File "statistics-and-probability-examination-questions-hl.html"
Path: /StudyIB/mathsanalysis/page/2910/statistics-and-probability-examination-questions-hlhtml
File size: 110.19 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html><html lang="EN"><head> <script>(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-PZBWZ8J');</script> <title>Statistics and Probability Examination Questions HL</title><!-- Removed by WebCopy --><!--<base href="/">--><!-- Removed by WebCopy --><meta http-equiv="x-ua-compatible" content="IE=Edge"><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1"><link href="../../../css/bootstrap.flatly.min.css" rel="stylesheet" media="screen"><link rel="stylesheet" href="../../../fonts/awesome/css/font-awesome.min.css"><link href="../../../js/jquery-fancybox/jquery.fancybox.min.css" type="text/css" rel="stylesheet"><link rel="stylesheet" href="../../../css/style.min.css?v=202301301645"><meta name="robots" content="index, follow"><meta name="googlebot" content="noarchive"><meta prefix="og: http://ogp.me/ns#" property="og:title" name="og:title" content="StudyIB Maths: Analysis & Approaches: Statistics and Probability Examination Questions HL"> <meta prefix="og: http://ogp.me/ns#" property="og:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta prefix="og: http://ogp.me/ns#" property="og:description" name="og:description" content="On this page you can find examination questions from the topic of statistics and probability"> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsanalysis/page/2910/statistics-and-probability-examination-questions-hl"> <meta prefix="og: http://ogp.me/ns#" property="og:site_name" name="og:site_name" content="StudyIB - Your IB learning companion"> <meta prefix="og: http://ogp.me/ns#" property="og:locale" name="og:locale" content="en"> <meta prefix="og: http://ogp.me/ns#" property="og:type" name="og:type" content="website"> <meta name="description" content="On this page you can find examination questions from the topic of statistics and probability"> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="IB, IBDP, InThinking, International Baccalaureate, Revision, Revision websites, Student Sites, Students, Learning, Guidance, Exam, Questions, Practice problems, Diagnose, Help, Notes"> <meta itemprop="name" content="StudyIB Maths: Analysis & Approaches: Statistics and Probability Examination Questions HL"> <meta itemprop="description" content="On this page you can find examination questions from the topic of statistics and probability"> <meta itemprop="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:title" content="StudyIB Maths: Analysis & Approaches: Statistics and Probability Examination Questions HL"> <meta name="twitter:description" content="On this page you can find examination questions from the topic of statistics and probability"> <meta name="twitter:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:creator" content="@inthinker"><link rel="stylesheet" href="../../../css/snippets.min.css?v=202209111300"><link rel="stylesheet" href="../../../css/article.min.css?v=202211221000"><style type="text/css">.filter-sl-only { display: none; }</style><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script src="../../../js/ifvisible.min.js"></script><script>ifvisible.setIdleDuration(300);</script><link rel="apple-touch-icon-precomposed" sizes="57x57" href="../../../img/favicon/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon-precomposed" sizes="114x114" href="../../../img/favicon/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon-precomposed" sizes="72x72" href="../../../img/favicon/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="../../../img/favicon/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon-precomposed" sizes="60x60" href="../../../img/favicon/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon-precomposed" sizes="120x120" href="../../../img/favicon/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon-precomposed" sizes="76x76" href="../../../img/favicon/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon-precomposed" sizes="152x152" href="../../../img/favicon/apple-touch-icon-152x152.png"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-196x196.png" sizes="196x196"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-16x16.png" sizes="16x16"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-128.png" sizes="128x128"> <meta name="application-name" content="StudyIB: Your IB Learning Companion"> <meta name="msapplication-TileColor" content="#A5BED5"> <meta name="msapplication-TileImage" content="/img/favicon/mstile-144x144.png"> <meta name="msapplication-square70x70logo" content="/img/favicon/mstile-70x70.png"> <meta name="msapplication-square150x150logo" content="/img/favicon/mstile-150x150.png"> <meta name="msapplication-wide310x150logo" content="/img/favicon/mstile-310x150.png"> <meta name="msapplication-square310x310logo" content="/img/favicon/mstile-310x310.png"><script>var stdHash = "6da675cbcb3d25f050b05557349abc79", stdTicket = "082b9c9c4ae3624d";</script><script src="../../../js/user/local-stats.min.js?v=202205311700"></script><link href="../../../css/subjects-frontpage.min.css?v=202302031000" rel="stylesheet"></head><body class="public mathsanalysis"> <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PZBWZ8J" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <div id="top-header"> <div class="wmap general hidden-sm hidden-xs subjects"> <div class="layout-wrapper"> <div class="container-fluid"> <a href=""> <img class="header-thinker" src="../../../img/header-thinker.svg"> </a> <h1> <a href="../../../mathsanalysis.html"> <span style="font-family: 'Helvetica Narrow','Arial Narrow',Tahoma,Arial,Helvetica,sans-serif;font-stretch: condensed;letter-spacing: -1px">IBDP Maths: Analysis & Approaches</span> </a> <a href="https://inthinking.net" title="inthinking.net" class="inthinking-logo pull-right"> <img src="../../../img/header-logo.svg" style="height: 80px; width: auto;"> </a> </h1> <p class="slogan">InThinking Revision Sites for students</p> <p class="author">Website by <strong>Richard Wade</strong></p> <p class="updated">Updated 3 February 2023</p> </div> </div> </div> <nav id="public-topnav" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <a class="navbar-brand hidden-md hidden-lg" href="../../../index.htm"> <img class="header-xs-thinker" src="../../../img/header-thinker.svg"> </a> <button type="button" class="collapsed navbar-toggle" data-toggle="collapse" data-target="#subject-navbar-collapse" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <i class="fa fa-fw fa-2x fa-user"></i> </button> <div class="brand hidden-lg hidden-md hidden-sm"> <a class="brand-xs" href="../../../mathsanalysis.html"> <strong class="title" style="white-space: nowrap;font-size: 18px;">Maths: A&A</strong> </a> </div> </div> <div class="collapse navbar-collapse" id="subject-navbar-collapse"> <ul class="nav navbar-bar navbar-userbox hidden-md hidden-lg"><li class="dropdown"><a href="#" class="dropdown-toggle" data-toggle="dropdown"><i class="fa fa-fw fa-lg fa-user-circle" style="margin-right: 5px;"></i><span style="font-size: 16px;"></span></a><ul class="dropdown-menu"><li class="dropdown-submenu"><a class="dropdown-toggle" href="#" data-toggle="dropdown"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="menu-item"><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li class="menu-item"><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li class="menu-item"><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li><li class="menu-item hidden-md hidden-lg mt-xs-3"><a href="../../../index.htm?logout=1" class="btn btn-primary btn-xs-block"><i class="fa fa-fw fa-lg fa-power-off text-danger"></i>Log out</a></li></ul></li></ul> <ul class="nav navbar-bar"><li class=""><a class="home" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-home"></i> Home</a></li><li class=""><a class="topics" href="#"><i class="fa fa-fw fa-th-large"></i> Topics</a></li><li class=""><a class="favorites" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-star"></i> My favorites</a></li><li class=""><a class="qbank" href="../../test-your-knowledge.html"><i class="fa fa-fw fa-question"></i> Question bank</a></li><li class=""><a class="sitemap" href="../../sitemap.html"><i class="fa fa-fw fa-sitemap"></i> Sitemap</a></li><li class=""><a class="activity" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-calendar-check-o"></i> My activity</a></li></ul> <ul class="nav navbar-bar navbar-right"> <li> <form class="navbar-form hidden-md hidden-lg" role="search" method="get" action="mathsanalysis/search"> <div class="input-group"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search..."> <span class="input-group-btn"> <button type="submit" class="btn bg-blue"> <i class="fa fa-search"> </i> </button> </span> </div> </form> <a href="#" class="toggle-menu-search hidden-sm hidden-xs" data-toggle="dropdown"> <i class="fa fa-lg fa-search"></i> </a> </li></ul> </div> </div></nav><nav id="nav-menu-search" class="shadow-bottom hidden-sm hidden-xs" style="display: none;"> <div class="container-fluid"> <form class="" role="search" method="get" action="mathsanalysis/search"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search Maths: A&A..."> <button class="btn btn-sm btn-primary" type="submit"> Search </button> </form> </div></nav><div class="modal fade" tabindex="-1" role="dialog" id="modal-mini-login"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-body"> <button aria-hidden="true" data-dismiss="modal" class="close hidden-xs login-button-close" type="button">×</button> <form method="post"> <div class="form-group"> <input name="user-email" class="form-control" type="text" placeholder="Email"> </div> <div class="form-group"> <input name="user-password" class="form-control" type="password" placeholder="Password"> <input name="user-fp" class="fp" value="d8c893da5db7501669ffeeac46273ba6" type="hidden"> </div> <div class="form-group"> <a href="../../../reset-password.html" class="reset-password" style="font-weight: normal;"> Forgot your password? </a> </div> <div class="form-group"> <button type="submit" name="submit-login" value="1" class="btn btn-primary btn-block text-center"> <i class="fa fa-user fa-fw"></i> Log in </button> </div> </form> </div> </div> </div></div></div><div class="modal fade" id="show-topics" tabindex="-1" role="dialog"><div id="dialog-topics" class="modal-dialog modal-dialog-topics"><div class="modal-content"><div class="modal-body modal-body-topics"><div id="new-frontpage-toplevels-topics" class="frontpage-box"><div class="row"><div class="col-md-4 col-sm-4"><div class="item"><a href="../3016/free-access-weekend.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Free Access Weekend</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/free.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2696/start-here.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Start Here</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/start-here.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2902/examination-questions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Examination Questions</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/exam.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2563/question-bank.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Question Bank</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/questionbank.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../537/algebra.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Algebra</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/algebra.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../539/functions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Functions</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/function_s.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../540/geometry-trigonometry.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Geometry & Trigonometry</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/photo-by-a-href=httpsunsplash.com@pawel_czerwinskiutm_source=unsplash-utm_medium=referral-utm_content=creditcopytextpawel-czerwinskia-on-a-href=httpsunsplash.comutm_source=unsplash-utm_medium=referral-utm_co.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../549/stats-probability.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Stats & Probability</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/probability.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../550/calculus.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Calculus</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/calculus1.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../857/exam-tips.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Exam Tips</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/files/exam-tips/exam-tips.jpg'); height: 10vw;"></div></div></a></div></div></div></div></div></div></div></div><div class="container-fluid shadow-ith"><div class="row frontpage"> <div class="col-md-3" id="left-column"> <p class="visible-xs-block"></p><div class="dropdown user-menu hidden-xs" style="margin-bottom: 20px;"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" style="border-radius: 0; padding: 6px 18px;"> <i class="fa fa-fw fa-user-circle"></i> <small><i class="fa fa-caret-down"></i></small> </button> <ul class="dropdown-menu"> <li class="dropdown-submenu"><a href="#"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul id="user-subjects-menu" class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="divider"></li><li><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li> <li class="divider"></li> <li> <a href="../../../index.htm?logout=1"> <i class="fa fa-fw fa-power-off"></i> Log out </a> </li> </ul></div> <div id="side-nav"><h4 class="side-nav-title dropdown"><a href="../2902/examination-questions.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../2902/examination-questions.html">Examination Questions</a><ul class="dropdown-menu"><li><a href="../../../mathsanalysis.html" style="padding-left: 5px; border-bottom: solid 1px #ddd"><i class="fa fa-fw fa-home"></i> Home</a></li><li><a href="../3016/free-access-weekend.html"><i class="fa fa-fw fa-caret-right"></i> Free Access Weekend</a></li><li><a href="../2696/start-here.html"><i class="fa fa-fw fa-caret-right"></i> Start Here</a></li><li><a href="../2563/question-bank.html"><i class="fa fa-fw fa-caret-right"></i> Question Bank</a></li><li><a href="../537/algebra.html"><i class="fa fa-fw fa-caret-right"></i> Algebra</a></li><li><a href="../539/functions.html"><i class="fa fa-fw fa-caret-right"></i> Functions</a></li><li><a href="../540/geometry-trigonometry.html"><i class="fa fa-fw fa-caret-right"></i> Geometry & Trigonometry</a></li><li><a href="../549/stats-probability.html"><i class="fa fa-fw fa-caret-right"></i> Stats & Probability</a></li><li><a href="../550/calculus.html"><i class="fa fa-fw fa-caret-right"></i> Calculus</a></li><li><a href="../857/exam-tips.html"><i class="fa fa-fw fa-caret-right"></i> Exam Tips</a></li></ul><div class="pull-right"><a class="sidenav-expand" title="Expand all" href="#"><i class="fa fa-plus-circle"></i></a> <a class="sidenav-compress" title="Compress all" href="#"><i class="fa fa-minus-circle"></i></a></div></h4><ul class="side-nav level-0"><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2904/algebra-examination-questions-hl.html">Algebra Examination Questions HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2906/functions-examination-questions-hl.html">Functions Examination Questions HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2908/geometry-and-trigonometry-examination-questions-hl.html">Geometry and Trigonometry Examination Questions HL</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="statistics-and-probability-examination-questions-hl.html">Statistics and Probability Examination Questions HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2912/calculus-examination-questions-hl.html">Calculus Examination Questions HL</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Statistics and Probability Examination Questions HL <a href="#" class="mark-page-favorite pull-right" data-pid="2910" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../2902/examination-questions.html">Examination Questions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Statistics and Probability Examination Questions HL</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p>On this page you can find examination questions from the topic of statistics and probability</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Representing and Measuring Data</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="877"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">During week 1, a group of 60 athletes were asked to record the amount of water, X litres, that they consumed in that week. Here are the results</span></p> <p><span class="math-tex">\(\sum x=1470\)</span></p> <p><span class="math-tex">\(\sum x^2=36\ 132.6\)</span></p> <p>Calculate</p> <p style="margin-left: 40px;">a. the mean of X</p> <p style="margin-left: 40px;">b. the standard deviation of X</p> <p>During week 2, as part of a programme to improve their performance, the athletes were instructed to drink 10% more water. Assuming that they do this, find for week 2</p> <p style="margin-left: 40px;">c. the new mean</p> <p style="margin-left: 40px;">d. the new standard deviation</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>If every athlete drinks 10% more, then each piece of data is multiplied by 1.1</p> <p>Mean is multiplied by 1.1</p> Standard deviation is multiplied by 1.1</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/esq_representing_and_measuring_data1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/esq_representing_and_measuring_data1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="878"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The graph below shows the scores in the SL Mathematics Analysis and Approaches for at Goodenough High School. There are 20 students, the minimum score was 2 and the mean score is 4.45</p> <p><img alt="" src="../../files/data-handling/esq2.bmp" style="width: 500px; height: 390px;"></p> <p>How many grade 4s and how many grade 7s were there?</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let x be the number of 4s and y be the number of 7s.</p> <p><content>Write 2 equations for the information given.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let x be the number of 4s and y be the number of 7s.</p> <p>There are 20 students altogether: 2 + 3 + x + 4 + 3 + y = 20</p> <p>x + y = 8</p> <p>The mean score is 4.45 which mean that the sum of all the grades is 4.45x20 = 89</p> <p>4 + 9 + 4x + 20 + 18 + 7y = 89</p> <p>4x + 7y = 38</p> <p>Solve the simultaneous equations:</p> <p>4x + 7(8 - x) = 38</p> <p>4x + 56 - 7x = 38</p> <p>-3x = -18</p> <p>x = 6</p> <p>y = 2</p> <p>6 students got a grade 4</p> <p>2 students go a grade 7</p> <p><img alt="" src="../../files/data-handling/esq2ans.bmp" style="width: 400px; height: 312px;"></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="879"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span style="font-size:12.0pt;line-height:115%; font-family:" timesnewromanpsmt",sans-serif;mso-fareast-font-family:calibri;="" mso-bidi-font-family:timesnewromanpsmt;mso-ansi-language:en-gb;mso-fareast-language:="" en-gb;mso-bidi-language:ar-sa"="">The cumulative frequency graph gives information about the lengths, in minutes, of 80 telephone calls.</span></p> <p><img alt="" src="../../files/data-handling/esq3a.png" style="width: 750px; height: 500px;"></p> <p>a. Find the median length of a phone call</p> <p>b. Find the interquartile range of the length of a phone call</p> <p>c. Find the number of phone calls that were more than 10 minutes in length</p> <p>d. The frequency table below shows the lengths of the 80 phone calls. Find values a, b and c.</p> <p><img alt="" src="../../files/data-handling/esq3b.png" style="width: 300px; height: 344px;"></p> <div> <p>e. These data contain some outliers. How many outliers are there?</p> </div> <p>f. Calculate an estimate of the mean length of a phone call</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Read values from the graph carefully<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/esq_representing_and_measuring_data3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/esq_representing_and_measuring_data3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Linear Correlation</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="882"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span lang="EN-US" style="font-size:11.0pt;line-height: 115%;font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-us;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa;mso-no-proof:yes"="">The world record times in seconds for the women’s 100m sprint from 1970 onwards are given below</span></p> <p><img alt="" src="../../files/data-handling/bivariate-data/esq1.png" style="width: 600px; height: 77px;"></p> <p>Use your calculator to write down</p> <p>a) <span class="math-tex">\(\bar{x}\)</span> , the mean year</p> <p>b) <span class="math-tex">\(\bar{y}\)</span> , the mean time</p> <p>c) <span class="math-tex">\(r\)</span>, <span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">Pearson’s product-moment correlation coefficient</span></p> <p>The equation of the regression line y on x is y = ax + b</p> <p>d) Find the values of a and b for these data</p> <p>e) Show that <span class="math-tex">\((\bar{x},\bar{y})\)</span> lies on this line</p> <p>f) Use the regression line to estimate the world record time in 2024</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Enter the data carefully into your calculator. Then double check that you have typed them in correctly.<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/bivariate-data/esq_bivariate_data1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/bivariate-data/esq_bivariate_data1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="883"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The table below shows the test scores of SL students following a six-week revision period using studyib.</span></p> <p><img alt="" src="../../files/data-handling/bivariate-data/esq2.png" style="width: 600px; height: 77px;"></p> <p>a) Work out r, Pearson’s correlation coefficient</p> <p>The y on x regression line is y = ax + b</p> <p>b) Find a and b</p> <p>c) A student scores 80 marks before revision. Use the regression line to estimate the score after revision</p> <p>The x on y regression line is x = cx + d</p> <p>d) Find c and d</p> <p>e) A student scores 90 marks after revision. Use the regression line to estimate the score before revision</p> <p>f) Find the point of intersection of the y on x and the x on y regression lines</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>Enter the data carefully into your graphical calculator.</p> <p>Give answers to 3 significant figures, but use a higher degree of accuracy for calculations.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/bivariate-data/esq_bivariate_data2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/bivariate-data/esq_bivariate_data2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="881"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The following table shows the hand spans and the heights of eight basketball players on a team</span></p> <p><img alt="" src="../../files/data-handling/bivariate-data/esq3.png" style="width: 600px; height: 73px;"></p> <p>The relationship between x and y can be modelled by the x on y line of regression x = ay + b</p> <ul> <li>Find the values of a and b</li> <li>Write down the correlation coefficient</li> <li>Another basketball player is 193cm tall. Use this regression line to estimate the handspan of this player.</li> </ul> <p>Another player is 180cm tall. Use this regression line to estimate the handspan of this player.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Take care how you set up your calculator to find the <strong>x on y</strong> repression line<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/bivariate-data/esq_bivariate_data3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/bivariate-data/esq_bivariate_data3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="884"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The following table shows the age in days (x) and length of new-born babies in cm (y).</span></p> <p><img alt="" src="../../files/data-handling/bivariate-data/esq4.png" style="width: 600px; height: 78px;"></p> <p>The relationship between the variables is modelled by the regression line with equation y = ax + b</p> <p>a) Find the values of a and b</p> <p>b) Write down the correlation coefficient</p> <p>c) Use your equation to estimate the length of a baby that is 40 days old</p> <p>d) Use your equation to estimate the length of a baby that is 150 days old</p> <p>e) Use your equation to estimate the age of a child that is 54 cm long.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Think carefully about when you can use y on x regression line for predictions</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/data-handling/bivariate-data/esq_bivariate_data4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/data-handling/bivariate-data/esq_bivariate_data4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Probability Calculations</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="546"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Two events <strong><em>A</em></strong> and <strong><em>B </em></strong> are such that P(A) = 0.35 and P(B) = 0.6 and <span class="math-tex">\(P(A\cup B)\)</span> = 0.74</p> <p>a) Find <span class="math-tex">\(P(A\cap B)\)</span></p> <p>b) Determine whether A and B are independent</p> <p>c) Find <span class="math-tex">\(P(A'\cap B)\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) Put the information in a Venn diagram or use the rule <span class="math-tex">\(P(A\cup B)=P(A) +P(B)-P(A\cap B)\)</span></p> <p>b) If A and B are independent <span class="math-tex">\(P(A\cap B)=P(A)\times P(B)\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/probability_rules/esq_probability-rules1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="547"><p><img alt="" height="222" src="../../files/probability/probability_rules/esq2.png" style="float: right;" width="108"><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <div> <p>Katniss is practising archery. She fires three arrows at a target. The probability that she hits the target with her first arrow is 0.7. Whenever she hits the target, her confidence increases so that the probability that she hits the target on her next attempt increases by 0.1. Whenever she misses the target, the probability reduces by 0.1.</p> <p><img alt="" src="../../files/probability/probability_rules/esq2tree.jpg" style="width: 700px; height: 418px; float: left;">a) Complete the probability tree for Katniss’s three attempts.</p> <hr class="hidden-separator"></div> <div> <p>b) Calculate the probability that she hits the target with two attempts.</p> <p>c) Find the probability the she hits the target on at least one attempt.</p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Fill out the probability tree carefully, the probability of hits and misses changes each time</p> <p><content>c) P(at least one hit) = 1 - P(zero hits)</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/probability_rules/esq_probability-rules2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="548"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Events <strong><em>A</em></strong> and <strong><em>B</em></strong> are such that P(A) = 0.4 and <span class="math-tex">\(P(A\cup B)\)</span> = 0.7</p> <p>Find P(B) if <strong><em>A</em></strong> and <strong><em>B</em></strong> are independent</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>If <strong><em>A</em></strong> and <strong><em>B</em></strong> are independent, then <span class="math-tex">\(P(A\cap B) =P(A)\times P(B)\)</span><content> </content></p> <p>Write an equation and consider writing P(B) = x</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/probability_rules/esq_probability-rules3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="549"><p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <div> <p>Alphonse and Bettina are playing a game. A bag contains 2 yellow beads and 3 red beads. They take it in turns to pick a bead from the bag at random. Alphonse goes first. If Alphonse picks a yellow bead, he wins and the game stops. If he picks a red bead, he replaces the bead and it is Bettina’s turn. If Bettina picks a red bead, she wins and the game stops. If she picks a yellow bead, she replaces the bead and it is Alphonse’s turn again.</p> <p style="text-align: center;"><img alt="" src="../../files/probability/probability_rules/esq4.jpg" style="width: 300px; height: 71px;"></p> <p>a) Find the probability that Alphonse wins on his first turn.</p> <p>b) Show that the probability that Alphonse wins on his second turn is <span class="math-tex">\(\frac{12}{125}\)</span></p> <p>c) The game continues until one of the player wins. What is the probability that Alphonse wins the game?</p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>b) For Alphonse to win on his second turn, Alphonse must lose on his first turn, and then Bettina must lose on her first turn and then Alphonse must win on his second turn.</p> <p>c) The probability that Alphonse wins is the sum of an infinite geometric series <span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ 1-r } \)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/probability_rules/esq_probability-rules4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="550"><p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <div> <p>A box contains 25 tickets. <strong><em>x</em></strong> tickets are gold, the rest are silver. <strong>Two </strong>tickets are selected at random.</p> <p>a) Show that the probability of selecting two gold tickets is <span class="math-tex">\(\frac{x^2-x}{600}\)</span></p> <p>b) Find the probability of selecting two tickets of the same colour.</p> <p>c) The probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour. Find how many gold tickets there are.</p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) There are 25 - <strong><em>x</em></strong> silver tickets</p> <p>b) Find probability of GG or SS</p> <p>c) If the probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour, then the probability of selecting two tickets of the same colour = <span class="math-tex">\(\frac{2}{3}\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/probability_rules/esq_probability-rules5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules5.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Conditional Probability</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="520"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Henri Tarr travels to school by bike <span class="math-tex">\(\frac{3}{5}\)</span> of the week and by car the rest of the time.</p> <p>If he travels by bike, the probability that he is late is <span class="math-tex">\(\frac{1}{8}\)</span>.</p> <p>If he travels by car, the probability that he is late is <span class="math-tex">\(\frac{3}{8}\)</span>.</p> <p>a) Copy and complete the following tree diagram</p> <p><img alt="" src="../../files/probability/conditional-probability/esq1.jpg" style="width: 507px; height: 357px;"></p> <p>b) Find the probability that Henri goes to school by car and is late for school.</p> <p>c) Find the probability that he is late for school.</p> <p>d) Given that he is late, find the probability that he travels to school by car.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) Probability Late = P(car AND late) OR P(bike AND late)</p> <p><content>d) This is a question about conditional probability. Use the previous parts.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="521"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A and B are independent events. P(A) = 0.3 and P(B) = 0.4</p> <p>a) Find <span class="math-tex">\(P(A'\cap B')\)</span></p> <p>b) Hence find P(A'|B')</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>A Venn diagram would be helpful.</p> <p>Since A and B are independent, <span class="math-tex">\(P(A\cap B)=P(A)\times P(B)\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="522"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <div> <p>In the Olympic Games, it is believed that 5% of sprinters are taking anabolic steroids. A test is developed that gives a positive result for an athlete who <strong><u>has </u></strong>taken anabolic steroids in the last month in 90% of cases. The test gives a positive result for an athlete who <strong><u>has not </u></strong>taken anabolic steroids in the last month in 15% of cases. A sprinter in chosen at random.</p> <p>a) Find the probability that she/he tests positive anabolic steroids</p> <p>b) Given that the athlete tests positive, find the probability that the athlete has taken anabolic steroids in the last month.</p> <hr class="hidden-separator"></div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Put the information in a tree diagram</p> <p>This is Bayes' Thereom.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="523"><p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A and B are independent events.</p> <p>P(A) = P(B) = p</p> <p>Show that P(A'|B')=1-p</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>A and B are independent, therefore <span class="math-tex">\(P(A\cap B)=P(A)\times P(B)\)</span></p> <p><content>Put all the information in a Venn diagram</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq4.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Discrete Random Variables</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The discrete random variable X has probability function</p> <p style="margin-left: 80px;"> P(X = x) = k(16 – x²) for x = 0, 1, 2, 3</p> <ol style="list-style-type:lower-alpha;"> <li>Find the value of the constant k</li> <li>Find P(1 ≤ X < 3)</li> </ol> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>The sum of the probabilities = 1</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/discrete-random-variables/esq1_discretervs.pdf" target="_blank">here</a></p> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The random variable <em>X </em>has probability function</p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq2.png" style="width: 300px; height: 57px;"></p> <p>a) Find the value of <strong><em>k</em></strong></p> <p>b) Work out <span class="math-tex">\(P(X\ge 2)\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Work out some probabilities by substituting the different values of x into the function</p> <p><content>e.g. P(X=1) = 1x</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/discrete-random-variables/esq2_discretervs.pdf" target="_blank">here</a></p> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>Two boxes each contain three cards. </p> <p>The first box contains cards labelled 1,3 and 5</p> <p>The second box contains cards labelled 2, 4 and 6. </p> <p>In a game, a player draws one card at random from each box and his score, X, is the <strong>sum</strong> of the numbers on the two cards.</p> <p>a) Complete the following probability distribution</p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq3.png" style="width: 600px; height: 76px;"></p> <p>b) Work out E(X)</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Work out all the possible outcomes</p> <p><content><img alt="" src="../../files/probability/discrete-random-variables/esq3expl.png" style="width: 300px; height: 189px;"></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/discrete-random-variables/esq3_discretervs.pdf" target="_blank">here</a></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The following table shows the probability distribution of a discrete random variable <strong><em>X</em></strong></p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq4.png" style="width: 600px; height: 89px;"></p> <p>If <strong><em>X </em></strong>represents the return from a game. Find <strong><em>a</em></strong> and <strong><em>b</em></strong> if the game is<strong> fair</strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>We know that <span class="math-tex">\(\sum P(X)=1\)</span><content> </content></p> <p>Fair game means E(X) = 0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/discrete-random-variables/esq4_discretervs.pdf" target="_blank">here</a></p> </section> <h4> </h4> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The discrete random variable <em>X </em>has probability function</p> <p><span class="math-tex">\(P(X=x)=k(\frac{2}{5})^x\)</span> , for <span class="math-tex">\(x\in Z ,\quad x>0\)</span></p> <p>Work out the value of <strong><em>k</em></strong></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Work out the first few probabilities...they make an infinite geometric series<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/discrete-random-variables/esq5_discretervs.pdf" target="_blank">here</a></p> </section> <h4> </h4> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Binomial Distribution</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="525"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/probability/binomial-distribution/esq1.png" style="float: right; width: 100px; height: 200px;">On average, it is found that 5% of AirPods* made on a production line are faulty.</p> <div> <p>a) Find the probability that in a random sample of 10, there are</p> <p>i) No faulty AirPods</p> <p>ii) more than one faulty set of AirPods</p> <p>b) A sample of <strong><em>n </em></strong>sets of AirPods is taken from the production line. If the probability that there is at least one faulty AirPod is <strong>more than 75%,</strong> find the smallest possible value of <strong><em>n</em></strong></p> <p><strong><em>* </em></strong><em>Fictitious data. I’m sure that AirPods production is very reliable!</em></p> <hr class="hidden-separator"></div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) ii) <span class="math-tex">\(P(X>1)=P(X\ge 2)\)</span></p> <p>b) Find <strong><em>n</em></strong> such that <span class="math-tex">\(P(X\ge 1)>0.75\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/binomial-distribution/esq1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/esq1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="526"><p><img alt="" src="../../files/probability/binomial-distribution/monster.jpg" style="float: right; width: 200px; height: 267px;"><img class="sibico" height="19" src="../../../img/sibico/hl-green.svg" style="vertical-align: text-bottom;" title="HL easy" width="19"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Monster Energy drink cans are advertised to contain 500ml. During the production, the volume <strong><em>X </em></strong>in ml in cans of the drink can be modelled by a normal distribution with mean 504 and variance 10.</p> <div> <p>a) For a randomly selected can, work out P(X > 500).</p> <p>b) Cans are put in packs of 6. Find the probability that at least 5 cans are have a volume of at least 500ml.</p> <hr class="hidden-separator"></div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>part a) of this question is about the Normal Distribution</p> <p><content>part b) is about the Binomial Distribution uses the probability found in part a)</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/binomial-distribution/esq2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/esq2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="524"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><img alt="" src="../../files/probability/binomial-distribution/esq2.jpg" style="float: right; width: 250px; height: 245px;">A glass contains 5 green sweets and <strong><em>m</em></strong> sweets of other colours. A sweet is taken at random.</p> <p>a) Write down the probability that the sweet is green.</p> <p>b) The sweet is replaced in the glass and the process is repeated a further three times. Each time, it is noted whether a green sweet is taken. The variance of the number of green sweets taken over the whole process in calculated to be 0.75.</p> <p>Work out how many sweets in total there are in the glass.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>For the Binomial distribution <span class="math-tex">\(X\sim B(n,p)\)</span></p> <p>Var(X) = <strong><em>npq q = 1 - p</em></strong><content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/binomial-distribution/esq3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/esq3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Continuous Random Variables</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="887"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The probability density function of X is given by </span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}ax^n\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<1\\otherwise\\\end{matrix}\)</span></p> <p>a) Show that <em><strong>a = n + 1</strong></em></p> <p>b) Given that E(X) = 0.75, find <em><strong>a</strong></em> and <em><strong>n</strong></em></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">For continuous random variables, the total area under the function = 1</span></p> <p><content><span class="math-tex">\(\int_{-\infty}^{\infty}{f\left(x\right)dx=1}\)</span> </content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="885"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The continuous random variable X has a probability density function given by</span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}a\\asin\left(\frac{\pi x}{4}\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<2\\2\le x<4\\otherwise\\\end{matrix}\)</span></p> <p>a) <span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">Draw a sketch of </span><strong><em>f</em></strong></p> <p>b) <span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">Show that the value of </span> <span class="math-tex">\(a=\frac{\pi}{2\pi+4}\)</span></p> <p>c) Find E(X)</p> <p>d) <span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">Find the exact value of the median of X</span></p> <p>e) Find <span class="math-tex">\(P(X\le2|X\le3)\)</span></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>b) The area under the graph of <strong><em>f</em></strong> equals 1 (sum of probabilities=1). Use this fact to find <strong><em>a</em></strong></p> <p><content>c) <span class="math-tex">\(E\left(X\right)=\int_{-\infty}^{\infty}xf\left(x\right)dx\)</span></content></p> <p>d) <span class="math-tex">\(\int_{-\infty}^{m}{f\left(x\right)dx=0.5}\)</span></p> <p>From the graph it is obvious that the median occurs between <span class="math-tex">\(0 \le x<2\)</span>, so there is no need to use Calculus</p> <p><img alt="" src="../../files/probability/continuousrvs/esq1c" style="width: 400px; height: 383px;"></p> <p>e) <span class="math-tex">\(P(X\le2|X\le3)=\frac{P(X\le2)}{P(X\le3)}\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="886"><p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span style="font-size:11.0pt;line-height:115%; font-family:" calibri",sans-serif;mso-fareast-font-family:"times="" new="" roman";="" mso-bidi-font-family:"times="" new="" roman";mso-ansi-language:en-gb;mso-fareast-language:="" en-us;mso-bidi-language:ar-sa"="">The continuous random variable X has a probability density function given by</span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}k\bullet a r c o s\left(x\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}-1\le x<1\\otherwise\\\end{matrix}\)</span></p> <p>a) Draw a sketch of <strong><em>f</em></strong></p> <p>b) State the mode of <strong><em>X</em></strong></p> <p>c) Find <span class="math-tex">\(\int a r c o s\left(x\right)dx\)</span></p> <p>d) Find E(X)</p> <p>e) Find Var(X)</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) You will need to use Integration by parts and make the 'other part' 1</p> <p><span class="math-tex">\(\int{1\bullet a r c o s\left(x\right)dx}\)</span><content></content></p> <p>d) and e) You can use your graphical calculator to work these out.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Normal Distribution</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="517"><p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> </p> <p>The random variable <strong><em>X</em></strong> is normally distributed with a mean of 100 and a standard deviation of 15.</p> <p>a) Write down P(X < 120)</p> <p>b) Write down P(95 < X < 105)</p> <p>c) P(X<strong> </strong><<strong> c</strong>) = 0.6. Write down the value of <strong>c</strong></p> <p>d) Find the interquartile range of <em><strong>X</strong></em></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) and b) use Normalcdf</p> <p><content>c) use InvNorm</content></p> <p>d) Interquartile Range = Q<sub>3</sub> - Q<sub>1</sub> , where</p> <p>P(X < <strong>Q<sub>1</sub></strong>) = 0.25</p> <p>P(X < <strong>Q<sub>3</sub></strong>) = 0.75</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/normal-distribution/esq_normal_distribution1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/normal-distribution/esq_normal_distribution1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="518"><p><img alt="" src="../../files/probability/normal-distribution/penguins.jpg" style="float: right; width: 400px; height: 300px;"><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The weights of female adult penguins found on a remote island are normally distributed with a mean of 3.15 kg. It is found that 81% of these penguins weigh more than 3 kg. The weights of the penguins have a standard deviation of <span class="math-tex">\(\sigma\)</span>.</p> <p>a) Find the standardized value for 3kg.</p> <p>b) Hence, find the value of <span class="math-tex">\(\sigma\)</span>.</p> <p>It is estimated that there are 4000 female adult penguins on the island. Penguins are considered underweight if they weigh less than 2.9 kg.</p> <p>c) Find the estimated number of underweight female penguins on the island.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Since you use the answers from one part of the question oto the next, it is important to keep a high degree of accuracy in your answers.</p> <p>a) Use InvNorm</p> <p>b) <em>'Hence' </em>means that this should follow on from the previous part. Use your z value to find <span class="math-tex">\(\sigma\)</span> using standardizing formula <span class="math-tex">\(Z=\frac{X- \mu }{\sigma}\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/normal-distribution/esq_normal_distribution2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/normal-distribution/esq_normal_distribution2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-border panel-has-colored-body panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="516"><p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img alt="" src="../../files/probability/normal-distribution/lightbulb.jpg" style="float: right; width: 300px; height: 200px;"></p> <p>It is known that 2% of Filips lightbulbs have a life of less than 1000 hours and 10% have a life less than 1100 hours. It can be assumed that lightbulb life is normally distributed with a mean of <span class="math-tex">\(\mu\)</span> and a standard deviation of <span class="math-tex">\(\sigma\)</span></p> <p>a) Find the value of <span class="math-tex">\(\mu\)</span> and the value of <span class="math-tex">\(\sigma\)</span>.</p> <p>b) Find the probability that a randomly selected Filips lightbulb will have a life of at least 1200 hours.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) In this question we need to standardize the normal distribution.</p> <p>Use InvNorm to find the corresponding z values for 1000 hours and 1100 hours.</p> <p>Solve 2 simultaneous equations to find <span class="math-tex">\(\mu\)</span> and <span class="math-tex">\(\sigma\)</span>.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/normal-distribution/esq_normal_distribution3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/normal-distribution/esq_normal_distribution3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="519"><p><img alt="" src="../../files/probability/normal-distribution/cupoftea.jpeg" style="float: right; width: 300px; height: 200px;"><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The amount of caffeine, <em><strong>X</strong></em> in mg, found in a cup of tea can be modelled by the normal distribution <span class="math-tex">\(X\sim N(26,5)\)</span></p> <p>a) Find the probability that a randomly selected cup of tea contains at least 27 mg of caffeine.</p> <p>b) 10 cups of tea are randomly selected. Find the probability that more than 5 of these cups has at least 24 mg of caffeine.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) Careful, the standard deviation =<span class="math-tex">\(\sqrt{5}\)</span></p> <p>b) this question actually uses the binomial distribution: the probability of 'success' is found in part a)<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/normal-distribution/esq_normal_distribution4.pdf">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/normal-distribution/esq_normal_distribution4.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2910"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Statistics and Probability Examination Questions HL</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> </div> </div></div> <div id="modal-feedback" class="modal fade" tabindex="-1" role="dialog"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <h4 class="modal-title">Feedback</h4> <button type="button" class="close hidden-xs hidden-sm" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> <div class="modal-body"> <div class="errors"></div> <p><strong>Which of the following best describes your feedback?</strong></p> <form method="post" style="overflow: hidden"> <div class="form-group"> <div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Recommendation"> Recommend</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Problem"> Report a problem</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Improvement"> Suggest an improvement</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Other"> Other</label></div> </div> <hr> <div class="row"> <div class="col-md-6"> <div class="form-group"> <label for="feedback-name">Name</label> <input type="text" class="form-control" name="feedback-name" placeholder="Name" value=" "> </div> </div> <div class="col-md-6"> <div class="form-group"> <label for="feedback-email">Email address</label> <input type="email" class="form-control" name="feedback-email" placeholder="Email" value="@airmail.cc"> </div> </div> </div> <div class="form-group"> <label for="feedback-comments">Comments</label> <textarea class="form-control" name="feedback-comments" style="resize: vertical;"></textarea> </div> <input type="hidden" name="feedback-ticket" value="082b9c9c4ae3624d"> <input type="hidden" name="feedback-url" value="https://studyib.net/mathsanalysis/page/2910/statistics-and-probability-examination-questions-hl"> <input type="hidden" name="feedback-subject" value="11"> <input type="hidden" name="feedback-subject-name" value="Maths: Analysis & Approaches"> <div class="pull-left"> </div> </form> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary btn-xs-block feedback-submit mb-xs-3 pull-right"> <i class="fa fa-send"></i> Send </button> <button type="button" class="btn btn-default btn-xs-block m-xs-0 pull-left" data-dismiss="modal"> Close </button> </div> </div> </div></div> </article> <hr class="hidden-md hidden-lg"> <div class="hidden-md hidden-lg mt-xs-3"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <input type="hidden" id="user-id" value="38342"></div><input id="ticket" type="hidden" value="082b9c9c4ae3624d"><input id="tzoffset" type="hidden" value="new"><input id="fp" class="fp" type="hidden" value=""></div><div id="std-footer"> <div class="wmap"> <div class="layout-wrapper"> <p> <a href="https://www.inthinking.net"> © <span id="footer-year"></span> <em>InThinking</em> <script>document.getElementById("footer-year").innerHTML = new Date().getFullYear();</script> </a> | <a target="_self" href="../../../about.html"> About us </a> | <a target="_self" href="../../../terms-and-conditions.html"> Legal </a> | <a target="_self" href="../../../contact.html"> Contact </a> </p> <p> <a class="social" target="_blank" href="https://twitter.com/#!/inthinker"> <img src="../../../img/social/twitter-square.svg"> Twitter </a> <a class="social" target="_blank" href="https://www.facebook.com/inthinking.net"> <img src="../../../img/social/facebook-square.svg"> Facebook </a> <a class="social" target="_blank" href="https://www.linkedin.com/profile/view?id=139741989"> <img src="../../../img/social/linkedin-square.svg"> LinkedIn </a> </p> </div> </div></div><script src="../../../js/jquery-1.11.3.min.js"></script><script src="../../../js/bootstrap.min.js"></script><script src="../../../js/jquery-fancybox/jquery.fancybox.min.js"></script><script>var pageId = 2910;</script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.min.js"></script><script src="../../../js/user/page-my-progress.min.js?v=202211221000"></script><script>var sAJAX='/pages/subjects/activity/user-stats-page.php?t=082b9c9c4ae3624d&p=2910&s=11&x=11520';var sData='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';var loopSecs = 30;var minSecsLog = 60;var lsKey = '11-83608-2910';</script><script src="../../../js/subjects/activity/user-stats-page.min.js?v=202205311700"></script><script src="../../../js/subjects/activity/my-favorites.min.js?v=20220610"></script><script src="../../../js/subjects/feedback.min.js?v=202211221000"></script><script>var sessionUpdateSecs = 300;</script><script type="text/javascript" src="../../../js/session-updater.js"></script><script>(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');ga('create', 'UA-98480796-4', 'auto');ga('send', 'pageview');</script><script src="../../../js/devicefp/devicefp.min.js"></script><script src="../../../js/jq-quickfit/jq-quickfit.min.js?v=20220201"></script><script src="../../../js/frontpage-subjects.min.js?v=202211161300"></script><script type="text/javascript" src="../../../js/studyib-jq.min.js?v=202211241300"></script><script type="text/javascript">$(document).ready(function(){ $('#trigger-modal-video-overview').click(function() { if( $('#modal-video-overview .modal-body iframe').attr('src').length == 0 ) { $('#modal-video-overview .modal-body iframe').attr('src', 'https://www.youtube.com/embed/hxxtdiLdTFk?rel=0'); } }); $('#modal-video-overview').on('hidden.bs.modal', function() { $('#modal-video-overview .modal-body iframe').attr('src', ''); }); });</script></body></html>