File "normal-distribution.html"
Path: /StudyIB/mathsanalysis/page/693/normal-distributionhtml
File size: 59.51 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>Normal Distribution</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: Normal Distribution"> <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="The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough s"> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsanalysis/page/693/normal-distribution"> <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="The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough s"> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="Normal distribution, Gaussian, Gauss, Bell curve, 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: Normal Distribution"> <meta itemprop="description" content="The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough s"> <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: Normal Distribution"> <meta name="twitter:description" content="The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough s"> <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"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick.min.css"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick-theme.min.css"><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="../549/stats-probability.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../549/stats-probability.html">Stats & Probability</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="../2902/examination-questions.html"><i class="fa fa-fw fa-caret-right"></i> Examination Questions</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="../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="../687/representing-and-measuring-data.html">Representing and Measuring Data</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1926/linear-correlation.html">Linear Correlation</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../919/probability-calculations.html">Probability Calculations</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../689/conditional-probability.html">Conditional Probability</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../690/discrete-random-variables.html">Discrete Random Variables</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../692/binomial-distribution.html">Binomial Distribution</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../691/continuous-random-variables.html">Continuous Random Variables</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="normal-distribution.html">Normal Distribution</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"> Normal Distribution <a href="#" class="mark-page-favorite pull-right" data-pid="693" 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="../549/stats-probability.html">Stats & Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Normal Distribution</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><img alt="" src="../../files/probability/normal-distribution/main-1.jpg" style="float: left; width: 100px; height: 100px;">The normal distribution is an extremely useful statistical distribution. Data for heights and weights of adults and objects produced by machines, for example, tend to follow the normal (bell-shaped) distribution. More importantly, if we take large enough samples of any data, the means of the samples can be modelled by a normal distribution. Psychologists use the normal distribution to classify people's test results, including intelligence tests. Don't be a dummy in your IB exam and make sure you understand this topic!</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p>On this page, you should learn about</p> <ul> <li>the normal distribution and its curve</li> <li>probability calculations with the normal distribution</li> <li>standardization of the normal variables</li> <li>finding mean and/or variance using inverse normal calculations</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/probability/normal-distribution/revision-notes_normal_distribution.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/normal-distribution/revision-notes_normal_distribution.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#67e9f45e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="67e9f45e"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Normal Distribution <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-219-693" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> A random variable <strong>X</strong> is distributed normally with a mean of 10 and a standard deviation of 2</p><p>Which graph best represents P(X<12)?</p><table border="0" cellpadding="0" cellspacing="0" height="23" width="695"><tbody><tr><td><p style="text-align: center;">A</p><p style="text-align: center;"><img alt="" src="../../files/probability/normal-distribution/quiz/q1a.jpg" style="width: 150px; height: 150px;"></p></td><td><p style="text-align: center;">B</p><p style="text-align: center;"><img alt="" src="../../files/probability/normal-distribution/quiz/q1b.jpg" style="width: 150px; height: 150px;"></p></td><td><p style="text-align: center;">C</p><p style="text-align: center;"><img alt="" src="../../files/probability/normal-distribution/quiz/q1c.jpg" style="width: 150px; height: 150px;"></p></td><td><p style="text-align: center;">D</p><p style="text-align: center;"><img alt="" src="../../files/probability/normal-distribution/quiz/q1d.jpg" style="width: 150px; height: 150px;"></p></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/normal-distribution/quiz/q1answer.jpg" style="width: 250px; height: 250px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> A random variable X is distributed such that X~N(80 , 10²). Work out the unknown values in the following situations</p><p><em>For a normal distribution, we know that </em></p><ul><li><em>about <strong>68%</strong> of the values are within <strong>one standard distribution</strong> away from the mean.</em></li><li><em>about <strong>95%</strong> of the values are within <strong>two standard distribution</strong>s away from the mean.</em></li><li><em>about <strong>97.7%</strong> of the values are within <strong>three standard distributions</strong> away from the mean.</em></li></ul></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p>a = <input type="text" style="height: auto;" data-c="60"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="100"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q2a.jpg" style="width: 200px; height: 200px;"></p></td><td><p>c = <input type="text" style="height: auto;" data-c="70"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="90"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q2b.jpg" style="width: 200px; height: 200px;"></p></td><td><p>e =<input type="text" style="height: auto;" data-c="50"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="110"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q2c.jpg" style="width: 200px; height: 200px;"> </p></td></tr></tbody></table></div><div class="q-explanation"><p>There is no need to a use a graphical calculator in this question. Use the geometry of the shapes and t to work out the missing values</p><ul><li><em><span class="math-tex">\(80 \pm 20\)</span></em></li><li><em><span class="math-tex">\(80 \pm 10\)</span></em></li><li><em><span class="math-tex">\(80 \pm 30\)</span></em></li></ul></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> A random variable X is distributed such that X~N(100 , 15²). Work out the probabilities of the following situations</p><p><em>For a normal distribution, we know that about <strong>68%</strong> of the values are within <strong>one standard distribution</strong> away from the mean.</em></p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p>p = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3a.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>p = <input type="text" style="height: auto;" data-c="0.68"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3b.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td><td><p>p = <input type="text" style="height: auto;" data-c="0.16"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3c.jpg" style="width: 200px; height: 200px;"></p><hr class="hidden-separator"></td></tr><tr><td><p>p = <input type="text" style="height: auto;" data-c="0.84"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3d.jpg" style="width: 200px; height: 200px;"></p></td><td><p>p = <input type="text" style="height: auto;" data-c="0.34"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3e.jpg" style="width: 200px; height: 200px;"></p></td><td><p>p = <input type="text" style="height: auto;" data-c="0.32"> <span class="review"></span></p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q3f.jpg" style="width: 200px; height: 200px;"></p></td></tr></tbody></table></div><div class="q-explanation"><p>There is no need to a use a graphical calculator in this question.</p><p>Use the fact that <em>we know that about <strong>68%</strong> of the values are within <strong>one standard distribution</strong> away from the mean </em>and to work out the areas of each part</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The random variable X is normally distributed with mean of 50. The following diagram shows the curve for X.</p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q4.jpg" style="width: 300px; height: 300px;"></p><p>The area of the shaded region R is 0.19</p><p>Work out the following probabilities</p></div><div class="q-answer"><p>P(X < 58) = <input type="text" style="height: auto;" data-c="0.81"> <span class="review"></span></p><p>P(50 < X < 58) = <input type="text" style="height: auto;" data-c="0.31"> <span class="review"></span></p><p>P(42 < X < 58) = <input type="text" style="height: auto;" data-c="0.62"> <span class="review"></span></p></div><div class="q-explanation"><p>P(X < 58) = 1 - 0.19</p><p>P(50 < X < 58) = 0.81 - 0.5</p><p>P(42 < X < 58) = 1 - 2x0.19 or 0.81 - 0.19</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> The random variable X is normally distributed with mean of 16. The following diagram shows the curve for X.</p><p><img alt="" src="../../files/probability/normal-distribution/quiz/q5.jpg" style="width: 300px; height: 300px;"></p><p>The area of the shaded region R is 0.37</p><p>Work out the following probabilities</p></div><div class="q-answer"><p>P(X < 11.5) = <input type="text" style="height: auto;" data-c="0.13"> <span class="review"></span></p><p>P(11.5 < X < 20.5) = <input type="text" style="height: auto;" data-c="0.74"> <span class="review"></span></p><p>P(X < 20.5) = <input type="text" style="height: auto;" data-c="0.61"> <span class="review"></span></p></div><div class="q-explanation"><p>P(X < 11.5) = 0.5 - 0.37</p><p>P(11.5 < X < 20.5) = 2x0.37</p><p>P(X < 20.5) = 0.74 - 0.13</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Find all the unknowns below</p></div><div class="q-answer"><p>The random variable X is normally distributed with mean of 10.</p><p>Given that P(X < 7) = P(X > <em><strong>a</strong></em>)</p><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span></p><hr class="hidden-separator"><p>The random variable X is normally distributed with mean of 18.</p><p>P(X < 16.5) <span class="math-tex">\(\approx\)</span> 0.309</p><p>P(16.5 < X < 19.5) <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.382"> <span class="review"></span></p><hr class="hidden-separator"><p>The random variable X is normally distributed with mean of 100.</p><p>P(88 < X < 112) <span class="math-tex">\(\approx\)</span> 0.576</p><p>P(X > 112) <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.212"> <span class="review"></span></p><hr class="hidden-separator"><p>The random variable X is normally distributed with mean of 35.</p><p>P(X > 30) <span class="math-tex">\(\approx\)</span> 0.841</p><p>P(X > 40) <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.159"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A random variable X is normally distributed. Work out the unknown values in the following situations to 3 significant figures</p></div><div class="q-answer"><p>X~N(25 , 3²). P(X < 27) = <input type="text" style="height: auto;" data-c="0.748"> <span class="review"></span></p><hr class="hidden-separator"><p>X~N(-3 , 2²). P(X > 0) = <input type="text" style="height: auto;" data-c="0.0668"> <span class="review"></span></p><hr class="hidden-separator"><p>Z~N(0 , 1²). P(-0.2 < Z < 0.3) = <input type="text" style="height: auto;" data-c="0.197"> <span class="review"></span></p></div><div class="q-explanation"><p>All of the above are probability calculations. Use the normal cumulative function on your calculator.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A random variable X is normally distributed. Work out the unknown values in the following situations to 3 significant figures</p></div><div class="q-answer"><p>X~N(100 , 15²). P(X < a) = 0.95, a <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="125"> <span class="review"></span></p><hr class="hidden-separator"><p>X~N(58 , 8²). P(X > b) = 0.67, b <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="54.5"> <span class="review"></span></p><hr class="hidden-separator">Z~N(0 , 1²). P(|Z| < c) = 0.75 c <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="1.15"> <span class="review"></span> </div><div class="q-explanation"><p>All of the above calculations use the inverse normal function.</p><p>You could think of the last one as P(-c < Z < c) = 0.75</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> The heights of adults can be modelled using the normal distribution H~N(165 , 12²).</p><p>Calculate the following to 3 significant figures.</p></div><div class="q-answer"><p>Upper quartile height = <input type="text" style="height: auto;" data-c="173"> <span class="review"></span></p><p>Interquartile range of the heights= <input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p>90th percentile height = <input type="text" style="height: auto;" data-c="180"> <span class="review"></span></p></div><div class="q-explanation"><p>Upper quartile P(H < Q3) = 0.75, P(H < 173) = 0.75,</p><p>Using symmetry, lower quartile, Q1 = 149 Interquartile range = Q3 - Q1 = 173 - 157 = 16</p><p>90th percentile P(H < a) = 0.9</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A random variable X is distributed such that X~N(14 , 2²). Work out these conditional probabilities to 3 significant figures</p></div><div class="q-answer"><p>P(X >15 / X > 13) <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.446"> <span class="review"></span></p><p>P(X >15 / 14 < X < 16) <span class="math-tex">\(\approx\)</span> <input type="text" style="height: auto;" data-c="0.439"> <span class="review"></span></p></div><div class="q-explanation"><p>These are questions about conditional probability</p><p><span class="math-tex">\(P(A/ B)=\frac{P(A \cap B )}{P(B)}\)</span></p><p>P(X >15 / X > 13) = <span class="math-tex">\(\frac{P(X >15)}{P(X > 13)}\approx\)</span><span class="math-tex">\(\frac{0.3085}{0.6915}\approx 0.446\)</span></p><p>P(X >15 / 14 < X < 16) =<span class="math-tex">\(\frac{P(15<X <16)}{P(14<X< 16)}\approx\)</span> <span class="math-tex"><x><span class="math-tex">\(\frac{0.1499}{0.3413}\approx 0.439\)</span></x></span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </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>Exam-style Questions</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="693"> <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>Normal Distribution</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/693/normal-distribution"> <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 = 693;</script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui-custom.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.js"></script><script type="text/javascript" src="../../../js/std-quizzes-helpers.min.js?v=20220406"></script><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script type="text/javascript" src="../../../js/slick-carousel/slick.min.js"></script><script type="text/javascript" src="../../../js/std-slide-quizzes.min.js?v=20220406"></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=693&s=11&x=17998';var sData='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';var loopSecs = 30;var minSecsLog = 60;var lsKey = '11-83608-693';</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>