Y\) was obtained and the value of \(r\)</span>, the sample product moment correlation coefficient, was calculated to be 0.486.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State suitable hypotheses to investigate whether or not \(X\)<span class="s1">, </span>\(Y\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Determine the \(p\)-value.</p>
<p class="p1">(ii) State your conclusion at the <span class="s1">5% </span>significance level.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the equation of the regression line of \(y\) on \(x\) should not be used to predict the value of \(y\) corresponding to \(x = {x_0}\), where \({x_0}\) lies within the range of values of \(x\) in the sample.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">John rings a church bell <span class="s1">120 </span>times. The time interval, \({T_i}\), between two successive rings is a random variable with mean of <span class="s1">2 </span>seconds and variance of \(\frac{1}{9}{\text{ second}}{{\text{s}}^2}\).</p>
<p class="p1">Each time interval, \({T_i}\), is independent of the other time intervals. Let \(X = \sum\limits_{i = 1}^{119} {{T_i}} \) be the total time between the first ring and the last ring.</p>
</div>
<div class="specification">
<p class="p1">The church vicar subsequently becomes suspicious that John has stopped coming to ring the bell and that he is letting his friend Ray do it. When Ray rings the bell the time interval, \({T_i}\) has a mean of <span class="s1">2 </span>seconds and variance of \(\frac{1}{{25}}{\text{ second}}{{\text{s}}^2}\).</p>
<p class="p1">The church vicar makes the following hypotheses:</p>
<p class="p1"><span class="s1">\({H_0}\): </span>Ray is ringing the bell; \({H_1}\)<span class="s1">: </span>John is ringing the bell.</p>
<p class="p1">He records four values of \(X\). He decides on the following decision rule:</p>
<p class="p1">If \(236 \leqslant X \leqslant 240\) for all four values of \(X\) he accepts \({H_0}\), otherwise he accepts \({H_1}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p2">(i) <span class="Apple-converted-space"> \({\text{E}}(X)\)</span>;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> \({\text{Var}}(X)\)</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why a normal distribution can be used to give an approximate model for \(X\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Use this model to find the values of \(A\) </span>and \(B\) such that \({\text{P}}(A < X < B) = 0.9\), where \(A\) and \(B\) are symmetrical about the mean of \(X\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the probability that he makes a Type <span class="s1">II </span><span class="s2">error.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variables <em>X</em> , <em>Y</em> follow a bivariate normal distribution with product moment correlation coefficient <em>ρ</em>.</p>
</div>
<div class="specification">
<p>A random sample of 11 observations on <em>X</em>, <em>Y</em> was obtained and the value of the sample product moment correlation coefficient, <em>r</em>, was calculated to be −0.708.</p>
</div>
<div class="specification">
<p>The covariance of the random variables <em>U</em>, <em>V</em> is defined by</p>
<p style="text-align: center;">Cov(<em>U</em>, <em>V</em>) = E((<em>U</em> − E(<em>U</em>))(<em>V</em> − E(<em>V</em>))).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to investigate whether or not a negative linear association exists between <em>X</em> and <em>Y</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the <em>p</em>-value.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State your conclusion at the 1 % significance level.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Cov(<em>U</em>, <em>V</em>) = E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that if <em>U</em>, <em>V</em> are independent random variables then the population product moment correlation coefficient, <em>ρ</em>, is zero.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A smartphone’s battery life is defined as the number of hours a fully charged battery can be used before the smartphone stops working. A company claims that the battery life of a model of smartphone is, on average, 9.5 hours. To test this claim, an experiment is conducted on a random sample of 20 smartphones of this model. For each smartphone, the battery life, \(b\) hours, is measured and the sample mean, \({\bar b}\), calculated. It can be assumed the battery lives are normally distributed with standard deviation 0.4 hours.</p>
</div>
<div class="specification">
<p>It is then found that this model of smartphone has an average battery life of 9.8 hours.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses for a two-tailed test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the critical region for testing \({\bar b}\) at the 5 % significance level.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type II error.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Another model of smartphone whose battery life may be assumed to be normally distributed with mean <em>μ</em> hours and standard deviation 1.2 hours is tested. A researcher measures the battery life of six of these smartphones and calculates a confidence interval of [10.2, 11.4] for <em>μ</em>.</p>
<p>Calculate the confidence level of this interval.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has the negative binomial distribution NB(5, <em>p</em>), where <em>p</em> < 0.5, and \({\text{P}}(X = 10) = 0.05\). By first finding the value of <em>p</em>, find the value of \({\text{P}}(X = 11)\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of adult monkeys of a certain species are known to be normally distributed, the males with mean 30 kg and standard deviation 3 kg and the females with mean 20 kg and standard deviation 2.5 kg.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the weight of a randomly selected male is more than twice the weight of a randomly selected female.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two males and five females stand together on a weighing machine. Find the probability that their total weight is less than 175 kg.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A teacher decides to use the marks obtained by a random sample of 12 students in Geography and History examinations to investigate whether or not there is a positive association between marks obtained by students in these two subjects. You may assume that the distribution of marks in the two subjects is bivariate normal.</p>
</div>
<div class="specification">
<p>He gives the marks to Anne, one of his students, and asks her to use a calculator to carry out an appropriate test at the 5% significance level. Anne reports that the \(p\)-value is 0.177.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses for this investigation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in context, what conclusion should be drawn from this \(p\)-value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The teacher then asks Anne for the values of the \(t\)-statistic and the product moment correlation coefficient \(r\) produced by the calculator but she has deleted these. Starting with the \(p\)-value, calculate these values of \(t\) and \(r\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The heating in a residential school is to be increased on the third frosty day during the term. If the probability that a day will be frosty is 0.09, what is the probability that the heating is increased on the \({25^{{\text{th}}}}\) day of the term?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) On which day is the heating most likely to be increased?</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(X\) and \(Y\) are two random variables such that \({\text{E}}(X) = {\mu _X}\) and \({\text{E}}(Y) = {\mu _Y}\) then \({\text{Cov}}(X,{\text{ }}Y) = {\text{E}}\left( {(X - {\mu _X})(Y - {\mu _Y})} \right)\).</p>
<p class="p1">Prove that if \(X\) and \(Y\) are independent then \({\text{Cov}}(X,{\text{ }}Y) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In a particular company, it is claimed that the distance travelled by employees to work is independent of their salary. To test this, 20 randomly selected employees are asked about the distance they travel to work and the size of their salaries. It is found that the product moment correlation coefficient, \(r\), for the sample is \( - 0.35\).</p>
<p class="p1">You may assume that both salary and distance travelled to work follow normal distributions.</p>
<p class="p1">Perform a one-tailed test at the \(5\% \) significance level to test whether or not the distance travelled to work and the salaries of the employees are independent.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the recurrence relation</p>
<p style="text-align: center;">\({u_n} = 5{u_{n - 1}} - 6{u_{n - 2}},{\text{ }}{u_0} = 0\) and \({u_1} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for \({u_n}\) in terms of \(n\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For every prime number \(p > 3\), show that \(p|{u_{p - 1}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The students in a class take an examination in Applied Mathematics which consists of two papers. Paper 1 is in Mechanics and Paper 2 is in Statistics. The marks obtained by the students in Paper 1 and Paper 2 are denoted by \((x,{\text{ }}y)\) respectively and you may assume that the values of \((x,{\text{ }}y)\) form a random sample from a bivariate normal distribution with correlation coefficient \(\rho \) . The teacher wishes to determine whether or not there is a positive association between marks in Mechanics and marks in Statistics.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State suitable hypotheses.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The marks obtained by the 12 students who sat both papers are given in the following table.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 24px;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the product moment correlation coefficient for these data and state its <em>p</em>-value.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Interpret your <em>p</em>-value in the context of the problem.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">George obtained a mark of 63 on Paper 1 but was unable to sit Paper 2 because of illness. Predict the mark that he would have obtained on Paper 2.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Another class of 16 students sat examinations in Physics and Chemistry and the product moment correlation coefficient between the marks in these two subjects was calculated to be 0.524. Using a 1 % significance level, determine whether or not this value suggests a positive association between marks in Physics and marks in Chemistry.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has a Poisson distribution with unknown mean \(\mu \) . It is required to test the hypotheses</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({H_0}:\mu = 3\) against \({H_1}:\mu \ne 3\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>S</em> denote the sum of 10 randomly chosen values of <em>X</em> . The critical region is defined as \((S \leqslant 22) \cup (S \geqslant 38)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the significance level of the test.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the value of \(\mu \) is actually 2.5, determine the probability of a Type II error.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table gives the average yield of olives per tree, in kg, and the rainfall, in cm, for nine separate regions of Greece. You may assume that these data are a random sample from a bivariate normal distribution, with correlation coefficient \(\rho \).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-11_om_08.55.24.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A scientist wishes to use these data to determine whether there is a positive correlation between rainfall and yield.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State suitable hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the product moment correlation coefficient for these data.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Determine the associated <em>p</em>-value and comment on this value in the context of the question.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the equation of the regression line of <em>y </em>on <em>x</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Hence, estimate the yield per tree in a tenth region where the rainfall was 19 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Determine the angle between the regression line of <em>y </em>on <em>x </em>and that of <em>x </em>on <em>y </em>. Give your answer to the nearest degree.</span></p>
</div>
<br><hr><br><div class="specification">
<p>A farmer sells bags of potatoes which he states have a mean weight of 7 kg . An inspector, however, claims that the mean weight is less than 7 kg . In order to test this claim, the inspector takes a random sample of 12 of these bags and determines the weight, \(x\) kg , of each bag. He finds that \[\sum {x = 83.64;{\text{ }}\sum {{x^2} = 583.05.} } \] You may assume that the weights of the bags of potatoes can be modelled by the normal distribution \({\text{N}}(\mu ,{\text{ }}{\sigma ^2})\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to test the inspector’s claim.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Carry out an appropriate test and state the \(p\)-value obtained.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using a 10% significance level and justifying your answer, state your conclusion in context.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> represents the lifetime in hours of a battery. The lifetime may be assumed to be a continuous random variable <em>X</em> with a probability density function given by \(f(x) = \lambda {{\text{e}}^{ - \lambda x}}\), where \(x \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the cumulative distribution function, \(F(x)\), of <em>X</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the lifetime of a particular battery is more than twice the mean.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the median of <em>X</em> in terms of \(\lambda \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the lifetime of a particular battery lies between the median and the mean.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Anne is a farmer who grows and sells pumpkins. Interested in the weights of pumpkins produced, she records the weights of eight pumpkins and obtains the following results in kilograms.</p>
<p>\[{\text{7.7}}\quad {\text{7.5}}\quad {\text{8.4}}\quad {\text{8.8}}\quad {\text{7.3}}\quad {\text{9.0}}\quad {\text{7.8}}\quad {\text{7.6}}\]</p>
<p>Assume that these weights form a random sample from a \(N(\mu ,{\text{ }}{\sigma ^2})\) distribution. </p>
<p> </p>
</div>
<div class="specification">
<p>Anne claims that the mean pumpkin weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis \({{\text{H}}_0}:\mu = 7.5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use a two-tailed test to determine the \(p\)-value for the above results.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your \(p\)-value at the 5% level of significance, justifying your conclusion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has probability distribution Po(8).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(X = 6)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find \({\text{P}}(X = 6|5 \leqslant X \leqslant 8)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar X\) denotes the sample mean of \(n > 1\) independent observations from \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down \({\text{E}}(\bar X)\) and \({\text{Var}}(\bar X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, give a reason why \(\bar X\) is not a Poisson distribution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample of \(40\) observations is taken from the distribution for \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(7.1 < \bar X < 8.5)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({\text{P}}\left( {\left| {\bar X - 8} \right| \leqslant k} \right) = 0.95\), find the value of \(k\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weight of tea in <em>Supermug</em> tea bags has a normal distribution with mean 4.2 g and standard deviation 0.15 g. The weight of tea in <em>Megamug</em> tea bags has a normal distribution with mean 5.6 g and standard deviation 0.17 g.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen <em>Supermug</em> tea bag contains more than 3.9 g of tea.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that, of two randomly chosen <em>Megamug</em> tea bags, one contains more than 5.4 g of tea and one contains less than 5.4 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that five randomly chosen <em>Supermug</em> tea bags contain a total of less than 20.5 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the total weight of tea in seven randomly chosen <em>Supermug</em> tea bags is more than the total weight in five randomly chosen <em>Megamug</em> tea bags.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable \(X\) has the following probability distribution.</p>
<p style="padding-left: 90px;">\({\text{P}}(X = x) = \left\{ {\begin{array}{*{20}{l}}<br> {p{q^{\frac{x}{2}}}}&{{\text{for }}x = 0,{\text{ }}2,{\text{ }}4,{\text{ }}6 \ldots {\text{ where }}p + q = 1,{\text{ }}0 < p < 1.} \\ <br> 0&{{\text{otherwise}}} <br>\end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability generating function for \(X\) is given by \(G(t) = \frac{P}{{1 - q{t^2}}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine \({\text{E}}(X)\) in terms of \(p\) and \(q\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable \(Y\) is given by \(Y = 2X + 1\). Find the probability generating function for \(Y\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable \(X\) follows a Poisson distribution with mean \(\lambda \). The probability generating function of \(X\) is given by \({G_X}(t) = {{\text{e}}^{\lambda (t - 1)}}\).</p>
</div>
<div class="specification">
<p>The random variable \(Y\), independent of \(X\), follows a Poisson distribution with mean \(\mu \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find expressions for \({G’_X}(t)\) and \({G’’_X}(t)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \({\text{Var}}(X) = \lambda \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\), show that \(X + Y\) follows a Poisson distribution with mean \(\lambda + \mu \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{P}}(X = x|X + Y = n) = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){\left( {\frac{\lambda }{{\lambda + \mu }}} \right)^x}{\left( {1 - \frac{\lambda }{{\lambda + \mu }}} \right)^{n - x}}\), where \(n\), \(x\) are non-negative integers and \(n \geqslant x\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Identify the probability distribution given in part (c)(i) and state its parameters.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider an unbiased tetrahedral (four-sided) die with faces labelled 1, 2, 3 and 4 respectively.</p>
<p>The random variable <em>X</em> represents the number of throws required to obtain a 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the distribution of <em>X</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability generating function, \(G\left( t \right)\), for <em>X</em> is given by \(G\left( t \right) = \frac{t}{{4 - 3t}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(G'\left( t \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean number of throws required to obtain a 1.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Alun answers mathematics questions and checks his answer after doing each one.</p>
<p class="p1">The probability that he answers any question correctly is always \(\frac{6}{7}\), independently of all other questions. He will stop for coffee immediately following a second incorrect answer. Let \(X\) be the number of questions Alun answers before he stops for coffee.</p>
</div>
<div class="specification">
<p class="p1">Nic answers mathematics questions and checks his answer after doing each one.</p>
<p class="p1">The probability that he answers any question correctly is initially \(\frac{6}{7}\). After his first incorrect answer, Nic loses confidence in his own ability and from this point onwards, the probability that he answers any question correctly is now only \(\frac{4}{7}\).</p>
<p class="p1">Both before and after his first incorrect answer, the result of each question is independent of the result of any other question. Nic will also stop for coffee immediately following a second incorrect answer. Let \(Y\) be the number of questions Nic answers before he stops for coffee.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the distribution of \(X\), <span class="s1">including its parameters.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate \({\text{E}}(X)\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Calculate \({\text{P}}(X = 5)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate \({\text{E}}(Y)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate \({\text{P}}(Y = 5)\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Two independent discrete random variables \(X\) and \(Y\) have probability generating functions \(G(t)\) and \(H(t)\) respectively. Let \(Z = X + Y\) have probability generating function \(J(t)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down an expression for \(J(t)\) <span class="s1">in terms of \(G(t)\) and \(H(t)\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By differentiating \(J(t)\), <span class="s1">prove that</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> \({\text{E}}(Z) = {\text{E}}(X) + {\text{E}}(Y)\)</span>;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> \({\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)\)</span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>n </em>independent random variables \({X_1},{X_2},…,{X_n}\) all have the distribution \({\text{N}}(\mu ,\,{\sigma ^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the mean and the variance of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({X_1} + {X_2}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(3{X_1}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({X_1} + {X_2} - {X_3}\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) \(\bar X = \frac{{({X_1} + {X_2} + ... + {X_n})}}{n}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{E}}(X_1^2)\) in terms of \(\mu \) and \(\sigma \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Anna has a fair cubical die with the numbers 1, 2, 3, 4, 5, 6 respectively on the six faces. When she tosses it, the score is defined as the number on the uppermost face. One day, she decides to toss the die repeatedly until all the possible scores have occurred at least once.</span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Having thrown the die once, she lets \({X_2}\) denote the number of additional throws required to obtain a different number from the one obtained on the first throw. State the distribution of \({X_2}\) and hence find \({\text{E}}({X_2})\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) She then lets \({X_3}\) denote the number of additional throws required to obtain a different number from the two numbers already obtained. State the distribution of \({X_3}\) and hence find \({\text{E}}({X_3})\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) By continuing the process, show that the expected number of tosses needed to obtain all six possible scores is 14.7.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A coin was tossed 200 times and 115 of these tosses resulted in ‘heads’. Use a two-tailed test with significance level 1 % to investigate whether or not the coin is biased.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>Y</em> is such that \({\text{E}}(2Y + 3) = 6{\text{ and Var}}(2 - 3Y) = 11\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) E(<em>Y</em>) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{Var}}(Y)\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{E}}({Y^2})\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Independent random variables <em>R</em> and <em>S</em> are such that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[R \sim {\text{N}}(5,{\text{ 1}}){\text{ and }}S \sim {\text{N(8, 2).}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>V</em> is defined by <em>V</em> = 3<em>S</em> – 4<em>R</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate P(<em>V</em> > 5).</span></p>
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<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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A baker produces loaves of bread that he claims weigh on average 800 g each. Many customers believe the average weight of his loaves is less than this. A food inspector visits the bakery and weighs a random sample of 10 loaves, with the following results, in grams:</span></p>
<p style="font: normal normal normal 12px/normal Times; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">783, 802, 804, 785, 810, 805, 789, 781, 800, 791.</span></p>
<p style="font: normal normal normal 12px/normal Times; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that these results are taken from a normal distribution.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates for the mean and variance of the distribution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">In spite of these results the baker insists that his claim is correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Stating appropriate hypotheses, test the baker’s claim at the 10 % level of significance.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
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<p>Two species of plant, \(A\) and \(B\), are identical in appearance though it is known that the mean length of leaves from a plant of species \(A\) is \(5.2\) cm, whereas the mean length of leaves from a plant of species \(B\) is \(4.6\) cm. Both lengths can be modelled by normal distributions with standard deviation \(1.2\) cm.</p>
<p>In order to test whether a particular plant is from species \(A\) or species \(B\), \(16\) leaves are collected at random from the plant. The length, \(x\), of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, \(\bar X\), is then performed at the \(5\% \) level, with the hypotheses: \({H_0}:\mu = 5.2\) and \({H_1}:\mu < 5.2\).</p>
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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;">Let \(X\) and \(Y\) be independent random variables with \(X \sim {P_o}{\text{ (3)}}\) and \(Y \sim {P_o}{\text{ (2)}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(S = 2X + 3Y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the mean and variance of \(S\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence state with a reason whether or not \(S\) follows a Poisson distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(T = X + Y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \({\text{P}}(T = 3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that \({\text{P}}(T = t) = \sum\limits_{r = 0}^t {{\text{P}}(X = r){\text{P}}(Y = t - r)} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Hence show that \(T\) follows a Poisson distribution with mean 5.</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">.</div>
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<p class="p1">Find the probability of a Type II error if the leaves are in fact from a plant of species B.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Alan and Brian are athletes specializing in the long jump. When Alan jumps, the length of his jump is a normally distributed random variable with mean 5.2 metres and standard deviation 0.1 metres. When Brian jumps, the length of his jump is a normally distributed random variable with mean 5.1 metres and standard deviation 0.12 metres. For both athletes, the length of a jump is independent of the lengths of all other jumps. During a training session, Alan makes four jumps and Brian makes three jumps. Calculate the probability that the mean length of Alan’s four jumps is less than the mean length of Brian’s three jumps.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Colin joins the squad and the coach wants to know the mean length, \(\mu \) metres, of his jumps. Colin makes six jumps resulting in the following lengths in metres.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">5.21, 5.30, 5.22, 5.19, 5.28, 5.18</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate an unbiased estimate of both the mean \(\mu \) and the variance of the lengths of his jumps.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Assuming that the lengths of these jumps are independent and normally distributed, calculate a 90 % confidence interval for \(\mu \) .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When Ben shoots an arrow, he hits the target with probability 0.4. Successive shots are independent.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) he hits the target exactly 4 times in his first 8 shots;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) he hits the target for the \({4^{{\text{th}}}}\) time with his \({8^{{\text{th}}}}\) shot.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ben hits the target for the \({10^{{\text{th}}}}\) time with his \({X^{{\text{th}}}}\) shot.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the expected value of the random variable <em>X</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down an expression for \({\text{P}}(X = x)\) and show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{P}}(X = x)}}{{{\text{P}}(X = x - 1)}} = \frac{{3(x - 1)}}{{5(x - 10)}}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence, or otherwise, find the most likely value of <em>X</em>.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The mean weight of a certain breed of bird is believed to be 2.5 kg. In order to test this belief, it is planned to determine the weights \({x_1}{\text{ , }}{x_2}{\text{ , }}{x_3}{\text{ , }} \ldots {\text{, }}{x_{16}}\) (in kg) of sixteen of these birds and then to calculate the sample mean \({\bar x}\) . You may assume that these weights are a random sample from a normal distribution with standard deviation 0.1 kg.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State suitable hypotheses for a two-tailed test.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the critical region for \({\bar x}\) having a significance level of 5 %.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that the mean weight of birds of this breed is actually 2.6 kg, find the probability of making a Type II error.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The apple trees in a large orchard have, for several years, suffered from a disease for which the outward sign is a red discolouration on some leaves.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The fruit grower knows that the mean number of discoloured leaves per tree is 42.3. The fruit grower suspects that the disease is caused by an infection from a nearby group of cedar trees. He cuts down the cedar trees and, the following year, counts the number of discoloured leaves on a random sample of seven apple trees. The results are given in the table below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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byHJT14V7U2ev2sv9dMrDGau6sYlbtFoQswFX4hXCDtb0N0EbQkVk6YMktBuAULRUeS7emJ2km+Yyra6aIn3QLFXI5brLTNvRuZYsNoPKQubPNRO97vH4Jn49tF+3BLuDpSy9exjCdD7Zu2v+Amt74+UpLhecuMps+DyN7FSrVuUfQCTIVTCNhF/3jgVxoZbuF/sZRSEG7BAoekZZ2y7nSk/jAr0IQsxy02+eTKcDr3zk11dzWT2NLvFJvgS8qoalPK15npf/OeasL2hwd7zYymEiNlXeCJaRob72KiUrcofAGmwicEdzoavA76x9a6BSmlFIRbsMCrRqZPWXz25vGLgmdqiW2LxKl5aekHANoWnpK7Ky865NANRfjXUynliy19P55Ydzrcu7nmvY1XpawLfM0MgsKTCyiqcwseF2AqPEJIfZYoe6ZsCaUg3IIFHkk7N9VOoicKz83H/4h6IdzPJ/pHhlvjska5HbPfiHednZvqXwpWatzaFoPv6CYOj24owreeSi/fVRvmkNg45bUtHgwybs/X7GKiIrfgdAGmUkaXZm7bgnspCLdggSVp+GzS222rPyB74fW278ffXBs+DB9t614MzlkbTWov02JuHLf8deYX9vRpt1O0w4RRKrbR63fIu6qx/fPjwUOTnr3CpRuKXeEqa8rXnQ73msHTedidnXS3YExTToJt9OY18mZrYfvjSNXCinXNLi5U4Rb8LsBUKnCLCkqhJLeIvVAv5a1E4fqv8hd4oyoPmJIWvNnJm35qoGiaZPCGqOTGOD2maIEmVg6gLphw3ZRmYHuFYHWL8E2OO13thZkyffbS0lUuc1r4XB155RvMegC98oc/puKdvS+px1wyd3GB9yMjK0XA9QJMpRK3KL0USmxbBMvHprlFVGx/HreoCMjaKOBLha8wFxECBOoRQpk9UdQLJJLQb5u4VkAue8jaKOBLha8wFxECBOoRgnALFiCXPWRtFPClwleYiwgBAvUIYY1bRMuAb2vTP7zlfP2Rz9jqvnvqSbwHPuyB3RtMzGdXcYusY2LXOn0VrOcc7YpGPoKDJN7JSjI62b0FJjddmXxN4ph+VwWQtVHAlwpfYS4iBAjUI4S1bYvwBUF3H776xbb0A0W6NUSOOzvpNuRWz5hjb/JMMIXD//5u3zyPLWawiVvgrGN6gx8t1XSixXi9gR3/J/GR2CU6+W9/eoz/4qN7xhn2pkB4q744Zr/hfR+71vuh9k64xRcEvlT4CnMRIUCgHiGs74nyXz7qPf2BLX2/MTCdf5vqbngL71X62xpaJt6ghwm5Uk/UecYxg9cYNAamg4OfhAcMVpfzn309Nx+/8J9S8Sf2xN/temuIXF9S9jp0myeuyM9LBbI2CvhS4SvMRYQAgXqEsJFbJOYmU6+Zi7p0km/nJldxi8xjBriWqY++U+9SjZVg2tVu3zwnzqfJ6Xny4PHNsyL/peKx9+4xJo7thxUAWRsFfKnwFeYiQoBAPULYyC224y9a9Wv21UdXgnEOdrdIfRzGG37YuWecLei2BQnfldZS3/3r9YvT4Gmy4OCpM6YXNvqhv9dUwpd7Xx3IZQ9ZGwV8qfAV5iJCgEA9Qri6W4SDGfT671k9UekrWyXcIvuYLhq1payeqPgfvdV9FvvUb3OkLvLqfcN7wivdyXKBXPaQtVHAlwpfYS4iBAjUI4TNRrkTT8mHo807Xf2zSwi20T+9eUrudLjX9OtoPD993Ptr5gPb4dN53u1/1jHDeVkHY8sOOpGSVbwvMtmMCB/ODFeqQCayF8R+N9QMb+2NuXHcYn0THOSyh6yNAr5U+ApzESFAoB4hbDSDdmWyaTg5lXqHWjQLtnX49Gf0471D7ZXxLmWJiPj8V7/nKvWY4cSnTl83Z9b7B95bPhKvhcpYDNW1fgrm3UarKdifTOvfljFor8z9vWrimH5XBZC1UcCXCl9hLiIECNQjBLGqIAuQkwZZGwV8qfAV5iJCgEA9QhBuwQLkpEHWRgFfKnyFuYgQIFCPEIRbsAA5aZC1UcCXCl9hLiIECNQjBOEWLEBOGmRtFPClwleYiwgBAvUIQbgFC5CTBlkbBXyp8BXmIkKAQD1CEG7BAuSkQdZGAV8qfIW5iBAgUI8QhFuwADlpkLVRwJcKX2EuIgQI1CME4RYsQE4aZG0U8KXCV5iLCAEC9QjhRvT8ndjEJjaxiU1sGdv/A0sscxpkLFw1AAAAAElFTkSuQmCC" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) From these data calculate an unbiased estimate of the population variance.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Stating null and alternative hypotheses, carry out an appropriate test at the 10 % level to justify the cutting down of the cedar trees.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The discrete random variable <em>X</em> has the following probability distribution, where \(0 < \theta < \frac{1}{3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine \({\text{E}}(X)\) and show that \({\text{Var}}(X) = 6\theta - 16{\theta ^2}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In order to estimate \(\theta \), a random sample of <em>n</em> observations is obtained from the distribution of <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \({\bar X}\) denotes the mean of this sample, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\hat \theta }_1} = \frac{{3 - \bar X}}{4}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is an unbiased estimator for \(\theta \) and write down an expression for the variance of \({{\hat \theta }_1}\) in terms of <em>n</em> and \(\theta \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let <em>Y</em> denote the number of observations that are equal to 1 in the sample. Show that <em>Y</em> has the binomial distribution \({\text{B}}(n,{\text{ }}\theta )\) and deduce that \({{\hat \theta }_2} = \frac{Y}{n}\) is another unbiased estimator for \(\theta \). Obtain an expression for the variance of \({{\hat \theta }_2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that \({\text{Var}}({{\hat \theta }_1}) < {\text{Var}}({{\hat \theta }_2})\) and state, with a reason, which is the more </span><span style="font-family: 'times new roman', times; font-size: medium;">efficient estimator, \({{\hat \theta }_1}\) or \({{\hat \theta }_2}\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Consider the random variable \(X\) for which \({\text{E}}(X) = a\lambda + b\), where \(a\) and \(b\)are constants and \(\lambda \) is a parameter.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{X - b}}{a}\) is an unbiased estimator for \(\lambda \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The continuous random variable <em>Y </em>has probability density function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(y) = \left\{ \begin{array}{r}{\textstyle{2 \over 9}}(3 + y - \lambda ),\\0,\end{array} \right.\begin{array}{*{20}{l}}{{\rm{ for}}\, \lambda - 3 \le y \le \lambda }\\{{\rm{ otherwise}}}\end{array}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(\lambda \) is a parameter.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Verify that \(f(y)\) is a probability density function for all values of \(\lambda \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Determine \({\text{E}}(Y)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Write down an unbiased estimator for \(\lambda \).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> is normally distributed with unknown mean \(\mu \) and unknown variance \({\sigma ^2}\). A random sample of 20 observations on <em>X</em> gave the following results.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\sum {x = 280,{\text{ }}\sum {{x^2} = 3977.57} } \]</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find unbiased estimates of \(\mu \) and \({\sigma ^2}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a 95 % confidence interval for \(\mu \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the hypotheses</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\text{H}}_0}:\mu = 15;{\text{ }}{{\text{H}}_1}:\mu \ne 15,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the <em>p</em>-value of the above results and state your conclusion at the 1 % significance level.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the oranges produced by a farm may be assumed to be normally distributed with mean 205 grams and standard deviation 10 grams.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen orange weighs more than 200 grams.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Five of these oranges are selected at random to be put into a bag. Find the probability that the combined weight of the five oranges is less than 1 kilogram.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The farm also produces lemons whose weights may be assumed to be normally distributed with mean 75 grams and standard deviation 3 grams. Find the probability that the weight of a randomly chosen orange is more than three times the weight of a randomly chosen lemon.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The continuous random variable <em>X</em> has probability density function <em>f</em> given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {\frac{{3{x^2} + 2x}}{{10}},}&{{\text{for }}1 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine an expression for \(F(x)\), valid for \(1 \leqslant x \leqslant 2\), where <em>F</em> denotes the cumulative distribution function of <em>X</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, determine the median of <em>X</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the central limit theorem.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) A random sample of 150 observations is taken from the distribution of <em>X</em> and \(\bar X\) denotes the sample mean. Use the central limit theorem to find, approximately, the probability that \(\bar X\) is greater than 1.6.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) A random variable, <em>X</em> , has probability density function defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{l}}<br> {100,}&{{\text{for }} - 0.005 \leqslant x < 0.005} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine <em>E</em>(<em>X</em>) and Var(<em>X</em>) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) When a real number is rounded to two decimal places, an error is made.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this error can be modelled by the random variable <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) A list contains 20 real numbers, each of which has been given to two decimal places. The numbers are then added together.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down bounds for the resulting error in this sum.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Using the central limit theorem, estimate to two decimal places the probability that the absolute value of the error exceeds 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) State clearly any assumptions you have made in your calculation.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When Andrew throws a dart at a target, the probability that he hits it is \(\frac{1}{3}\) ; when Bill throws a dart at the target, the probability that he hits the it is \(\frac{1}{4}\) . Successive throws are independent. One evening, they throw darts at the target alternately, starting with Andrew, and stopping as soon as one of their darts hits the target. Let <em>X</em> denote the total number of darts thrown.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \({\text{P}}(X = 1)\) and show that \({\text{P}}(X = 2) = \frac{1}{6}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the probability generating function for <em>X</em> is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[G(t) = \frac{{2t + {t^2}}}{{6 - 3{t^2}}}.\]</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence determine \({\text{E}}(X)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>If \(X\) is a random variable that follows a Poisson distribution with mean \(\lambda > 0\) then the probability generating function of \(X\) is \(G(t) = {e^{\lambda (t - 1)}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Prove that \({\text{E}}(X) = \lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Prove that \({\text{Var}}(X) = \lambda \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(Y\) is a random variable, independent of \(X\), that also follows a Poisson distribution with mean \(\lambda \).</p>
<p class="p1">If \(S = 2X - Y\) find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({\text{E}}(S)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}(S)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(T = \frac{Y}{2} + \frac{Y}{2}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(T\) is an unbiased estimator for \(\lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(T\) is a more efficient unbiased estimator of \(\lambda \) than \(S\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Could either \(S\) or \(T\) model a Poisson distribution? Justify your answer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By consideration of the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\), prove that \(X + Y\) follows a Poisson distribution with mean \(2\lambda \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({G_{X + Y}}(1)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({G_{X + Y}}( - 1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the probability that \(X + Y\) is an even number.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Jenny tosses seven coins simultaneously and counts the number of tails obtained. She repeats the experiment 750 times. The following frequency table shows her results.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.31.35.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Explain what can be done with this data to decrease the probability of making a type I error.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the meaning of a type II error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down how to proceed if it is required to decrease the probability of making both a type I and type II error.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> has a binomial distribution with parameters \(n\) and \(p\).</p>
</div>
<div class="specification">
<p>Let \(U = nP\left( {1 - P} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(P = \frac{X}{n}\) is an unbiased estimator of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down an unbiased estimator of Var(<em>X</em>).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Two students are selected at random from a large school with equal numbers of boys and girls. The boys’ heights are normally distributed with mean \(178\) cm and standard deviation \(5.2\) cm, and the girls’ heights are normally distributed with mean \(169\) cm and standard deviation \(5.4\) cm<span class="s1">.</span></p>
<p class="p2">Calculate the probability that the taller of the two students selected is a boy.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A hospital specializes in treating overweight patients. These patients have weights that are independently, normally distributed with mean 200 kg and standard deviation 15 kg. The elevator in the hospital will break if the total weight of people inside it exceeds 1150 kg. Six patients enter the elevator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the elevator breaks.</span></p>
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<br><hr><br><div class="specification">
<p class="p1">Eleven students who had under-performed in a philosophy practice examination were given extra tuition before their final examination. The differences between their final examination marks and their practice examination marks were</p>
<p class="p1">\[10,{\text{ }} - 1,{\text{ }}6,{\text{ }}7,{\text{ }} - 5,{\text{ }} - 5,{\text{ }}2,{\text{ }} - 3,{\text{ }}8,{\text{ }}9,{\text{ }} - 2.\]</p>
<p class="p1">Assume that these differences form a random sample from a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine unbiased estimates of \(\mu \) and \({\sigma ^2}\)<span class="s1">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State suitable hypotheses to test the claim that extra tuition improves examination marks.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Calculate the \(p\)-value of the sample.</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Determine whether or not the above claim is supported at the \(5\% \) significance level.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two species of plant, \(A\) and \(B\), are identical in appearance though it is known that the mean length of leaves from a plant of species \(A\) is \(5.2\) cm, whereas the mean length of leaves from a plant of species \(B\) is \(4.6\) cm. Both lengths can be modelled by normal distributions with standard deviation \(1.2\) cm.</p>
<p>In order to test whether a particular plant is from species \(A\) or species \(B\), \(16\) leaves are collected at random from the plant. The length, \(x\), of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, \(\bar X\), is then performed at the \(5\% \) level, with the hypotheses: \({H_0}:\mu = 5.2\) and \({H_1}:\mu < 5.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the critical region for this test.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">It is now known that in the area in which the plant was found \(90\% \) of all the plants are of species \(A\) and \(10\% \) are of species \(B\).</p>
<p class="p1">Find the probability that \(\bar X\) will fall within the critical region of the test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If, having done the test, the sample mean is found to lie within the critical region, find the probability that the leaves came from a plant of species \(A\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A factory makes wine glasses. The manager claims that on average 2 % of the glasses are imperfect. A random sample of 200 glasses is taken and 8 of these are found to be imperfect.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Test the manager’s claim at a 1 % level of significance using a one-tailed test.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ten friends try a diet which is claimed to reduce weight. They each weigh themselves before starting the diet, and after a month on the diet, with the following results.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img 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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates of the mean and variance of the loss in weight achieved over the month by people using this diet.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State suitable hypotheses for testing whether or not this diet causes a mean loss in weight.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine the value of a suitable statistic for testing your hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the 1 % critical value for your statistic and state your conclusion.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The owner of a factory is asked to produce bricks of weight <span class="s1">2.2 kg</span>. The quality control manager wishes to test whether or not, on a particular day, the mean weight of bricks being produced is <span class="s1">2.2 kg</span>.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">He therefore collects a random sample of 20 </span>of these bricks and determines the weight, \(x\) <span class="s1">kg</span>, of each brick. He produces the following summary statistics.</p>
<p class="p1">\[\sum {x = 42.0,{\text{ }}\sum {{x^2} = 89.2} } \]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State hypotheses to enable the quality control manager to test the mean weight using a two-tailed test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate unbiased estimates of the mean and the variance of the weights of the bricks being produced.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Assuming that the weights of the bricks are normally distributed, determine the \(p\)<span class="s1">-value of the above results and state the conclusion in context using a 5% </span>significance level.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The owner is more familiar with using confidence intervals. Determine a <span class="s1">95% </span>confidence interval for the mean weight of bricks produced on that particular day.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The continuous random variable \(X\) <span class="s1">takes values in the interval \([0,{\text{ }}\theta ]\) and</span></p>
<p class="p2" style="text-align: center;">\({\text{E}}(X) = \frac{\theta }{2}\) and \({\text{Var}}(X) = \frac{{{\theta ^2}}}{{24}}\).</p>
<p class="p1">To estimate the unknown parameter \(\theta \), a random sample of size \(n\) is obtained from the distribution of \(X\). The sample mean is denoted by \(\overline X \) and \(U = k\overline X\) is an unbiased estimator for \(\theta \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate an unbiased estimate for \(\theta \)<span class="s1">, using the random sample,</span></p>
<p class="p2">8.3, 4.2, 6.5, 10.3, 2.7, 1.2, 3.3, 4.3<span class="s2">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Explain briefly why this is not a good estimate for \(\theta \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \({\text{Var}}(U) = \frac{{{\theta ^2}}}{{6n}}\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Show that \({U^2}\) is not an unbiased estimator for \({\theta ^2}\).</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>Find an unbiased estimator for \({\theta ^2}\) in terms of \(U\) and \(n\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) After a chemical spillage at sea, a scientist measures the amount, <em>x</em> units, of the chemical in the water at 15 randomly chosen sites. The results are summarised in the form \(\sum {x = 18} \) and \(\sum {{x^2} = 28.94} \). Before the spillage occurred the mean level of the chemical in the water was 1.1. Test at the 5 % significance level the hypothesis that there has been an increase in the amount of the chemical in the water.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Six months later the scientist returns and finds that the mean amount of the chemical in the water at the 15 randomly chosen sites is 1.18. Assuming that this sample came from a normal population with variance 0.0256, find a 90 % confidence interval for the mean level of the chemical.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A shopper buys 12 apples from a market stall and weighs them with the following results (in grams).</span></p>
<p style="font: normal normal normal 27px/normal Helvetica; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">117, 124, 129, 118, 124, 116, 121, 126, 118, 121, 122, 129</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">You may assume that this is a random sample from a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine unbiased estimates of \(\mu \) and \({\sigma ^2}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a 99 % confidence interval for \(\mu \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The stallholder claims that the mean weight of apples is 125 grams but the shopper claims that the mean is less than this.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State suitable hypotheses for testing these claims.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the <em>p</em>-value of the above sample.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Giving a reason, state which claim is supported by your <em>p</em>-value using a 5 % significance level.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Engine oil is sold in cans of two capacities, large and small. The amount, in millilitres, in each can, is normally distributed according to Large \( \sim {\text{N}}(5000,{\text{ }}40)\) and Small \( \sim {\text{N}}(1000,{\text{ }}25)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can is selected at random. Find the probability that the can contains at least \(4995\) millilitres of oil.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and a small can are selected at random. Find the probability that the large can contains at least \(30\) milliliters more than five times the amount contained in the small can.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and five small cans are selected at random. Find the probability that the large can contains at least \(30\) milliliters less than the total amount contained in the small cans.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of machine breakdowns occurring in a day in a certain factory may be assumed to follow a Poisson distribution with mean \(\mu \). The value of \(\mu \) is known, from past experience, to be 1.2. In an attempt to reduce the value of \(\mu \), all the machines are fitted with new control units. To investigate whether or not this reduces the value of \(\mu \), the total number of breakdowns, <em>x</em>, occurring during a 30-day period following the installation of these new units is recorded.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">State suitable hypotheses for this investigation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is decided to define the critical region by \(x \leqslant 25\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the significance level.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Assuming that the value of \(\mu \) was actually reduced to 0.75, determine the probability of a Type II error.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the probability generating function for \(X \sim {\text{B}}(1,{\text{ }}p)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the probability generating function for \({\text{B}}(n,{\text{ }}p)\) is a polynomial of degree \(n\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Two independent random variables \({X_1}\) and \({X_2}\) are such that \({X_1} \sim {\text{B}}(1,{\text{ }}{p_1})\) <span class="s1">and \({X_2} \sim {\text{B}}(1,{\text{ }}{p_2})\)</span>. Prove that if \({X_1} + {X_2}\) has a binomial distribution then \({p_1} = {p_2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Ahmed and Brian live in the same house. Ahmed always walks to school and Brian always cycles to school. The times taken to travel to school may be assumed to be independent and normally distributed. The mean and the standard deviation for these times are shown in the table below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that on a particular day Ahmed takes more than 35 minutes to walk to school.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Brian cycles to school on five successive mornings. Find the probability that the total time taken is less than 70 minutes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the probability that, on a particular day, the time taken by Ahmed to walk to school is more than twice the time taken by Brian to cycle to school.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A manufacturer of stopwatches employs a large number of people to time the winner of a \(100\) </span>metre sprint. It is believed that if the true time of the winner is \(\mu \) seconds, the times recorded are normally distributed with mean \(\mu \) <span class="s1">seconds and standard deviation \(0.03\) seconds.</span></p>
<p class="p2">The times, in seconds, recorded by six randomly chosen people are</p>
<p class="p2">\[9.765,{\text{ }}9.811,{\text{ }}9.783,{\text{ }}9.797,{\text{ }}9.804,{\text{ }}9.798.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate a \(99\% \) <span class="s1">confidence interval for \(\mu \)</span>. Give your answer correct to three decimal places.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Interpret the result found in (a).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the confidence level of the interval that corresponds to halving the width of the \(99\% \) confidence interval. Give your answer as a percentage to the nearest whole number.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A random variable \(X\) </span>has a population mean \(\mu \)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain briefly the meaning of</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>an estimator of \(\mu \);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>an unbiased estimator of \(\mu \)<span class="s1">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A random sample \({X_1},{\text{ }}{X_2},{\text{ }}{X_3}\) of three independent observations is taken from the distribution of \(X\).</p>
<p>An unbiased estimator of \(\mu ,{\text{ }}\mu \ne 0\), is given by \(U = \alpha {X_1} + \beta {X_2} + (\alpha - \beta ){X_3}\),</p>
<p>where \(\alpha ,{\text{ }}\beta \in \mathbb{R}\).</p>
<p>(i) Find the value of \(\alpha \).</p>
<p>(ii) Show that \({\text{Var}}(U) = {\sigma ^2}\left( {2{\beta ^2} - \beta + \frac{1}{2}} \right)\) where \({\sigma ^2} = {\text{Var}}(X)\).</p>
<p>(iii) Find the value of \(\beta \) which gives the most efficient estimator of \(\mu \) of this form.</p>
<p>(iv) Write down an expression for this estimator and determine its variance.</p>
<p>(v) Write down a more efficient estimator of \(\mu \) than the one found in (iv), justifying your answer.</p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A shop sells apples, pears and peaches. The weights, in grams, of these three types of fruit may be assumed to be normally distributed with means and standard deviations as given in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Alan buys 1 apple and 1 pear while Brian buys 1 peach. Calculate the probability that the combined weight of Alan’s apple and pear is greater than twice the weight of Brian’s peach.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A traffic radar records the speed, \(v\) kilometres per hour (\({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\)), of cars on a section of a road.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table shows a summary of the results for a random sample of 1000 cars whose speeds were recorded on a given day.</span></p>
<p style="font: normal normal normal 20.5px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.17.39.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Using the data in the table,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) show that an estimate of the mean speed of the sample is 113.21 \({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find an estimate of the variance of the speed of the cars on this section of the road.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the 95% confidence interval, \(I\), for the mean speed.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \(J\) be the 90% confidence interval for the mean speed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Without calculating \(J\), explain why \(J \subset I\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">As soon as Sarah misses a total of 4 lessons at her school an email is sent to her parents. The probability that she misses any particular lesson is constant with a value of \(\frac{1}{3}\). Her decision to attend a lesson is independent of her previous decisions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that an email is sent to Sarah’s parents after the \({8^{{\text{th}}}}\) lesson that Sarah was scheduled to attend.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If an email is sent to Sarah’s parents after the \({X^{{\text{th}}}}\) lesson that she was scheduled to attend, find \({\text{E}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) If after 6 of Sarah’s scheduled lessons we are told that she has missed exactly 2 lessons, find the probability that an email is sent to her parents after a total of 12 scheduled lessons.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) If we know that an email was sent to Sarah’s parents immediately after her \({6^{{\text{th}}}}\) scheduled lesson, find the probability that Sarah missed her \({2^{{\text{nd}}}}\) scheduled lesson.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has a Poisson distribution with mean \(\mu \). The value of \(\mu \) is known to be either 1 or 2 so the following hypotheses are set up.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\text{H}}_0}:\mu = 1;{\text{ }}{{\text{H}}_1}:\mu = 2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample \({x_1},{\text{ }}{x_2},{\text{ }} \ldots ,{\text{ }}{x_{10}}\) of 10 observations is taken from the distribution of <em>X</em> and the following critical region is defined.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\sum\limits_{i = 1}^{10} {{x_i} \geqslant 15} \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the probability of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) a Type I error;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) a Type II error.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In a game there are <em>n</em> players, where \(n > 2\) . Each player has a disc, one side of which is red and one side blue. When thrown, the disc is equally likely to show red or blue. All players throw their discs simultaneously. A player wins if his disc shows a different colour from all the other discs. Players throw repeatedly until one player wins.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>X</em> be the number of throws each player makes, up to and including the one on which the game is won.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State the distribution of <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find \({\text{P}}(X = x)\) in terms of <em>n</em> and <em>x</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \({\text{E}}(X)\) in terms of <em>n</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Given that <em>n</em> = 7 , find the least number, <em>k</em> , such that \({\text{P}}(X \leqslant k) > 0.5\) .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The length of time, <em>T</em>, in months, that a football manager stays in his job before he is removed can be approximately modelled by a normal distribution with population mean \(\mu \) and population variance \({\sigma ^2}\). An independent sample of five values of <em>T</em> is given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">6.5, 12.4, 18.2, 3.7, 5.4</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that \({\sigma ^2} = 9\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find the smallest number of values of <em>T</em> that would be required in a sample for the total width of the 90 % confidence interval for \(\mu \) to be less than 2 months.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If the value of \({\sigma ^2}\) is unknown, use the above sample to find the 95 % confidence interval for \(\mu \), giving the bounds of the interval to two decimal places.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>is normally distributed with unknown mean \(\mu \) and unknown variance \({\sigma ^2}\) . A random sample of 10 observations on <em>X </em>was taken and the following 95 % confidence interval for \(\mu \) was correctly calculated as [4.35, 4.53] .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Calculate an unbiased estimate for</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) \(\mu \) ,</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \({\sigma ^2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The value of \(\mu \) is thought to be 4.5, so the following hypotheses are defined.\[{{\text{H}}_0}:\mu = 4.5;{\text{ }}{{\text{H}}_1}:\mu < 4.5\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Find the <em>p</em>-value of the observed sample mean.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) State your conclusion if the significance level is</span></p>
<p style="margin: 0px 0px 0px 60px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (a) 1 %,</span></p>
<p style="margin: 0px 0px 0px 60px; font: 19px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (b) 10 %.</span></p>
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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;">Consider the random variable \(X \sim {\text{Geo}}(p)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State \({\text{P}}(X < 4)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the probability generating function for <em>X </em>is given by \({G_X}(t) = \frac{{pt}}{{1 - qt}}\), where \(q = 1 - p\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let the random variable \(Y = 2X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Show that the probability generating function for <em>Y </em>is given by \({G_Y}(t) = {G_X}({t^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) By considering \({G'_Y}(1)\), show that \({\text{E}}(Y) = 2{\text{E}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let the random variable \(W = 2X + 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) (i) Find the probability generating function for <em>W </em>in terms of the probability generating function of <em>Y</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Hence, show that \({\text{E}}(W) = 2{\text{E}}(X) + 1\).</span></p>
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<p class="p1">The continuous random variable \(X\) has probability density function</p>
<p class="p1">\[f(x) = \left\{ {\begin{array}{*{20}{c}} {{{\text{e}}^{ - x}}}&{x \geqslant 0} \\ 0&{x < 0} \end{array}.} \right.\]</p>
<p class="p1">The discrete random variable \(Y\) is defined as the integer part of \(X\), that is the largest integer less than or equal to \(X\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the probability distribution of \(Y\) <span class="s1">is given by \({\text{P}}(Y = y) = {{\text{e}}^{ - y}}(1 - {{\text{e}}^{ - 1}}),{\text{ }}y \in \mathbb{N}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(G(t)\), the probability generating function of \(Y\), is given by \(G(t) = \frac{{1 - {{\text{e}}^{ - 1}}}}{{1 - {{\text{e}}^{ - 1}}t}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence determine the value of \({\text{E}}(Y)\) correct to three significant figures.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
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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;">Francisco and his friends want to test whether performance in running 400 metres improves if they follow a particular training schedule. The competitors are tested before and after the training schedule.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The times taken to run 400 metres, in seconds, before and after training are shown in the following table.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.37.33.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Apply an appropriate test at the 1% significance level to decide whether the training schedule improves competitors’ times, stating clearly the null and alternative hypotheses. (It may be assumed that the distributions of the times before and after training are normal.)</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A population is known to have a normal distribution with a variance of 3 and an unknown mean \(\mu \) . It is proposed to test the hypotheses \({{\text{H}}_0}:\mu = 13,{\text{ }}{{\text{H}}_1}:\mu > 13\) using the mean of a sample of size 2.</span></p>
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<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the appropriate critical regions corresponding to a significance level of</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) 0.05;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that the true population mean is 15.2, calculate the probability of making a Type II error when the level of significance is</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) 0.05;</span></p>
<p style="margin: 0px 0px 0px 30px; font: 28px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) 0.01.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) How is the change in the probability of a Type I error related to the change in the probability of a Type II error?</span></p>
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<br><hr><br><div class="specification">
<p class="p1">Eric plays a game at a fairground in which he throws darts at a target. Each time he throws a dart, the probability of hitting the target is \(0.2\). He is allowed to throw as many darts as he likes, but it costs him \($1\) a throw. If he hits the target a total of three times he wins \($10\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability he has his third success of hitting the target on his sixth throw.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the expected number of throws required for Eric to hit the target three times.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down his expected profit or loss if he plays until he wins the \($10\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If he has just \($8\), find the probability he will lose all his money before he hits the target three times.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A teacher wants to determine whether practice sessions improve the ability to memorize digits.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">He tests a group of 12 children to discover how many digits of a twelve-digit number could be repeated from memory after hearing them once. He gives them test 1, and following a series of practice sessions, he gives them test 2 one week later. The results are shown in the table below.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><img 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" alt></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State appropriate null and alternative hypotheses.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Test at the 5 % significance level whether or not practice sessions improve ability to memorize digits, justifying your choice of test.</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The continuous random variable <em>X </em>has probability density function <em>f </em>given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x,}&{0 \leqslant x \leqslant 0.5,} \\ <br> {\frac{4}{3} - \frac{2}{3}x,}&{0.5 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the function <em>f </em>and show that the lower quartile is 0.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine E(<em>X </em>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{E}}({X^2})\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Two independent observations are made from <em>X </em>and the values are added.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The resulting random variable is denoted <em>Y </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine \({\text{E}}(Y - 2X)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{Var}}\,(Y - 2X)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the cumulative distribution function for <em>X </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, find the median of the distribution.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) has probability density function</p>
<p class="p1">\(f(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{1}{2}}&{0 \le x < 1} \\ {\frac{1}{4}}&{1 \le x < 3} \\ 0&{x \ge 3} \end{array}} \right.\)</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the cumulative distribution function for \(X\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the interquartile range for \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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