File "SL-paper2.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Further Mathematics/Topic 3/SL-paper2html
File size: 40.36 KB
MIME-type: text/x-tex
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-212ef6a30de2a281f3295db168f85ac1c6eb97815f52f785535f1adfaee1ef4f.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-13d27c3a5846e837c0ce48b604dc257852658574909702fa21f9891f7bb643ed.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../index.html">Home</a>
</li>
<li class="active dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Questionbanks
<b class="caret"></b>
</a><ul class="dropdown-menu">
<li>
<a href="../../geography.html" target="_blank">DP Geography</a>
</li>
<li>
<a href="../../physics.html" target="_blank">DP Physics</a>
</li>
<li>
<a href="../../chemistry.html" target="_blank">DP Chemistry</a>
</li>
<li>
<a href="../../biology.html" target="_blank">DP Biology</a>
</li>
<li>
<a href="../../furtherMath.html" target="_blank">DP Further Mathematics HL</a>
</li>
<li>
<a href="../../mathHL.html" target="_blank">DP Mathematics HL</a>
</li>
<li>
<a href="../../mathSL.html" target="_blank">DP Mathematics SL</a>
</li>
<li>
<a href="../../mathStudies.html" target="_blank">DP Mathematical Studies</a>
</li>
</ul></li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="images/logo.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>SL Paper 2</h2><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The random variable \(X\) has cumulative distribution function\[F(x) = \left\{ {\begin{array}{*{20}{c}}<br> 0&{x < 0} \\ <br> {{{\left( {\frac{x}{a}} \right)}^3}}&{0 \leqslant x \leqslant a} \\ <br> 1&{x > a} <br>\end{array}} \right.\]where \(a\) is an unknown parameter. You are given that the mean and variance of \(X\) </span><span style="font-family: times new roman,times; font-size: medium;">are \(\frac{{3a}}{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(\frac{{3{a^2}}}{{80}}\) </span><span style="font-family: times new roman,times; font-size: medium;">respectively. To estimate the value of \(a\) , a random sample of \(n\) </span><span style="font-family: times new roman,times; font-size: medium;">independent observations, \({X_1},{X_2}, \ldots {X_n}\) is taken from the distribution of \(X\) .</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for \(c\) in terms of \(n\) such that \(U = c\sum\limits_{i = 1}^n {{X_i}} \) is an unbiased </span><span style="font-family: times new roman,times; font-size: medium;">estimator for \(a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Determine an expression for \({\text{Var}}(U)\) in this case.</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><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({\rm{P}}(Y \le y) = {\left( {\frac{y}{a}} \right)^{3n}},0 \le y \le a\) and deduce an expression for the </span><span style="font-family: times new roman,times; font-size: medium;">probability density function of \(Y\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{E}}(Y)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Show that \({\rm{Var}}(Y) = \frac{{3n{a^2}}}{{(3n + 2){{(3n + 1)}^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iv) Find an expression for \(d\) in terms of \(n\) such that \(V = dY\) is an unbiased </span><span style="font-family: times new roman,times; font-size: medium;">estimator for \(a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (v) Determine an expression for \({\rm{Var}}(V)\) in this case.</span></p>
<div class="marks">[16]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(\frac{{{\rm{Var}}(U)}}{{{\rm{Var}}(V)}} = \frac{{3n + 2}}{5}\) and hence state, with a reason, which of \(U\) or \(V\) is </span><span style="font-family: times new roman,times; font-size: medium;">the more efficient estimator for \(a\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Gillian is throwing a ball at a target. The number of throws she makes before hitting the target follows a geometric distribution, \(X \sim {\text{Geo}}(p)\). When she uses a cricket ball the number of throws she makes follows a geometric distribution with \(p = \frac{1}{4}\). When she uses a tennis ball the number of throws she makes follows a geometric distribution with \(p = \frac{3}{4}\)<span class="s1">. There is a box containing a large number of balls, \(80\%\)</span> of which are cricket balls and the remainder are tennis balls. The random variable \(A\) is the number of throws needed to hit the target when a single ball is chosen at random from this box and used for all throws.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{E}}(A)\).</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">Show that \({\text{P}}(A = r) = \frac{1}{5} \times {\left( {\frac{3}{4}} \right)^{r - 1}} + \frac{3}{{20}} \times {\left( {\frac{1}{4}} \right)^{r - 1}}\).</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">Find \({\text{P}}(A \leqslant 5|A > 3)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable \(X\) has the following probability distribution.</p>
<p style="text-align: center;">\({\text{P}}(X = x) = \frac{{kx}}{{{3^x}}}\), where \(x \in {\mathbb{Z}^ + }\) and \(k\) is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three terms of the infinite series for \(G(t)\), the probability generating function for \(X\).</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 radius of convergence of this infinite series.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering \(\left( {1 - \frac{t}{3}} \right)G(t)\), show that</p>
<p>\[G(t) = \frac{{3kt}}{{{{(3 - t)}^2}}}.\]</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(k = \frac{4}{3}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\ln G(t) = \ln 4 + \ln t - 2\ln (3 - t)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating both sides of this equation, determine the values of \(G’(1)\) and \(G’’(1)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find \({\text{Var}}(X)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A machine fills containers with grass seed. Each container is supposed to weigh \(28\) kg. </span><span style="font-family: times new roman,times; font-size: medium;">However the weights vary with a standard deviation of \(0.54\) kg. A random sample of </span><span style="font-family: times new roman,times; font-size: medium;">\(24\)</span><span style="font-family: times new roman,times; font-size: medium;"> bags is taken to check that the mean weight is \(28\) kg.</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;">Assuming the series for \({{\rm{e}}^x}\) , find the first five terms of the Maclaurin series for\[\frac{1}{{\sqrt {2\pi } }}{{\rm{e}}^{\frac{{ - {x^2}}}{2}}} {\rm{ .}}\]</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Use your answer to (a) to find an approximate expression for the cumulative distributive </span><span style="font-family: times new roman,times; font-size: medium;">function of \({\rm{N}}(0,1)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong> find an approximate value for \({\rm{P}}( - 0.5 \le Z \le 0.5)\) , where </span><span style="font-family: times new roman,times; font-size: medium;">\(Z \sim {\rm{N}}(0,1)\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">A.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">State and justify an appropriate test procedure giving the null and alternate </span><span style="font-family: times new roman,times; font-size: medium;">hypotheses.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">What is the critical region for the sample mean if the probability of a Type I error </span><span style="font-family: times new roman,times; font-size: medium;">is to be \(3.5\%\)?</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">B.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If the mean weight of the bags is actually \(28\).1 kg, what would be the probability </span><span style="font-family: times new roman,times; font-size: medium;">of a Type II error?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The weights, \(X\) grams, of tomatoes may be assumed to be normally distributed </span><span style="font-family: times new roman,times; font-size: medium;">with mean \(\mu \) grams and standard deviation \(\sigma \) grams. Barry weighs \(21\) tomatoes </span><span style="font-family: times new roman,times; font-size: medium;">selected at random and calculates the following statistics.\[\sum {x = 1071} \) ; \(\sum {{x^2} = 54705} \]</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Determine unbiased estimates of \(\mu \) and \({\sigma ^2}\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Determine a \(95\%\) confidence interval for \(\mu \) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The random variable \(Y\) has variance \({\sigma ^2}\) , where \({\sigma ^2} > 0\) . A random sample </span><span style="font-family: times new roman,times; font-size: medium;">of \(n\) observations of \(Y\) is taken and \(S_{n - 1}^2\)</span><span style="font-family: times new roman,times; font-size: medium;"> denotes the unbiased estimator for \({\sigma ^2}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">By considering the expression </span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Var}}({S_{n - 1}}) = {\rm{E}}(S_{n - 1}^2) - {\left\{ {E\left. {({S_{n - 1}})} \right\}} \right.^2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> ,</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">show that \(S_{n - 1}^{}\) is not an unbiased estimator for \(\sigma \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The weights of apples, in grams, produced on a farm may be assumed to be normally </span><span style="font-family: times new roman,times; font-size: medium;">distributed with mean \(\mu \) and variance \({\sigma ^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The farm manager selects a random sample of \(10\) apples and weighs them with </span><span style="font-family: times new roman,times; font-size: medium;">the following results, given in grams.\[82, 98, 102, 96, 111, 95, 90, 89, 99, 101\]</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Determine unbiased estimates for \(\mu \) and \({\sigma ^2}\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Determine a \(95\%\) confidence interval for \(\mu \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The farm manager claims that the mean weight of apples is \(100\) grams but </span><span style="font-family: times new roman,times; font-size: medium;">the buyer from the local supermarket claims that the mean is less than this. </span><span style="font-family: times new roman,times; font-size: medium;">To test these claims, they select a random sample of \(100\) apples and weigh them. </span><span style="font-family: times new roman,times; font-size: medium;">Their results are summarized as follows, where \(x\) is the weight of an apple </span><span style="font-family: times new roman,times; font-size: medium;">in grams.\[\sum {x = 9831;\sum {{x^2} = 972578} } \]</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) State suitable hypotheses for testing these claims.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Determine the \(p\)-value for this test.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (iii) At the \(1\%\) significance level, state which claim you accept and justify </span><span style="font-family: times new roman,times; font-size: medium;">your answer.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable <em>X</em> follows a geometric distribution Geo(\(p\)) where</p>
<p style="text-align: center;">\({\text{P}}\left( {X = x} \right) = \left\{ {\begin{array}{*{20}{c}}<br> {p{q^{x - 1}},\,{\text{for}}\,x = 1,\,2 \ldots } \\ <br> {0,\,{\text{otherwise}}} <br>\end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability generating function of <em>X</em> is given by</p>
<p>\[G\left( t \right) = \frac{{pt}}{{1 - qt}}.\]</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that \({\text{E}}\left( X \right) = \frac{1}{p}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two friends A and B play a ball game with the following rules.</p>
<p>Each player starts with zero points. Player A serves first and then the players have alternate pairs of serves so that the service order is A, B, B, A, A, … When player A serves, the probability of her scoring 1 point is \({p_A}\) and the probability of B scoring 1 point is \({q_A}\), where \({q_A} = 1 - {p_A}\).</p>
<p>When player B serves, the probability of her scoring 1 point is \({p_B}\) and the probability of A scoring 1 point is \({q_B}\), where \({q_B} = 1 - {p_B}\).</p>
<p>Show that, after the first 6 serves, the probability that each player has 3 points is</p>
<p>\(\sum\limits_{x = 0}^{x = 3} {{{\left( \begin{gathered}<br> 3 \hfill \\<br> x \hfill \\ <br>\end{gathered} \right)}^2}} {\left( {{p_A}} \right)^x}{\left( {{p_B}} \right)^x}{\left( {{q_A}} \right)^{3 - x}}{\left( {{q_B}} \right)^{3 - x}}\).</p>
<p> </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>After 6 serves the score is 3 points each. Play continues and the game ends when one player has scored two more points than the other player. Let <em>N</em> be the number of further serves required before the game ends. Given that \({p_A}\) = 0.7 and \({p_A}\) = 0.6 find P(<em>N</em> = 2).</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>Let \(M = \frac{1}{2}N\). Show that <em>M</em> has a geometric distribution and hence find the value of E(<em>N</em>).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The independent random variables <em>X</em> and <em>Y</em> are given by <em>X</em> ~ N\(\left( {{\mu _1},\,\sigma _1^2} \right)\) and <em>Y</em> ~ N\(\left( {{\mu _2},\,\sigma _2^2} \right)\).</p>
</div>
<div class="specification">
<p>Two independent random variables <em>X</em><sub>1</sub> and <em>X</em><sub>2</sub> each have a normal distribution with a mean 3 and a variance 9. Four independent random variables <em>Y</em><sub>1</sub>, <em>Y</em><sub>2</sub>, <em>Y</em><sub>3</sub>, <em>Y</em><sub>4</sub> each have a normal distribution with mean 2 and variance 25. Each of the variables <em>Y</em><sub>1</sub>, <em>Y</em><sub>2</sub>, <em>Y</em><sub>3</sub>, <em>Y</em><sub>4</sub> is independent of each of the variables <em>X</em><sub>1</sub>, <em>X</em><sub>2</sub>. Find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the distribution of <em>aX</em> + <em>bY</em> where <em>a</em>, <em>b </em>\( \in \mathbb{R}\).</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>P(<em>X</em><sub>1</sub> + <em>Y</em><sub>1</sub> < 11).</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>P(3<em>X</em><sub>1</sub> + 4<em>Y</em><sub>1</sub> > 15).</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>P(<em>X</em><sub>1</sub> + <em>X</em><sub>2</sub> + <em>Y</em><sub>1</sub> + <em>Y</em><sub>2</sub> + <em>Y</em><sub>3</sub> + <em>Y</em><sub>4</sub> < 30).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \({\bar X}\) and \({\bar Y}\) are the respective sample means, find \({\text{P}}\left( {\bar X > \bar Y} \right)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">An automatic machine is used to fill bottles of water. The amount delivered, \(</span><span style="font-family: times new roman,times; font-size: medium;">Y\) ml , may be assumed to be normally distributed with mean \(\mu \) ml and standard </span><span style="font-family: times new roman,times; font-size: medium;">deviation \(8\) ml . Initially, the machine is adjusted so that the value of \(\mu \) is \(500\). </span><span style="font-family: times new roman,times; font-size: medium;">In order to check that the value of \(\mu \) remains equal to \(500\), a random sample </span><span style="font-family: times new roman,times; font-size: medium;">of 10 bottles is selected at regular intervals, and the mean amount of water, \(\overline y \) , </span><span style="font-family: times new roman,times; font-size: medium;">in these bottles is calculated. The following hypotheses are set up.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({{\rm{H}}_0}:\mu = 500\) ; \({{\rm{H}}_1}:\mu \ne 500\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The critical region is defined to be \(\left( {\overline y < 495} \right) \cup \left( {\overline y > 505} \right)\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the significance level of this procedure.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Some time later, the actual value of \(\mu \) is \(503\). Find the probability of a </span><span style="font-family: times new roman,times; font-size: medium;">Type II error.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The function \(f\) is defined by \(f(x) = \ln (1 + \sin x)\) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When a scientist measures the concentration \(\mu \) of a solution, the measurement </span><span style="font-family: times new roman,times; font-size: medium;">obtained may be assumed to be a normally distributed random variable with mean </span><span style="font-family: times new roman,times; font-size: medium;">\(\mu \) and standard deviation \(1.6\).</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;">Show that \(f''(x) = \frac{{ - 1}}{{1 + \sin x}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Determine the Maclaurin series for \(f(x)\) as far as the term in \({x^4}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">A.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;">Deduce the Maclaurin series for \(\ln (1 - \sin x)\) as far as the term in \({x^4}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">By combining your two series, show that \(\ln \sec x = \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{12}} + \ldots \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">A.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, find \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \sec x}}{{x\sqrt x }}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">He makes 5 independent measurements of the concentration of a particular </span><span style="font-family: times new roman,times; font-size: medium;">solution and correctly calculates the following confidence interval for \(\mu \) .</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">[\(22.7\) , \(26.1\)]</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Determine the confidence level of this interval.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">He is now given a different solution and is asked to determine a \(95\%\) confidence </span><span style="font-family: times new roman,times; font-size: medium;">interval for its concentration. The confidence interval is required to have a width </span><span style="font-family: times new roman,times; font-size: medium;">less than \(2\). Find the minimum number of independent measurements required.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">B.b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The random variable \(X\) has probability density function given by</p>
<p class="p1">\[f(x) = \left\{ {\begin{array}{*{20}{l}}<br> {x{{\text{e}}^{ - x}},}&{{\text{for }}x \geqslant 0,} \\ <br> {0,}&{{\text{otherwise}}} <br>\end{array}} \right..\]</p>
</div>
<div class="specification">
<p class="p1">A sample of size <span class="s1">50 </span>is taken from the distribution of \(X\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use l’Hôpital’s rule to show that \(\mathop {\lim }\limits_{x \to \infty } \frac{{{x^3}}}{{{{\text{e}}^x}}} = 0\).</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">(i) <span class="Apple-converted-space"> </span>Find \({\text{E}}({X^2})\)<span class="s1">.</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Show that \({\text{Var}}(X) = 2\).</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the central limit theorem.</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">Find the probability that the sample mean is less than <span class="s1">2.3</span><span class="s2">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In a large population of sheep, their weights are normally distributed with mean \(\mu \) <span class="s1">kg </span>and standard deviation \(\sigma \) <span class="s1">kg. A random sample of \(100\) </span>sheep is taken from the population.</p>
<p class="p1">The mean weight of the sample is \(\bar X\) <span class="s1">kg</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the distribution of \(\bar X\) , giving its mean and standard deviation.</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">The sample values are summarized as \(\sum {x = 3782} \) and \(\sum {{x^2} = 155341} \) where \(x\) <span class="s1">kg </span>is the weight of a sheep.</p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>Find unbiased estimates for<span class="s2"> \(\mu \) </span>and<span class="s2"> \({\sigma ^2}\)</span><span class="s3">.</span></p>
<p class="p2"><span class="s3">(ii) <span class="Apple-converted-space"> </span>Find a \(95\%\) </span>confidence interval for<span class="s2"> \(\mu \)</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>Test, at the \(1\%\) level of significance, the null hypothesis \(\mu = 35\) against the alternative hypothesis that \(\mu > 35\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable \(X\) has the binomial distribution \({\text{B}}(n,{\text{ }}p)\), where \(n > 1\).</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;">Show that</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) \(\frac{X}{n}\) is an unbiased estimator for \(p\);</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;">(b) \({\left( {\frac{X}{n}} \right)^2}\) is <strong>not </strong>an unbiased estimator for \({p^2}\);</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;">(c) \(\frac{{X(X - 1)}}{{n(n - 1)}}\) is an unbiased estimator for \({p^2}\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The continuous random variable \(X\) takes values only in the interval [\(a\), \(b\)] and \(F\) </span><span style="font-family: times new roman,times; font-size: medium;">denotes its cumulative distribution function. Using integration by parts, show that:\[E(X) = b - \int_a^b {F(x){\rm{d}}x}. \]<br></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><span style="font-family: times new roman,times; font-size: medium;">The continuous random variable \(Y\) has probability density function \(f\) given by:\[\begin{array}{*{20}{c}}<br> {f(y) = \cos y,}&{0 \leqslant y \leqslant \frac{\pi }{2}} \\ <br> {f(y) = 0,}&{{\text{elsewhere}}{\text{.}}} <br>\end{array}\]</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Obtain an expression for the cumulative distribution function of \(Y\) , valid </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le y \le \frac{\pi }{2}\) . Use the result in (a) to determine \(E(Y)\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) The random variable \(U\) is defined by \(U = {Y^n}\) , where \(n \in {\mathbb{Z}^ + }\) . Obtain </span><span style="font-family: times new roman,times; font-size: medium;">an expression for the cumulative distribution function of \(U\) valid for \(0 \le u \le {\left( {\frac{\pi }{2}} \right)^n}\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) The medians of \(U\) and \(Y\) are denoted respectively by \({m_u}\) and \({m_y}\) . </span><span style="font-family: times new roman,times; font-size: medium;">Show that \({m_u} = m_y^n\) .</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A random variable \(X\) has probability density function \(f\) given by:\[f(x) = \left\{ {\begin{array}{*{20}{l}}<br> {\lambda {e^{ - \lambda x}},}&{{\text{for }}x \geqslant 0{\text{ where }}\lambda > 0} \\ <br> {0,}&{{\text{for }}x < 0.} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for \({\rm{P}}(X > a)\) , where \(a > 0\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A chicken crosses a road. It is known that cars pass the chicken’s crossing </span><span style="font-family: times new roman,times; font-size: medium;">route, with intervals between cars measured in seconds, according to the random </span><span style="font-family: times new roman,times; font-size: medium;">variable \(X\) , with \(\lambda = 0.03\) . The chicken, which takes \(10\) seconds to cross the </span><span style="font-family: times new roman,times; font-size: medium;">road, starts to cross just as one car passes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that the chicken will reach the other side of the road </span><span style="font-family: times new roman,times; font-size: medium;">before the next car arrives.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Later, the chicken crosses the road again just after a car has passed.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Show that the probability that the chicken completes both crossings is </span><span style="font-family: times new roman,times; font-size: medium;">greater than \(0.5\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A rifleman shoots at a circular target. The distance in centimetres from the </span><span style="font-family: times new roman,times; font-size: medium;">centre of the target at which the bullet hits, can be modelled by \(X\) with \(\lambda = 0.4\) . </span><span style="font-family: times new roman,times; font-size: medium;">The rifleman scores \(10\) points if \(X \le 1\) , \(5\) points if \(1 < X \le 5\) , \(1\) point if </span><span style="font-family: times new roman,times; font-size: medium;">\(5 < X \le 10\) and no points if \(X > 10\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the expected score when one bullet is fired at the target.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A second rifleman, whose shooting can also be modelled by \(X\) , wishes to find his </span><span style="font-family: times new roman,times; font-size: medium;">value of \(\lambda \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that his expected score is \(6.5\), find his value of \(\lambda \) .</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The discrete random variable \(X\) follows the distribution Geo(\(p\)).</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Arthur tosses a biased coin each morning to decide whether to walk or cycle to school; </span><span style="font-family: times new roman,times; font-size: medium;">he walks if the coin shows a head.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The probability of obtaining a head is \(0.55\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the mode of \(X\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the exact value of \(p\) if \({\rm{Var}}(X) = \frac{{28}}{9}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the smallest value of \(n\) for which the probability of Arthur walking to </span><span style="font-family: times new roman,times; font-size: medium;">school on the next \(n\) days is less than \(0.01\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that Arthur cycles to school for the third time on the </span><span style="font-family: times new roman,times; font-size: medium;">last of eight successive days.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>