{"id":1405,"date":"2023-06-22T02:22:48","date_gmt":"2023-06-22T02:22:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-learn-it-2\/"},"modified":"2025-05-16T23:38:02","modified_gmt":"2025-05-16T23:38:02","slug":"chi-square-test-for-goodness-of-fit-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-learn-it-2\/","title":{"raw":"Chi-Square Test for Goodness of Fit \u2013 Learn It 2","rendered":"Chi-Square Test for Goodness of Fit \u2013 Learn It 2"},"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]) Distribution<\/h2>\r\n<p>Unlike other sampling distributions we have studied, the chi-square model does not have a normal shape. It is skewed to the right. Like the [latex]t[\/latex]-model, the chi-square model is a family of curves that depend on degrees of freedom.<\/p>\r\n<ul>\r\n\t<li>For a chi-square goodness of fit test, the <strong>degrees of freedom<\/strong> is (number categories [latex]- 1[\/latex]).<\/li>\r\n\t<li>The <strong>mean<\/strong> of the chi-square distribution is equal to the degrees of freedom.<\/li>\r\n<\/ul>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>conditions for a [latex]\\chi^2[\/latex] goodness of fit test<\/h3>\r\n<p>A chi-square model is a good fit for the distribution of the chi-square test statistic only if the following conditions are met:<\/p>\r\n<ul>\r\n\t<li><strong>Random:<\/strong> Observed counts must come from a random sample (to ensure our conclusions are free from sampling bias).<\/li>\r\n\t<li><strong>10%:<\/strong> The sample size must be less than a tenth of the population size (to satisfy independence assumptions).<\/li>\r\n\t<li><strong>Large Sample:<\/strong> The sample is large enough such that the expected counts are all <em>five or greater<\/em> (to ensure our sampling distribution resembles a chi-square distribution).<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">Imagine that three different teams of researchers gathered birthdate data on three different random samples of professional Italian football players. The following tables show their results.\r\n\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Sample A<\/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>3<\/td>\r\n<td>4<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Sample B<\/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>3,000<\/td>\r\n<td>4,000<\/td>\r\n<td>2,000<\/td>\r\n<td>1,000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div 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 class=\"para\">The data in the tables are displayed in the following side-by-side bar chart:<\/p>\r\n<p><img class=\"alignnone wp-image-1473 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2023\/04\/24193809\/Picture1-5.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"1024\" height=\"298\" \/><\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2354[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2355[\/ohm2_question]<\/section>","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]) Distribution<\/h2>\n<p>Unlike other sampling distributions we have studied, the chi-square model does not have a normal shape. It is skewed to the right. Like the [latex]t[\/latex]-model, the chi-square model is a family of curves that depend on degrees of freedom.<\/p>\n<ul>\n<li>For a chi-square goodness of fit test, the <strong>degrees of freedom<\/strong> is (number categories [latex]- 1[\/latex]).<\/li>\n<li>The <strong>mean<\/strong> of the chi-square distribution is equal to the degrees of freedom.<\/li>\n<\/ul>\n<section class=\"textbox keyTakeaway\">\n<h3>conditions for a [latex]\\chi^2[\/latex] goodness of fit test<\/h3>\n<p>A chi-square model is a good fit for the distribution of the chi-square test statistic only if the following conditions are met:<\/p>\n<ul>\n<li><strong>Random:<\/strong> Observed counts must come from a random sample (to ensure our conclusions are free from sampling bias).<\/li>\n<li><strong>10%:<\/strong> The sample size must be less than a tenth of the population size (to satisfy independence assumptions).<\/li>\n<li><strong>Large Sample:<\/strong> The sample is large enough such that the expected counts are all <em>five or greater<\/em> (to ensure our sampling distribution resembles a chi-square distribution).<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Imagine that three different teams of researchers gathered birthdate data on three different random samples of professional Italian football players. The following tables show their results.<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Sample A<\/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>3<\/td>\n<td>4<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Sample B<\/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>3,000<\/td>\n<td>4,000<\/td>\n<td>2,000<\/td>\n<td>1,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div style=\"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 class=\"para\">The data in the tables are displayed in the following side-by-side bar chart:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1473 size-large\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2023\/04\/24193809\/Picture1-5.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"1024\" height=\"298\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2354\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2354&theme=lumen&iframe_resize_id=ohm2354&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2355\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2355&theme=lumen&iframe_resize_id=ohm2355&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":14,"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\/1405"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1405\/revisions"}],"predecessor-version":[{"id":6868,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1405\/revisions\/6868"}],"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\/1405\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1405"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1405"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1405"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}