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Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">4.11 Chi Squared Goodness of Fit and the T-Test</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"> <div class="intro-card" readonly="false"><img class="intro-image" readonly="true" src="../../images/dans-images/smarties-1.jpg" style="width: 200px; height: 200px;"> <div class="content" readonly="true"> <p class="text">This page continues and extends our understanding of hypothesis testing, the act of creating a hypothesis about what you believe about the association between variables, data sets or distributions and then running a mathematial process to test whether what you hypothesised is indeed the case. More specifically, we will look closely at two different hypothesis tests: the T-test and the Chi Squared Goodness of Fit Test.</p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>In this unit you should learn to…</p> <ul> <li>Complete a Chi-Squared Goodness of Fit Test. </li> <li>Complete a 2 sample T-test. </li> </ul> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-orange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Chi Squared Goodness of Fit (Part 1)</p> </div> </div> <div class="panel-body"> <p>What is the Chi-squared Goodness of Fit Test? How do you create the null and alternative hypothesis, the degrees of freedom and the expected frequencies?</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/649666918"></iframe></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Chi Squared Goodness of Fit (Part 2)</p> </div> </div> <div class="panel-body"> <div> <p>We will look at two of the same examples presented in the previous video and fully complete the Goodness of Fit test in each case. </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/649668309"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Chi Squared Goodness of Fit (Part 3)</p> </div> </div> <div class="panel-body"> <p>Conducting a Chi Squared GOF test involving the normal distribution and binomial distribution. </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/655109394"></iframe></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <p>4. T-Test: One tailed vs Two tailed </p> </div> <div class="panel-body"> <div> <p>What is a 2-sample t-test? How do you state the null and alternative hypothesis? and how do you know whether you have a one-tailed or two-tailed test?</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/650087075"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Completing the T-test</p> </div> </div> <div class="panel-body"> <div>In this video we will complete the t-test by comparing the significance level to the p-value for two examples. </div> <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/650557279"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-yellow"> <div class="panel-heading"> <div> <p>6. Using the GCD to find the p-value for a 2 sample t-test</p> </div> </div> <div class="panel-body"> <div>TI-84</div> <div> </div> <div> <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/548831893"></iframe></div> </section> <p>TI-Nspire</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/546954945"></iframe></div> </section> <p>Casio</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/663948792"></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/663942641"></iframe></div> </section> </div> <div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </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 id="carousel-273" class="dynamic-gallery carousel slide" data-id="273"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-273/gallery-2.png" data-fancybox="gallery-273" title="" data-caption=""><img alt="" src="../../../std-galleries/15-273/gallery-2.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-273/gallery-1.png" data-fancybox="gallery-273" title="" data-caption=""><img alt="" src="../../../std-galleries/15-273/gallery-1.png"></a></div></div><a class="left carousel-control" href="#carousel-273" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-273" 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/15-273/gallery-2-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-273/gallery-1-thumb128.jpg"><li></li></ol> <p> </p> <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"> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above<em>.</em></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="15-645-2413"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>A Chi-sqaured goodness of fit test has a p-value of 0.0231. If the test is conducted at a 5% significance level, what is the conclusion of the test. </p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Since <span class="math-tex">\(0.0231<0.05\)</span></label>, we reject the null hypothesis. </p><p><label class="radio"><input type="radio"> Since <span class="math-tex">\(0.0231<0.05\)</span></label>, we accept the null hypothesis. </p></div><div class="q-explanation"><p>If the p-value is less than or equal to the significance level, we reject the null hypothesis. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>A t-test is has a p-value of 0.516. If the test is conducted at the 5% significance level, what is the conclusion of the test. </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> Since <span class="math-tex">\(0.516>0.05\)</span></label>, we reject the null hypothesis.</p><p><label class="radio"><input class="c" type="radio"> Since <span class="math-tex">\(0.516>0.05\)</span></label>, we accept the null hypothesis. </p></div><div class="q-explanation"><p>If the p-value is greater than the significance level, we accept the null hypothesis. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The formula for the degrees of freedom of a Chi-Squared goodness of fit test is: </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> df = (r - 1)(c - 1), where r is the number of rows and c is the number of columns in a table. </label></p><p><label class="radio"><input class="c" type="radio"> df = n - 1, where n is the number of categories. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>When conducting a 2-sample t-test on your GDC, you start by inputting the data for each sample into your lists. The calculator will then ask you whether the data is Pooled. Which answer do you enter?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Yes</label></p><p><label class="radio"><input type="radio"> No</label></p><p><label class="radio"><input type="radio"> It depends on whether it is a one-tailed test or two-tailed test. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>When conducting a 2 sample t-test, what assumptions do you make in performing a test?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> The data sets follow a normal distribution. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> The variance of each data set is approximately equal. </label></p><p><label class="checkbox"><input type="checkbox"> The mean of each data set is approximately equal.</label></p></div><div class="q-explanation"><p>See answers above. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>A school decided to pilot a reward based system if students attended over 95% of the academic year. Any students in year 10 who attended over this percentage would be allowed to go on a reward trip. At the end of the year, the school decided to compare the year 10 attendance results with that of the previous year 10. It was assumed that students in the current year would be motivated by the reward trip and therefore attend school more regularly.</p><p>The analysis involved a t-test. What type of test would this involve?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> A one-tailed test, i.e. <span class="math-tex">\(\mu_1>\mu_2\)</span></label></p><p><label class="radio"><input type="radio"> A two-tailed test, i.e. <span class="math-tex">\(\mu_1 \neq \mu_2\)</span></label></p></div><div class="q-explanation"><p>This is a one-tailed test because we are hypothesising that the mean of the current group of year 10 students is going to be greater than the mean of the previous group of year 10 students. i.e. we have been given information in the question which suggests that one data set will have a higher mean than the other. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>A school decided to pilot a reward based system if students attended over 95% of the academic year. Any students in year 10 who attended over this percentage would be allowed to go on a reward trip. At the end of the year, the school decided to compare the current year 10 attendance results with that of the previous year 10. It was assumed that students in the current year would be motivated by the reward trip and therefore attend school more regularly. A sample was taken of eacy year group and the results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Current Year 10</td><td>90%</td><td>96%</td><td>96%</td><td>96%</td><td>92%</td><td>100%</td><td>100%</td><td>99%</td><td>97%</td><td>99%</td></tr><tr><td>Previous Year 10</td><td>90%</td><td>95%</td><td>89%</td><td>90%</td><td>96%</td><td>97%</td><td>100%</td><td>90%</td><td>93%</td><td>92%</td></tr></tbody></table><p>Conduct a t-test and identify the p-value for this data. Give your answer to 3 significant figures. </p></div><div class="q-answer"><p>p-value: <input type="text" style="height: auto;" data-c="0.0250"> <span class="review"></span> </p></div><div class="q-explanation"><p>Insert each data set into a list, then perform a 2 sample t-test, with <span class="math-tex">\(\mu_1>\mu_2\)</span>. Ensure you enter "Yes" on Pooled data. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>For a Chi-squared goodness of fit test, the critical value is given as 14.1. The <span class="math-tex">\(χ_2
\)</span> statistic iscalculated to be 10.5. Identify the conclusion for the test. </p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Since 10.5 < 14.1, we accept <span class="math-tex">\(H_0\)</span></label></p><p><label class="radio"><input type="radio"> Since 10.5 < 14.1, we reject <span class="math-tex">\(H_0\)</span></label></p></div><div class="q-explanation"><p>See answer above. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>For football stickers, the manufacturer claims the following distribution for the proportions of sticker type:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Percentage</td></tr><tr><td>Very Common</td><td>50%</td></tr><tr><td>Common</td><td>35%</td></tr><tr><td>Rare</td><td>10%</td></tr><tr><td>Very Rare</td><td>5%</td></tr></tbody></table><p>Sophie is skeptical of the manufacturers claim and tests the result on 200 stickers based on the sticker website information. Her results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Frequency </td></tr><tr><td>Very Common</td><td>110</td></tr><tr><td>Common</td><td>79</td></tr><tr><td>Rare</td><td>7</td></tr><tr><td>Very Rare</td><td>4</td></tr></tbody></table><p>Find the expected frequency of how rare each category of sticker is. </p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Expected Frequency</td></tr><tr><td>Very Common</td><td> <input type="text" style="height: auto;" data-c="100"> <span class="review"></span> </td></tr><tr><td>Common</td><td> <input type="text" style="height: auto;" data-c="70"> <span class="review"></span> </td></tr><tr><td>Rare</td><td> <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> </td></tr><tr><td>Very Rare</td><td> <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </td></tr></tbody></table></div><div class="q-explanation"><p>To find the expected frequencies, multiply the total number of trials by the expected proportions. i.e. </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Expected Frequency</td></tr><tr><td>Very Common</td><td><span class="math-tex">\(200\times0.5\)</span></td></tr><tr><td>Common</td><td><span class="math-tex">\(200\times0.35\)</span></td></tr><tr><td>Rare</td><td><span class="math-tex">\(200\times0.1\)</span></td></tr><tr><td>Very Rare</td><td><span class="math-tex">\(200\times0.05\)</span></td></tr></tbody></table></div><div class="remove-section" style="display: none;" title="Remove question field"> </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>For football stickers, the manufacturer claims the following distribution for the proportions of sticker type:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Percentage</td></tr><tr><td>Very Common</td><td>50%</td></tr><tr><td>Common</td><td>35%</td></tr><tr><td>Rare</td><td>10%</td></tr><tr><td>Very Rare</td><td>5%</td></tr></tbody></table><p>Sophie is skeptical of the manufacturers claim and tests the result on 200 stickers based on the sticker website information. Her results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Frequency </td></tr><tr><td>Very Common</td><td>110</td></tr><tr><td>Common</td><td>79</td></tr><tr><td>Rare</td><td>7</td></tr><tr><td>Very Rare</td><td>4</td></tr></tbody></table><p>Conduct a Chi-Squared goodness of fit test to determine whether the manufacturers claims are correct. Enter the p-value in the space below, given to 3 significant figures, and determine the conclusion of the test at a 1% significance level.</p></div><div class="q-answer"><p>Degrees of Freedom: <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>p-value: <input type="text" style="height: auto;" data-c="0.00264"> <span class="review"></span> </p></div><div class="q-explanation"><p>Since 0.00264 < 0.01, we reject the null hypothesis which means that the manufacturers claim about the proportions of sticker types is incorrect.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </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 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