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Rank Correlation Coefficient</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="411-chi-squared-independence-tests.html">4.11 Chi Squared Independence tests</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2413/411-chi-squared-goodness-of-fit-and-the-t-test.html">4.11 Chi Squared Goodness of Fit and the T-Test</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"> 4.11 Chi Squared Independence tests <a href="#" class="mark-page-favorite pull-right" data-pid="2259" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1065/statistics-and-probability.html">Statistics and Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">4.11 Chi Squared Independence tests</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"> <h2><img alt="" src="../../files/Stats/Chi-Squared/CS-1.jpeg" style="width: 150px; height: 150px; float: left;">Looking for relationships between categorical variables</h2> <p>The chi squared independence tests is a widely used technique for looking for a relationship between variables that are categorical. We can use a scatter graph to look for a relationship between GDP and life expectancy, but what about GDP and Continent? We cant put continent on a numerical axis. This tool gives us many new options and is an excellent bit of mathematics.</p> <hr> <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><span id="docs-internal-guid-933cf3af-7fff-5711-ada5-55a9c13020f0"></span></p> <p dir="ltr"><span id="docs-internal-guid-933cf3af-7fff-5711-ada5-55a9c13020f0">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation"><span id="docs-internal-guid-933cf3af-7fff-5711-ada5-55a9c13020f0">Understand the nature of the chi squared test of independence as a means of looking for relationships between categorical data</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Calculate expected frequencies assuming independence</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Calculate the chi squared statistic and p number for given data and use them to conclude</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Understand and use the language of hypothesis testing and carry out tests from start to finish</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Slide Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos that appear below.</p> <div id="carousel-249" class="dynamic-gallery carousel slide" data-id="249"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos003.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos004.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos005.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos005.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos006.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos006.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos007.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos007.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos008.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos008.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos009.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos009.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos010.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos010.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos011.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos011.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos012.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos012.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos013.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos013.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos001.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-249/chi2-no-logos002.jpeg" data-fancybox="gallery-249" title="" data-caption=""><img alt="" src="../../../std-galleries/15-249/chi2-no-logos002.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-249" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-249" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos003-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos004-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos005-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos006-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos007-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos008-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos009-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos010-thumb128.jpg"><li><li data-index="8"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos011-thumb128.jpg"><li><li data-index="9"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos012-thumb128.jpg"><li><li data-index="10"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos013-thumb128.jpg"><li><li data-index="11"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos001-thumb128.jpg"><li><li data-index="12"><img title="Click to view" src="../../../std-galleries/15-249/chi2-no-logos002-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. What are Chi squared tests of independence?</p> </div> </div> <div class="panel-body"> <div> <p>These two videos explain 1. What the purpose of a Chi squared independence test is and 2. What the fundamental principle behind them is.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592649312"></iframe></div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592649663"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Understand and calculate expected frequencies</p> </div> </div> <div class="panel-body"> <div> <p>This video lesson takes you through the idea of calculating expected frequencies, from understanding the concept to applying it in context.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592650232"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Calculating the Chi Squared Statistic</p> </div> </div> <div class="panel-body"> <div> <p>This video lesson goes through the process of calculating the chi squared statistic, both so that you can understand it and know how to calculate it.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592650882"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Degrees of freedom and significance levels</p> </div> </div> <div class="panel-body"> <div> <p>These two video lessons cover the key ideas of degrees of freedom and significance levels. Understand them and then how to work with them.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592651421"></iframe></div> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592652984"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Concluding Chi Squared tests</p> </div> </div> <div class="panel-body"> <div> <p>Here are two videos showing how you conclude these tests using 1. The critical value approach and 2. The p-number approach.</p> <p>Concluding with the critical value </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592653273"></iframe></div> <p>Concluding with the p-number </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592653679"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>6. An example test</p> </div> </div> <div class="panel-body"> <div> <p>The following video goes through an example question showing how to get both the calculated statistic and p number from both Casio and TI calculators.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592654387"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </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" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <h4>Flash Cards</h4> <p>Review these condensed 'key point' flashcards to help you check and keep ideas fresh in your mind.</p> <div id="carousel-250" class="dynamic-gallery carousel slide" data-id="250"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-250/chi2-no-logos001.jpeg" data-fancybox="gallery-250" title="" data-caption=""><img alt="" src="../../../std-galleries/15-250/chi2-no-logos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-250/chi2-no-logos002.jpeg" data-fancybox="gallery-250" title="" data-caption=""><img alt="" src="../../../std-galleries/15-250/chi2-no-logos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-250/chi2-no-logos003.jpeg" data-fancybox="gallery-250" title="" data-caption=""><img alt="" src="../../../std-galleries/15-250/chi2-no-logos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-250/chi2-no-logos004.jpeg" data-fancybox="gallery-250" title="" data-caption=""><img alt="" src="../../../std-galleries/15-250/chi2-no-logos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-250/chi2-no-logos005.jpeg" data-fancybox="gallery-250" title="" data-caption=""><img alt="" src="../../../std-galleries/15-250/chi2-no-logos005.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-250" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-250" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-250/chi2-no-logos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-250/chi2-no-logos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-250/chi2-no-logos003-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-250/chi2-no-logos004-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-250/chi2-no-logos005-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <h4>Self Checking Quiz</h4> <h4>Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-313-2259"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>A large mount of information is collected about a number of countires. Which of the following would be considered categorical data?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Continent</label></p><p><label class="checkbox"><input type="checkbox"> GDP</label></p><p><label class="checkbox"><input type="checkbox"> Life expectancy</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Predominent religion</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Type of Politics</label></p><p><label class="checkbox"><input type="checkbox"> Area</label></p><p><label class="checkbox"><input type="checkbox"> Infant mortality rate</label></p><p><label class="checkbox"><input class="c" type="checkbox"> MEDC of LEDC (more or less economically developed)</label></p></div><div class="q-explanation"><p>GDP, area, Life expectancy and Infant mortality rates can all be measured with numbers on a scale. The rest can't.</p></div><div class="actions"><span class="score" data-score="0"></span><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><p>........</p><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>We start chi squared tests by assuming that.....</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> The two variables are dependent</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The two variables are independent</label></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><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><p>The following is a contingency table of results fo a chi squared test to see if wage bracket is dependent the level of education reached. Use it to answer the next 3 questions.</p><p style="text-align: center;"><img alt="" src="../../files/Stats/Chi-Squared/wages.jpg" style="width: 500px; height: 197px;"></p><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the value for degrees of freedom for this table?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p></div><div class="q-explanation"><p>Degrees of freedom is (rows - 1) x (columns - 1) so (3 - 1) x (3 - 1) = 4</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the expected frequency of people who did not attend university in the low wage bracket?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="30.2"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="146 over 450 cross times 93" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»146«/mn»«mn»450«/mn»«/mfrac»«mo»§#215;«/mo»«mn»93«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-32.php?formula=c68536f87dbf43e26ab1eb6ad561cb09.png"></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the expected frequency of people with a post graduate degree in the high wage bracket?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="20.5"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="88 over 450 cross times 105" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»88«/mn»«mn»450«/mn»«/mfrac»«mo»§#215;«/mo»«mn»105«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-33.php?formula=78412fb1ee5fb1d3ee740e6459f5a8f9.png"></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the expected frequency of people with a university degree in the medium wage bracket?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="121"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="216 over 450 cross times 252" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»216«/mn»«mn»450«/mn»«/mfrac»«mo»§#215;«/mo»«mn»252«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-34.php?formula=4c4d61ee8b70068d24691a68fac250de.png"></p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Calculate the p-number using your GDC (start your answer 0. )</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0.835"> <span class="review"></span> </p></div><div class="q-explanation"><p>From GDC</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Calculate the chi squared statistic using your GDC </p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1.45"> <span class="review"></span> </p></div><div class="q-explanation"><p>From GDC</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use the p number to test at 10% significance level if these variables are dependent.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Accept H<sub>0</sub></label></p><p><label class="checkbox"><input type="checkbox"> Reject H<sub>0</sub> and accept H<sub>1</sub></label></p></div><div class="q-explanation"><p>Since the p number, 0.835 > 0.10 (10%), the probability of independence is high enough to accept the null hypothesis.</p></div><div class="actions"><span class="score" data-score="0"></span><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="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>To reject the null hypothesis, which of the following conditions should be met?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Chi squared statistic is greater than the critical value</label></p><p><label class="checkbox"><input type="checkbox"> Chi squared statistic is lower than the cirtical value</label></p><p><label class="checkbox"><input type="checkbox"> p - number is higher than the significance level</label></p><p><label class="checkbox"><input class="c" type="checkbox"> p number is lower than the significance level</label></p></div><div class="q-explanation"><p>To reject, the probability of independence must be below the signficance level (10%, 5%, 1%, 0.1, 05, 0.1) and the chi squared statistic (the measure of error) must be bigger the critical value (too big an error)</p></div><div class="actions"><span class="score" data-score="0"></span><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="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </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>Exam Style Questions</p> </div> </div> <div class="panel-body"> <div> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</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>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../files/Stats/Chi-Squared/Chi2Q1.jpg" style="width: 500px; height: 635px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/592654802"></iframe></div> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a 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