{"id":1422,"date":"2023-06-22T02:23:02","date_gmt":"2023-06-22T02:23:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/1909\/"},"modified":"2025-05-16T23:45:30","modified_gmt":"2025-05-16T23:45:30","slug":"1909","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/1909\/","title":{"raw":"Chi-Square Test of Independence \u2013 Learn It 3","rendered":"Chi-Square Test of Independence \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 of independence&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 of independence<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the conclusion of a chi-square test of independence 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}\">Write the conclusion of a chi-square test of independence in context of the problem<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>The mechanics of performing a chi-square test of independence are the same as those for the chi-square test of homogeneity.<\/p>\r\n<p class=\"para\">Since we are dealing with two variables here instead of just one, we can find the expected counts for each cell by focusing on the marginal distribution of either variable.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>marginal distribution<\/h3>\r\n<p>The <strong>marginal distribution <\/strong>of a variable gives the distribution of one of the variables with no regard to the other variable whatsoever.<\/p>\r\n<p>In the table, this will be either the total row or the total column. One way to remember this is that the \u201cmargins\u201d are on the outsides of a piece of paper (sides, top, and bottom), and the total row and column are the outside row and column of the table (on the side and bottom).<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\"><strong>Income level<\/strong><\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">\u00a0<\/td>\r\n<td style=\"width: 83.507px;\">\u00a0<\/td>\r\n<td style=\"width: 54.4792px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]&lt;$30,000[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]$30,000\u2013$74,999[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]$75,000[\/latex] and up<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">Total<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\"><strong>Education level<\/strong><\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">Post-Grad Degree<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]46[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]56[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">College Degree<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]39[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]113[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]202[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]354[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">Some College<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]131[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]138[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]120[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]389[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">HS Grad<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]175[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]129[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]65[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]369[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">No HS Degree<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]78[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]32[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]118[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\r\n<td style=\"height: 18px; width: 85.434px;\">Total<\/td>\r\n<td style=\"height: 18px; width: 87.6042px;\">[latex]425[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 91.2153px;\">[latex]420[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 83.507px;\">[latex]441[\/latex]<\/td>\r\n<td style=\"height: 18px; width: 54.4792px;\">[latex]1,286[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"para\">If <i>Income level<\/i> and <i>Education level<\/i> are independent, the proportion of people with incomes under [latex]$30,000[\/latex] should be the same regardless of education level, so it should match the overall proportion of individuals with incomes under [latex]$30,000[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{Total individuals with incomes under }$30,000}{\\text{Total individuals in the sample}}[\/latex][latex]=\\dfrac{425}{1286} = 0.33048212 \\text{ or } 33.048212\\%[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2871[\/ohm2_question]<\/section>\r\n<section>\r\n<p class=\"para\">The proportions you found in the previous table should be the proportions of income level for every value of the variable <i>Education level <\/i>and\u00a0<em>Income level.<\/em><\/p>\r\n<section class=\"textbox example\">For example, about [latex]33.05\\%[\/latex] of the [latex]56[\/latex] people with post-grad degrees should have an income level under [latex]$30,000[\/latex]:\r\n\r\n<p style=\"text-align: center;\">[latex]33.048212\\% \\text{ of } 56 = 0.33048212 \\times 56 = 18.507[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2877[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">For example, of the [latex]425[\/latex] individuals sampled with an income level under [latex]$30,000[\/latex], about [latex]4.35\\%[\/latex] of them should have post-graduate degrees, so there is an expected count of\r\n\r\n<p style=\"text-align: center;\">[latex]4.354588\\% \\text{ of } 425 = 0.04354588 \\times 425 = 18.507[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2878[\/ohm2_question]<\/section>\r\n<\/section>\r\n<p>Calculating each expected count for a table is tedious, but it is needed to calculate the [latex]\\chi^2[\/latex] value to make an inference about the population.<\/p>\r\n<p>However, we can utilize technology to help us conduct all of the calculations.<\/p>","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 of independence&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 of independence<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write the conclusion of a chi-square test of independence 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}\">Write the conclusion of a chi-square test of independence in context of the problem<\/span><\/li>\n<\/ul>\n<\/section>\n<p>The mechanics of performing a chi-square test of independence are the same as those for the chi-square test of homogeneity.<\/p>\n<p class=\"para\">Since we are dealing with two variables here instead of just one, we can find the expected counts for each cell by focusing on the marginal distribution of either variable.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>marginal distribution<\/h3>\n<p>The <strong>marginal distribution <\/strong>of a variable gives the distribution of one of the variables with no regard to the other variable whatsoever.<\/p>\n<p>In the table, this will be either the total row or the total column. One way to remember this is that the \u201cmargins\u201d are on the outsides of a piece of paper (sides, top, and bottom), and the total row and column are the outside row and column of the table (on the side and bottom).<\/p>\n<\/section>\n<section class=\"textbox example\">\n<table>\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 87.6042px;\"><strong>Income level<\/strong><\/td>\n<td style=\"height: 18px; width: 91.2153px;\">\u00a0<\/td>\n<td style=\"width: 83.507px;\">\u00a0<\/td>\n<td style=\"width: 54.4792px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]<$30,000[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]$30,000\u2013$74,999[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]$75,000[\/latex] and up<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">Total<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\"><strong>Education level<\/strong><\/td>\n<td style=\"height: 18px; width: 85.434px;\">Post-Grad Degree<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]8[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]46[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]56[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">College Degree<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]39[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]113[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]202[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]354[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">Some College<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]131[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]138[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]120[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]389[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">HS Grad<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]175[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]129[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]65[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]369[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">No HS Degree<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]78[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]32[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]8[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]118[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"height: 18px; width: 87.7604px;\">\u00a0<\/td>\n<td style=\"height: 18px; width: 85.434px;\">Total<\/td>\n<td style=\"height: 18px; width: 87.6042px;\">[latex]425[\/latex]<\/td>\n<td style=\"height: 18px; width: 91.2153px;\">[latex]420[\/latex]<\/td>\n<td style=\"height: 18px; width: 83.507px;\">[latex]441[\/latex]<\/td>\n<td style=\"height: 18px; width: 54.4792px;\">[latex]1,286[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"para\">If <i>Income level<\/i> and <i>Education level<\/i> are independent, the proportion of people with incomes under [latex]$30,000[\/latex] should be the same regardless of education level, so it should match the overall proportion of individuals with incomes under [latex]$30,000[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\text{Total individuals with incomes under }$30,000}{\\text{Total individuals in the sample}}[\/latex][latex]=\\dfrac{425}{1286} = 0.33048212 \\text{ or } 33.048212\\%[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2871\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2871&theme=lumen&iframe_resize_id=ohm2871&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<p class=\"para\">The proportions you found in the previous table should be the proportions of income level for every value of the variable <i>Education level <\/i>and\u00a0<em>Income level.<\/em><\/p>\n<section class=\"textbox example\">For example, about [latex]33.05\\%[\/latex] of the [latex]56[\/latex] people with post-grad degrees should have an income level under [latex]$30,000[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]33.048212\\% \\text{ of } 56 = 0.33048212 \\times 56 = 18.507[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2877\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2877&theme=lumen&iframe_resize_id=ohm2877&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">For example, of the [latex]425[\/latex] individuals sampled with an income level under [latex]$30,000[\/latex], about [latex]4.35\\%[\/latex] of them should have post-graduate degrees, so there is an expected count of<\/p>\n<p style=\"text-align: center;\">[latex]4.354588\\% \\text{ of } 425 = 0.04354588 \\times 425 = 18.507[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2878\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2878&theme=lumen&iframe_resize_id=ohm2878&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<p>Calculating each expected count for a table is tedious, but it is needed to calculate the [latex]\\chi^2[\/latex] value to make an inference about the population.<\/p>\n<p>However, we can utilize technology to help us conduct all of the calculations.<\/p>\n","protected":false},"author":8,"menu_order":29,"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\/1422"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions"}],"predecessor-version":[{"id":6880,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions\/6880"}],"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\/1422\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1422"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1422"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1422"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}