{"id":1406,"date":"2023-06-22T02:22:49","date_gmt":"2023-06-22T02:22:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-learn-it-3\/"},"modified":"2025-05-16T23:38:53","modified_gmt":"2025-05-16T23:38:53","slug":"chi-square-test-for-goodness-of-fit-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-learn-it-3\/","title":{"raw":"Chi-Square Test for Goodness of Fit \u2013 Learn It 3","rendered":"Chi-Square Test for Goodness of Fit \u2013 Learn It 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a chi-square test for goodness of fit and write its conclusion in context of the problem&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Complete a chi-square test for goodness of fit and write its conclusion in context of the problem<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Chi-Square ([latex]\\chi^2[\/latex]) Test Statistic<\/h2>\r\n<p>As with other hypothesis tests, we need to be able to model the variability we expect in samples if the null hypothesis is true. Then, we can determine whether the chi-square test statistic from the data is unusual or typical.<\/p>\r\n<p>An unusual [latex]\\chi^2[\/latex]\u00a0value suggests that there are statistically significant differences between the sample data and the null distribution and provides evidence against the null hypothesis. This is the same logic we have been applying with hypothesis testing.<\/p>\r\n<section class=\"textbox recall\">\r\n<p class=\"student12ptnumberlist\">The following is the formula for the chi-square test statistic:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\chi^2=\\sum\\dfrac{(\\text{Observed}-\\text{Expected})^2}{\\text{Expected}}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2356[\/ohm2_question]<\/section>\r\n<p>Sample C was the actual sample obtained by researchers in the study mentioned previously, and it satisfies the conditions for a chi-square goodness of fit test.<\/p>\r\n<div style=\"text-align: left;\" align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Sample C<\/strong><\/td>\r\n<td>\r\n<p><strong>Quarter 1<\/strong><\/p>\r\n<p><strong>(Jan. \u2013 March)<\/strong><\/p>\r\n<\/td>\r\n<td>\r\n<p><strong>Quarter 2 <\/strong><\/p>\r\n<p><strong>(April \u2013 June)<\/strong><\/p>\r\n<\/td>\r\n<td>\r\n<p><strong>Quarter 3<\/strong><\/p>\r\n<p><strong>(July \u2013 Sept.)<\/strong><\/p>\r\n<\/td>\r\n<td>\r\n<p><strong>Quarter 4<\/strong><\/p>\r\n<p><strong>(Oct. \u2013 Dec.)<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Observed number of football players<\/strong><\/td>\r\n<td>507<\/td>\r\n<td>534<\/td>\r\n<td>389<\/td>\r\n<td>273<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>Does this sample provide enough evidence to reject the null hypothesis and support the alternative hypothesis? Let\u2019s investigate.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Click the <strong>Find Probability<\/strong> tab at the top of the data analysis tool.<br \/>\r\n<strong>Step 2: <\/strong>Choose the appropriate degrees of freedom (number of categories - 1) and select the \u201cUpper Tail\u201d probability type.<br \/>\r\n<strong>Step 3: <\/strong>Enter the calculated chi-square statistic ([latex]\\chi^2=147[\/latex]).<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/chisqdist\/\" width=\"100%\" height=\"600\" frameborder=\"no\"><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/chisqdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2357[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">The chi-square goodness-of-fit test does not give information about the deviation for specific categories. It gives a more general conclusion of \u201cseems to fit the null distribution\u201d or \u201cdoes not fit the null distribution.\u201d<\/section>\r\n<\/section>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a chi-square test for goodness of fit and write its conclusion in context of the problem&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12801,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;arial&quot;,&quot;16&quot;:9}\">Complete a chi-square test for goodness of fit and write its conclusion in context of the problem<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Chi-Square ([latex]\\chi^2[\/latex]) Test Statistic<\/h2>\n<p>As with other hypothesis tests, we need to be able to model the variability we expect in samples if the null hypothesis is true. Then, we can determine whether the chi-square test statistic from the data is unusual or typical.<\/p>\n<p>An unusual [latex]\\chi^2[\/latex]\u00a0value suggests that there are statistically significant differences between the sample data and the null distribution and provides evidence against the null hypothesis. This is the same logic we have been applying with hypothesis testing.<\/p>\n<section class=\"textbox recall\">\n<p class=\"student12ptnumberlist\">The following is the formula for the chi-square test statistic:<\/p>\n<p style=\"text-align: center;\">[latex]\\chi^2=\\sum\\dfrac{(\\text{Observed}-\\text{Expected})^2}{\\text{Expected}}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2356\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2356&theme=lumen&iframe_resize_id=ohm2356&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Sample C was the actual sample obtained by researchers in the study mentioned previously, and it satisfies the conditions for a chi-square goodness of fit test.<\/p>\n<div style=\"text-align: left; margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Sample C<\/strong><\/td>\n<td>\n<p><strong>Quarter 1<\/strong><\/p>\n<p><strong>(Jan. \u2013 March)<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Quarter 2 <\/strong><\/p>\n<p><strong>(April \u2013 June)<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Quarter 3<\/strong><\/p>\n<p><strong>(July \u2013 Sept.)<\/strong><\/p>\n<\/td>\n<td>\n<p><strong>Quarter 4<\/strong><\/p>\n<p><strong>(Oct. \u2013 Dec.)<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Observed number of football players<\/strong><\/td>\n<td>507<\/td>\n<td>534<\/td>\n<td>389<\/td>\n<td>273<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Does this sample provide enough evidence to reject the null hypothesis and support the alternative hypothesis? Let\u2019s investigate.<\/p>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Click the <strong>Find Probability<\/strong> tab at the top of the data analysis tool.<br \/>\n<strong>Step 2: <\/strong>Choose the appropriate degrees of freedom (number of categories &#8211; 1) and select the \u201cUpper Tail\u201d probability type.<br \/>\n<strong>Step 3: <\/strong>Enter the calculated chi-square statistic ([latex]\\chi^2=147[\/latex]).<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/chisqdist\/\" width=\"100%\" height=\"600\" frameborder=\"no\"><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/chisqdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2357\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2357&theme=lumen&iframe_resize_id=ohm2357&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\">The chi-square goodness-of-fit test does not give information about the deviation for specific categories. It gives a more general conclusion of \u201cseems to fit the null distribution\u201d or \u201cdoes not fit the null distribution.\u201d<\/section>\n<\/section>\n<\/div>\n","protected":false},"author":8,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1388,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1406"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1406\/revisions"}],"predecessor-version":[{"id":6869,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1406\/revisions\/6869"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1388"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1406\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1406"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1406"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1406"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}