{"id":1424,"date":"2023-06-22T02:23:04","date_gmt":"2023-06-22T02:23:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-of-independence-learn-it-5\/"},"modified":"2023-10-20T20:46:57","modified_gmt":"2023-10-20T20:46:57","slug":"chi-square-test-of-independence-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/chi-square-test-of-independence-learn-it-5\/","title":{"raw":"Chi-Square Test of Independence \u2013 Learn It 5","rendered":"Chi-Square Test of Independence \u2013 Learn It 5"},"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<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2888[\/ohm2_question]<\/section>\r\n<p>We concluded from our hypothesis test that the variables <em>Income level<\/em> and <em>Education level<\/em> are not independent, but we do not know how they are associated.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>lurking variable<\/h3>\r\n<p>It could be that there is a third variable not included in our study that impacts the values of both of the variables we are considering. Such a variable is called a <strong>lurking variable<\/strong>.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2889[\/ohm2_question]<\/section>\r\n<p>Now that you\u2019ve seen both the chi-square test of homogeneity and the chi-square test of independence in action, let's summarize the difference between the two tests.<\/p>\r\n<section class=\"textbox connectIt\">\r\n<h4>Test of Independence for a Two-Way Table<\/h4>\r\n<ul>\r\n\t<li>In the test of independence, we consider <strong>one population and two categorical variables<\/strong>.<\/li>\r\n\t<li>We learned that two events are independent if [latex]P(A|B) = P(A)[\/latex], but we did not pay attention to variability in the sample. With the chi-square test of independence, we have a method for deciding whether our observed [latex]P(A|B)[\/latex] is \u201ctoo far\u201d from our observed [latex]P(A)[\/latex] to infer independence in the population.<\/li>\r\n\t<li>The null hypothesis says the two variables are independent (or not associated). The alternative hypothesis says the two variables are dependent (or associated).<\/li>\r\n\t<li>To test our hypotheses, we select a single random sample and gather data for two different categorical variables.<\/li>\r\n<\/ul>\r\n<h4>Test of Homogeneity for a Two-Way Table<\/h4>\r\n<ul>\r\n\t<li>In the test of homogeneity, we consider <strong>two or more populations<\/strong> (or two or more subgroups of a population) <strong>and a single categorical variable<\/strong>.<\/li>\r\n\t<li>The test of homogeneity expands on the test for a difference in two population proportions that we learned in\u00a0<em>Inference for Two Proportions<\/em>\u00a0by comparing the distribution of the categorical variable across multiple groups or populations.<\/li>\r\n\t<li>The null hypothesis says that the distribution of proportions for all categories is the same in each group or population. The alternative hypothesis says that the distributions differ.<\/li>\r\n\t<li>To test our hypotheses, we select a random sample from each population or subgroup independently. We gather data for one categorical variable.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1 ]2890[\/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 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<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2888\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2888&theme=lumen&iframe_resize_id=ohm2888&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We concluded from our hypothesis test that the variables <em>Income level<\/em> and <em>Education level<\/em> are not independent, but we do not know how they are associated.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>lurking variable<\/h3>\n<p>It could be that there is a third variable not included in our study that impacts the values of both of the variables we are considering. Such a variable is called a <strong>lurking variable<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2889\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2889&theme=lumen&iframe_resize_id=ohm2889&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Now that you\u2019ve seen both the chi-square test of homogeneity and the chi-square test of independence in action, let&#8217;s summarize the difference between the two tests.<\/p>\n<section class=\"textbox connectIt\">\n<h4>Test of Independence for a Two-Way Table<\/h4>\n<ul>\n<li>In the test of independence, we consider <strong>one population and two categorical variables<\/strong>.<\/li>\n<li>We learned that two events are independent if [latex]P(A|B) = P(A)[\/latex], but we did not pay attention to variability in the sample. With the chi-square test of independence, we have a method for deciding whether our observed [latex]P(A|B)[\/latex] is \u201ctoo far\u201d from our observed [latex]P(A)[\/latex] to infer independence in the population.<\/li>\n<li>The null hypothesis says the two variables are independent (or not associated). The alternative hypothesis says the two variables are dependent (or associated).<\/li>\n<li>To test our hypotheses, we select a single random sample and gather data for two different categorical variables.<\/li>\n<\/ul>\n<h4>Test of Homogeneity for a Two-Way Table<\/h4>\n<ul>\n<li>In the test of homogeneity, we consider <strong>two or more populations<\/strong> (or two or more subgroups of a population) <strong>and a single categorical variable<\/strong>.<\/li>\n<li>The test of homogeneity expands on the test for a difference in two population proportions that we learned in\u00a0<em>Inference for Two Proportions<\/em>\u00a0by comparing the distribution of the categorical variable across multiple groups or populations.<\/li>\n<li>The null hypothesis says that the distribution of proportions for all categories is the same in each group or population. The alternative hypothesis says that the distributions differ.<\/li>\n<li>To test our hypotheses, we select a random sample from each population or subgroup independently. We gather data for one categorical variable.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2890\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2890&theme=lumen&iframe_resize_id=ohm2890&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":31,"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\/1424"}],"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\/1424\/revisions"}],"predecessor-version":[{"id":3958,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1424\/revisions\/3958"}],"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\/1424\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1424"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1424"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1424"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}