File "markSceme-HL-paper3.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 4/markSceme-HL-paper3html
File size: 24.97 KB
MIME-type: text/html
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-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.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>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<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 class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</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="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<h2>HL Paper 3</h2><div class="specification">
<p>The random variables <em>X</em> , <em>Y</em> follow a bivariate normal distribution with product moment correlation coefficient <em>ρ</em>.</p>
</div>
<div class="specification">
<p>A random sample of 11 observations on <em>X</em>, <em>Y</em> was obtained and the value of the sample product moment correlation coefficient, <em>r</em>, was calculated to be −0.708.</p>
</div>
<div class="specification">
<p>The covariance of the random variables <em>U</em>, <em>V</em> is defined by</p>
<p style="text-align: center;">Cov(<em>U</em>, <em>V</em>) = E((<em>U</em> − E(<em>U</em>))(<em>V</em> − E(<em>V</em>))).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses to investigate whether or not a negative linear association exists between <em>X</em> and <em>Y</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the <em>p</em>-value.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State your conclusion at the 1 % significance level.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Cov(<em>U</em>, <em>V</em>) = E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that if <em>U</em>, <em>V</em> are independent random variables then the population product moment correlation coefficient, <em>ρ</em>, is zero.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>H<sub>0 </sub>: <em>ρ</em> = 0; H<sub>1 </sub>: <em>ρ</em> < 0 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = - 0.708\sqrt {\frac{{11 - 2}}{{1 - {{\left( { - 0.708} \right)}^2}}}} \,\, = \,\,\left( { - 3.0075 \ldots } \right)">
<mi>t</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.708</mn>
<msqrt>
<mfrac>
<mrow>
<mn>11</mn>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>0.708</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>3.0075</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>degrees of freedom = 9 <em><strong>(A1)</strong></em></p>
<p>P(<em>T</em> < −3.0075...) = 0.00739 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept any answer that rounds to 0.0074.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>reject H<sub>0</sub> or equivalent statement <em><strong> R1</strong></em></p>
<p><strong>Note:</strong> Apply follow through on the candidate’s <em>p</em>-value.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Cov(<em>U</em>, <em>V</em>) + E((<em>U</em> − E(<em>U</em>))(<em>V</em> − E(<em>V</em>)))</p>
<p>= E(<em>UV </em>− E(<em>U</em>)<em>V </em>− E(<em>V</em>)<em>U </em>+ E(<em>U</em>)E(<em>V</em>)) <strong>M1</strong></p>
<p>= E(<em>UV</em>) − E(E(<em>U</em>)<em>V</em>) − E(E(<em>V</em>)<em>U</em>) + E(E(<em>U</em>)E(<em>V</em>)) <em><strong>(A1)</strong></em></p>
<p>= E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>) − E(<em>V</em>)E(<em>U</em>) + E(<em>U</em>)E(<em>V</em>) <em><strong>A1</strong></em></p>
<p>Cov(<em>U</em>, <em>V</em>) = E(<em>UV</em>) − E(<em>U</em>)E(<em>V</em>) <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>E(<em>UV</em>) = E(<em>U</em>)E(<em>V</em>) (independent random variables) <em><strong>R1</strong></em></p>
<p>⇒Cov(<em>U</em>, <em>V</em>) = E(<em>U</em>)E(<em>V</em>) − E(<em>U</em>)E(<em>V</em>) = 0 <em><strong>A1</strong></em></p>
<p>hence, <em>ρ = </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{Cov}}\left( {U,\,V} \right)}}{{\sqrt {{\text{Var}}\left( U \right)\,{\text{Var}}\left( V \right)} }} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>Cov</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>U</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>V</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msqrt>
<mrow>
<mtext>Var</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>U</mi>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>Var</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1AG</strong></em></p>
<p><strong>Note:</strong> Accept the statement that Cov(<em>U</em>,<em>V</em>) is the numerator of the formula for <em>ρ</em>.</p>
<p><strong>Note:</strong> Only award the first <em><strong>A1</strong> </em>if the <em><strong>R1</strong> </em>is awarded.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Peter, the Principal of a college, believes that there is an association between the score in a Mathematics test, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, and the time taken to run 500 m, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> seconds, of his students. The following paired data are collected.</p>
<p><img src="data:image/png;base64,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"></p>
<p>It can be assumed that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {X{\text{, }}Y} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>Y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> follow a bivariate normal distribution with product moment correlation coefficient <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\rho ">
<mi>ρ<!-- ρ --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State suitable hypotheses <span class="mjpage"><math alttext="{H_0}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math alttext="{H_1}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> </math></span> to test Peter’s claim, using a two-tailed test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Carry out a suitable test at the 5 % significance level. With reference to the <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>-value, state your conclusion in the context of Peter’s claim.</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>Peter uses the regression line of <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> on <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> as <span class="mjpage"><math alttext="y = 0.248x + 83.0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mn>0.248</mn> <mi>x</mi> <mo>+</mo> <mn>83.0</mn> </math></span> and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math alttext="{H_0}\,{\text{:}}\,\rho = 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </math></span> <span class="mjpage"><math alttext="{H_1}\,{\text{:}}\,\rho \ne 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>ρ</mi> <mo>≠</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> It must be <span class="mjpage"><math alttext="\rho " xmlns="http://www.w3.org/1998/Math/MathML"> <mi>ρ</mi> </math></span>.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math alttext="p = 0.649" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> <mo>=</mo> <mn>0.649</mn> </math></span> <em><strong> A2</strong></em></p>
<p><strong>Note:</strong> Accept anything that rounds to 0.65</p>
<p>0.649 > 0.05 <em><strong>R1</strong></em></p>
<p>hence, we accept <span class="mjpage"><math alttext="{H_0}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> </mrow> </math></span> and conclude that Peter’s claim is wrong <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> The <em><strong>A</strong></em> mark depends on the <em><strong>R</strong></em> mark and the answer must be given in context. Follow through the <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>-value in part (b).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>a statement along along the lines of ‘(we have accepted that) the two variables are independent’ or ‘the two variables are weakly correlated’ <em><strong>R1</strong></em></p>
<p>a statement along the lines of ‘the use of the regression line is invalid’ or ‘it would give an inaccurate result’ <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award the second <strong><em>R1</em></strong> only if the first <em><strong>R1</strong></em> is awarded.</p>
<p><strong>Note:</strong> FT the conclusion in(a)(ii). If a candidate concludes that the claim is correct, mark as follows: (as we have accepted H<sub>1</sub>) the 2 variables are dependent and 73 lies in the range of <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> values <em><strong>R1</strong></em>, hence the use of the regression line is valid <em><strong>R1</strong></em>. </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>