{"id":1403,"date":"2023-06-22T02:22:47","date_gmt":"2023-06-22T02:22:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-chi-square-statistics-fresh-take\/"},"modified":"2025-05-16T23:37:19","modified_gmt":"2025-05-16T23:37:19","slug":"introduction-to-chi-square-statistics-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/introduction-to-chi-square-statistics-fresh-take\/","title":{"raw":"Introduction to Chi-Square Statistics \u2013 Fresh Take","rendered":"Introduction to Chi-Square Statistics \u2013 Fresh Take"},"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;Calculate and describe the value of a chi-square statistics in context of a real-world 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}\">Calculate and describe the value of a chi-square statistics in context of a real-world problem<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write a null and alternative hypothesis for a chi-square test&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}\">Write a null and alternative hypothesis for a chi-square test<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Chi-Square Test Statistic<\/h2>\r\n<p>Previously, we have calculated the value of the chi-square test statistic. Let's build upon this calculation using a simplified example in order to understand the statistic\u2019s purpose and meaning.<\/p>\r\n<p><img class=\"alignright wp-image-1461\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22022246\/download-1.png\" alt=\"Two dice\" width=\"189\" height=\"116\" \/>Imagine you are playing a dice gambling game. Each die has 6 sides. If you roll a 1 or 2, you win. If you roll a 3 or 4, it\u2019s a tie. If you roll a 5 or 6, your opponent wins.<\/p>\r\n<p>You play, and you end up losing money. You want to see if the die is unfairly weighted towards the higher numbers.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2349[\/ohm2_question]<\/section>\r\n<p>Let's calculate the chi-square statistic.<\/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 ]2350[\/ohm2_question]\u00a0<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2351[\/ohm2_question]<\/section>\r\n<p>If certain conditions are met (we will discuss these conditions soon!), we can compare our chi-square test statistic values to the chi-square distribution to get the probability of finding the set of rolls that we observed or one that differs more from our expectations by chance alone when the die is actually fair.<\/p>\r\n<p>Let\u2019s do this for the chi-square statistic value from Table D.<\/p>\r\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Click the <strong>Find Probability<\/strong> tab at the top of the 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=2.28[\/latex]) for Table D.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Table D<\/strong><\/td>\r\n<td><strong>You Win (1,2)<\/strong><\/td>\r\n<td><strong>Tie (3,4)<\/strong><\/td>\r\n<td><strong>You Lose (5,6)<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Number of Rolls<\/strong><\/td>\r\n<td>311<\/td>\r\n<td>342<\/td>\r\n<td>347<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n<\/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 ]2352[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate and describe the value of a chi-square statistics in context of a real-world 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}\">Calculate and describe the value of a chi-square statistics in context of a real-world problem<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write a null and alternative hypothesis for a chi-square test&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}\">Write a null and alternative hypothesis for a chi-square test<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Chi-Square Test Statistic<\/h2>\n<p>Previously, we have calculated the value of the chi-square test statistic. Let&#8217;s build upon this calculation using a simplified example in order to understand the statistic\u2019s purpose and meaning.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-1461\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22022246\/download-1.png\" alt=\"Two dice\" width=\"189\" height=\"116\" \/>Imagine you are playing a dice gambling game. Each die has 6 sides. If you roll a 1 or 2, you win. If you roll a 3 or 4, it\u2019s a tie. If you roll a 5 or 6, your opponent wins.<\/p>\n<p>You play, and you end up losing money. You want to see if the die is unfairly weighted towards the higher numbers.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2349\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2349&theme=lumen&iframe_resize_id=ohm2349&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Let&#8217;s calculate the chi-square statistic.<\/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=\"ohm2350\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2350&theme=lumen&iframe_resize_id=ohm2350&source=tnh\" width=\"100%\" height=\"150\"><\/iframe>\u00a0<\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2351\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2351&theme=lumen&iframe_resize_id=ohm2351&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>If certain conditions are met (we will discuss these conditions soon!), we can compare our chi-square test statistic values to the chi-square distribution to get the probability of finding the set of rolls that we observed or one that differs more from our expectations by chance alone when the die is actually fair.<\/p>\n<p>Let\u2019s do this for the chi-square statistic value from Table D.<\/p>\n<section class=\"textbox interact\"><strong>Step 1: <\/strong>Click the <strong>Find Probability<\/strong> tab at the top of the 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=2.28[\/latex]) for Table D.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Table D<\/strong><\/td>\n<td><strong>You Win (1,2)<\/strong><\/td>\n<td><strong>Tie (3,4)<\/strong><\/td>\n<td><strong>You Lose (5,6)<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Number of Rolls<\/strong><\/td>\n<td>311<\/td>\n<td>342<\/td>\n<td>347<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<\/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=\"ohm2352\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2352&theme=lumen&iframe_resize_id=ohm2352&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":12,"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":"fresh_take","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\/1403"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1403\/revisions"}],"predecessor-version":[{"id":6866,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1403\/revisions\/6866"}],"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\/1403\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1403"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1403"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1403"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1403"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}