{"id":1408,"date":"2023-06-22T02:22:51","date_gmt":"2023-06-22T02:22:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-apply-it-2\/"},"modified":"2025-05-16T23:39:59","modified_gmt":"2025-05-16T23:39:59","slug":"chi-square-test-for-goodness-of-fit-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-for-goodness-of-fit-apply-it-2\/","title":{"raw":"Chi-Square Test for Goodness of Fit \u2013 Apply It 2","rendered":"Chi-Square Test for Goodness of Fit \u2013 Apply 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>Conditions<\/h2>\r\n<p>The chi-square model is a family of curves that depend on degrees of freedom (number of categories minus 1). All chi-square curves are skewed to the right with a mean equal to the degrees of freedom.<\/p>\r\n<section class=\"textbox recall\">Here are the conditions for the chi-square test for goodness of fit to make sure it\u2019s valid:\r\n\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 five or greater (to ensure our sampling distribution resembles a chi-square distribution).<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2374[\/ohm2_question]<\/section>\r\n<h3>[latex]\\chi^2 [\/latex] Test Statistic, P-Value, &amp; Conclusions<\/h3>\r\n<p>If these conditions are met, we use the chi-square distribution to find the P-value. We use the same logic that we use in all hypothesis tests to draw a conclusion based on the P-value. If the P-value is at least as small as the significance level, we reject the null hypothesis and accept the alternative hypothesis.<\/p>\r\n<p>The P-value is the likelihood that results from random samples have a [latex]\\chi^2[\/latex] value equal to or greater than that calculated from the data if the null hypothesis is true. For different degrees of freedom, the same [latex]\\chi^2[\/latex] value gives different P-values.<\/p>\r\n<p>Using our mock data set, let's draw a conclusion and make an inference regarding the State of New York claims that each county receives a number of vaccines that is proportional to its population size.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Select the <strong>Goodness of Fit<\/strong> tab at the top of the data analysis tool.<strong><br \/>\r\nStep 2: <\/strong>Under \u201cEnter Data,\u201d choose \u201cContingency Table.\u201d<br \/>\r\n<strong>Step 3: <\/strong>Change each of the counts to the observed counts.<br \/>\r\n<strong>Step 4: <\/strong>Change each of the \u201cprops\u201d to the expected proportions.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>County<\/strong><\/td>\r\n<td>Queens<\/td>\r\n<td>Bronx<\/td>\r\n<td>Westchester<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Observed Count<\/strong><\/td>\r\n<td>204<\/td>\r\n<td>132<\/td>\r\n<td>164<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>%<\/strong><\/td>\r\n<td>48.8%<\/td>\r\n<td>30.6%<\/td>\r\n<td>20.6%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><strong>Step 5: <\/strong>Then press \u201cSubmit.\u201d<\/p>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/chisquaredtest\/\" width=\"100%\" height=\"1100\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/chisquaredtest\/\" 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]2377[\/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>Conditions<\/h2>\n<p>The chi-square model is a family of curves that depend on degrees of freedom (number of categories minus 1). All chi-square curves are skewed to the right with a mean equal to the degrees of freedom.<\/p>\n<section class=\"textbox recall\">Here are the conditions for the chi-square test for goodness of fit to make sure it\u2019s valid:<\/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 five or greater (to ensure our sampling distribution resembles a chi-square distribution).<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2374\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2374&theme=lumen&iframe_resize_id=ohm2374&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>[latex]\\chi^2[\/latex] Test Statistic, P-Value, &amp; Conclusions<\/h3>\n<p>If these conditions are met, we use the chi-square distribution to find the P-value. We use the same logic that we use in all hypothesis tests to draw a conclusion based on the P-value. If the P-value is at least as small as the significance level, we reject the null hypothesis and accept the alternative hypothesis.<\/p>\n<p>The P-value is the likelihood that results from random samples have a [latex]\\chi^2[\/latex] value equal to or greater than that calculated from the data if the null hypothesis is true. For different degrees of freedom, the same [latex]\\chi^2[\/latex] value gives different P-values.<\/p>\n<p>Using our mock data set, let&#8217;s draw a conclusion and make an inference regarding the State of New York claims that each county receives a number of vaccines that is proportional to its population size.<\/p>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Select the <strong>Goodness of Fit<\/strong> tab at the top of the data analysis tool.<strong><br \/>\nStep 2: <\/strong>Under \u201cEnter Data,\u201d choose \u201cContingency Table.\u201d<br \/>\n<strong>Step 3: <\/strong>Change each of the counts to the observed counts.<br \/>\n<strong>Step 4: <\/strong>Change each of the \u201cprops\u201d to the expected proportions.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>County<\/strong><\/td>\n<td>Queens<\/td>\n<td>Bronx<\/td>\n<td>Westchester<\/td>\n<\/tr>\n<tr>\n<td><strong>Observed Count<\/strong><\/td>\n<td>204<\/td>\n<td>132<\/td>\n<td>164<\/td>\n<\/tr>\n<tr>\n<td><strong>%<\/strong><\/td>\n<td>48.8%<\/td>\n<td>30.6%<\/td>\n<td>20.6%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Step 5: <\/strong>Then press \u201cSubmit.\u201d<\/p>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/chisquaredtest\/\" width=\"100%\" height=\"1100\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/chisquaredtest\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2377\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2377&theme=lumen&iframe_resize_id=ohm2377&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":17,"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":"apply_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\/1408"}],"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\/1408\/revisions"}],"predecessor-version":[{"id":6871,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1408\/revisions\/6871"}],"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\/1408\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1408"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1408"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1408"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1408"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}