{"id":2140,"date":"2023-07-27T21:13:14","date_gmt":"2023-07-27T21:13:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=2140"},"modified":"2025-05-16T03:44:24","modified_gmt":"2025-05-16T03:44:24","slug":"two-sample-test-for-proportions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/two-sample-test-for-proportions-learn-it-4\/","title":{"raw":"Two-Sample Test for Proportions: Learn It 4","rendered":"Two-Sample Test for Proportions: Learn It 4"},"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;Recognize when a one-sample z-test or a two-sample z-test is needed to answer a research question&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Recognize when a one-sample [latex]z[\/latex]-test or a two-sample [latex]z[\/latex]-test is needed to answer a research question.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a two-sample z-test for proportions from hypotheses to conclusions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Complete a two-sample [latex]z[\/latex]-test for proportions from hypotheses to conclusions.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Conditions needed for a two-sample [latex]z[\/latex]-test for proportions<\/h2>\r\n<p>When testing a claim that compares two populations, you must ensure that the two populations are independent. This is one of the conditions for a two-sample [latex]z[\/latex]-test for proportions. How about sample size? What are the necessary sample sizes?<\/p>\r\n<p>Recall that the\u00a0<strong>Central Limit Theorem<\/strong>\u00a0states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/p>\r\n<p>However, when we have two independent samples, and we need to compare the proportions between the two groups, we might have two different sample sizes and two different proportions. This means that the previous calculation [latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex] will not work. We need something called pooled proportion [latex]\\hat{p}_c[\/latex]. The <strong>pooled proportion<\/strong> is a combined proportion calculated from the two independent samples in the hypothesis test. It is used when assuming that the true proportions in both groups are equal under the null hypothesis.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Conditions for Two-Sample [latex]z[\/latex]-Test of Proportions<\/h3>\r\n<ol>\r\n\t<li><strong>Large Counts: <\/strong>For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that:\r\n\r\n<ul>\r\n\t<li>[latex]n_1\\hat{p}_c \\ge 10[\/latex],<\/li>\r\n\t<li>[latex]n_2\\hat{p}_c \\ge 10[\/latex],<\/li>\r\n\t<li>[latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and<\/li>\r\n\t<li>[latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Random Samples\/Assignment: <\/strong>Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.<\/li>\r\n\t<li><strong>10%<\/strong>:\u00a0Check that\u00a0[latex]n_1&lt;0.10(N_1)[\/latex] and [latex]n_2&lt;0.10(N_2)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1769[\/ohm2_question]<\/section>\r\n<section><strong>Note:<\/strong> The final condition is that the sample sizes are each less than a tenth of the size of the populations from which they\u2019re drawn [[latex]n_1&lt;0.10(N_1)[\/latex] and [latex]n_2&lt;0.10(N_2)[\/latex]]. This helps ensure our estimates for the standard errors are accurate. However, this condition does not need to be checked in the case of a randomized experiment.<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Recognize when a one-sample z-test or a two-sample z-test is needed to answer a research question&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Recognize when a one-sample [latex]z[\/latex]-test or a two-sample [latex]z[\/latex]-test is needed to answer a research question.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Complete a two-sample z-test for proportions from hypotheses to conclusions&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Complete a two-sample [latex]z[\/latex]-test for proportions from hypotheses to conclusions.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Conditions needed for a two-sample [latex]z[\/latex]-test for proportions<\/h2>\n<p>When testing a claim that compares two populations, you must ensure that the two populations are independent. This is one of the conditions for a two-sample [latex]z[\/latex]-test for proportions. How about sample size? What are the necessary sample sizes?<\/p>\n<p>Recall that the\u00a0<strong>Central Limit Theorem<\/strong>\u00a0states that, as the sample size gets larger, the distribution of the sample proportion will become closer to a normal distribution.<\/p>\n<p>However, when we have two independent samples, and we need to compare the proportions between the two groups, we might have two different sample sizes and two different proportions. This means that the previous calculation [latex]np \\geq 10[\/latex] and [latex]n(1-p) \\geq 10[\/latex] will not work. We need something called pooled proportion [latex]\\hat{p}_c[\/latex]. The <strong>pooled proportion<\/strong> is a combined proportion calculated from the two independent samples in the hypothesis test. It is used when assuming that the true proportions in both groups are equal under the null hypothesis.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Conditions for Two-Sample [latex]z[\/latex]-Test of Proportions<\/h3>\n<ol>\n<li><strong>Large Counts: <\/strong>For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that:\n<ul>\n<li>[latex]n_1\\hat{p}_c \\ge 10[\/latex],<\/li>\n<li>[latex]n_2\\hat{p}_c \\ge 10[\/latex],<\/li>\n<li>[latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and<\/li>\n<li>[latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Random Samples\/Assignment: <\/strong>Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.<\/li>\n<li><strong>10%<\/strong>:\u00a0Check that\u00a0[latex]n_1<0.10(N_1)[\/latex] and [latex]n_2<0.10(N_2)[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1769\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1769&theme=lumen&iframe_resize_id=ohm1769&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section><strong>Note:<\/strong> The final condition is that the sample sizes are each less than a tenth of the size of the populations from which they\u2019re drawn [[latex]n_1<0.10(N_1)[\/latex] and [latex]n_2<0.10(N_2)[\/latex]]. This helps ensure our estimates for the standard errors are accurate. However, this condition does not need to be checked in the case of a randomized experiment.<\/section>\n","protected":false},"author":12,"menu_order":35,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1205,"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\/2140"}],"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\/12"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2140\/revisions"}],"predecessor-version":[{"id":6772,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2140\/revisions\/6772"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1205"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/2140\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=2140"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=2140"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=2140"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=2140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}