{"id":1257,"date":"2023-06-22T02:09:42","date_gmt":"2023-06-22T02:09:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/two-sample-test-for-proportions-dig-deeper\/"},"modified":"2024-01-11T22:20:42","modified_gmt":"2024-01-11T22:20:42","slug":"two-sample-test-for-proportions-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/two-sample-test-for-proportions-dig-deeper\/","title":{"raw":"Two-Sample Test for Proportions: Fresh Take","rendered":"Two-Sample Test for Proportions: 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;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\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<\/ul>\r\n<\/section>\r\n<p class=\"para\" style=\"margin-left: 0in; text-indent: 0in;\">A one-sample test of proportions tests a claim about a population proportion. A <b>two-sample test of proportions<\/b> tests a claim about two population proportions. When testing a claim that compares two populations, you must also check that the two populations are independent.<\/p>\r\n<p>Let's see if we can distinguish between these situations.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1063[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1064[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1065[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1066[\/ohm2_question]<\/section>\r\n<section>Comparing two proportions is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions. Let's begin by stating null and alternative hypotheses.\r\n\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1067[\/ohm2_question]<\/section>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>two-sample [latex]z[\/latex]-test of proportions<\/h3>\r\n<ol>\r\n\t<li>[reveal-answer q=\"249099\"]Write out the null and alternative hypotheses[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"249099\"]\r\n\r\n<ul>\r\n\t<li>Null hypothesis: [latex]H_0: p_1=p_2[\/latex] or [latex]H_0: p_1-p_2=0[\/latex]<\/li>\r\n\t<li>Alternative hypothesis:\r\n\r\n<ul>\r\n\t<li>[latex]H_A: p_1\\lt p_2[\/latex] or [latex]H_A: p_1-p_2\\lt 0[\/latex]<\/li>\r\n\t<li>[latex]H_A: p_1&gt;p_2[\/latex] or [latex]H_A: p_1-p_2&gt;0[\/latex]<\/li>\r\n\t<li>[latex]H_A: p_1\\ne p_2[\/latex] or [latex]H_A: p_1-p_2\\ne0[\/latex][\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>[reveal-answer q=\"454056\"]Check the conditions for the hypothesis test.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"454056\"] For testing a one-sample [latex]z[\/latex]-test for proportions, we require:\r\n\r\n<ul>\r\n\t<li>Large Counts: For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that: [latex]n_1\\hat{p}_c \\ge 10[\/latex], [latex]n_2\\hat{p}_c \\ge 10[\/latex], [latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and [latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\r\n\t<li>Random samples\/assignment: 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>10%:\u00a0Check that\u00a0[latex]n_1&lt;0.10(N_1)[\/latex] and [latex]n_2&lt;0.10(N_2)[\/latex].[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Calculate a test statistic.<\/li>\r\n\t<li>Calculate a P-value.<\/li>\r\n\t<li>[reveal-answer q=\"919782\"]Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"919782\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Decision<\/strong><\/td>\r\n<td><strong>Conclusion<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\r\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/li>\r\n\t<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/ \" width=\"100%\" height=\"1200\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\r\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section class=\"textbox example\">Two types of medication for hives are being tested to determine if there is a\u00a0<strong>difference in the proportions of adult patient reactions<\/strong>.\r\n\r\n<ul>\r\n\t<li>Twenty\u00a0out of a random\u00a0sample of [latex]200[\/latex]\u00a0adults given medication A still had hives [latex]30[\/latex] minutes after taking the medication.<\/li>\r\n\t<li>Twelve out of another random sample of [latex]200[\/latex] adults given medication B still had hives [latex]30[\/latex] minutes after taking the medication.<\/li>\r\n<\/ul>\r\n<p>Test at a [latex]1\\%[\/latex] level of significance.<\/p>\r\n<p>[reveal-answer q=\"991590\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"991590\"]<\/p>\r\n<p>The problem asks for a difference in proportions, making it a test of two proportions.<\/p>\r\n<p>Let [latex]A[\/latex] and [latex]B[\/latex] be the subscripts for medication [latex]A[\/latex] and medication [latex]B[\/latex], respectively. Then [latex]p_A[\/latex] and [latex]p_B[\/latex] are the desired population proportions.<\/p>\r\n<p><strong>Step 1: Write out the hypotheses.<\/strong><\/p>\r\n<ul style=\"list-style-type: none;\">\r\n\t<li>[latex]H_0: p_A - p_B = 0[\/latex]<\/li>\r\n\t<li>[latex]H_A: p_A - p_B \\ne 0[\/latex]<\/li>\r\n<\/ul>\r\n<p><strong>Step 2: Check the conditions.<\/strong><\/p>\r\n<p>Our samples are random, so there is no problem there. Again, we want to determine whether the normal model is a good fit for the sampling distribution of sample proportions.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\hat{p}_c = \\frac{20+12}{200+200} = 0.08[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n_1\\hat{p}_c = n_2\\hat{p}_c = 200(0.08) = 16 \\ge 10[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]n_1(1-\\hat{p}_c) = n_2(1-\\hat{p}_c) = 200(1-0.08) = 200(0.92) = 184 \\ge 10[\/latex]<\/p>\r\n<p>Because these are all more than [latex]10[\/latex], we can use the normal model to find the P-value.<\/p>\r\n<p><strong>Step 3: Calculate the test statistic.<\/strong><\/p>\r\n<p>Since we can use the normal model, we need to calculate the [latex]z[\/latex]-test statistic for the difference in the sample proportion. Using the statistical tool, we get [latex]z= 1.47[\/latex].<\/p>\r\n<p>The observed sample difference is about [latex]1.47[\/latex] standard errors above\u00a0 the null hypothesis value of [latex]0[\/latex].<\/p>\r\n<p><strong>Step 4: Calculate the P-value.<\/strong><\/p>\r\n<p>Using our statistical tool, the P-value is [latex]0.1404[\/latex].<\/p>\r\n<p><strong>Step 5: Compare P-value with significance level.<\/strong><\/p>\r\n<p>Now we compare the P-value to the level of significance, [latex]\\alpha = 0.01[\/latex]. In this case, the P-value of [latex]0.1404[\/latex] is greater than [latex]0.01[\/latex], which means we do not have enough evidence to reject the null hypothesis.<\/p>\r\n<p><strong>Step 6: Write conclusion in context of the problem.<\/strong><\/p>\r\n<p>At a [latex]1\\%[\/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adult patients who did not react after [latex]30[\/latex] minutes to medication [latex]A[\/latex] and medication [latex]B[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\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<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<\/ul>\n<\/section>\n<p class=\"para\" style=\"margin-left: 0in; text-indent: 0in;\">A one-sample test of proportions tests a claim about a population proportion. A <b>two-sample test of proportions<\/b> tests a claim about two population proportions. When testing a claim that compares two populations, you must also check that the two populations are independent.<\/p>\n<p>Let&#8217;s see if we can distinguish between these situations.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1063\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1063&theme=lumen&iframe_resize_id=ohm1063&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1064\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1064&theme=lumen&iframe_resize_id=ohm1064&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1065\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1065&theme=lumen&iframe_resize_id=ohm1065&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1066\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1066&theme=lumen&iframe_resize_id=ohm1066&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>Comparing two proportions is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions. Let&#8217;s begin by stating null and alternative hypotheses.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1067\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1067&theme=lumen&iframe_resize_id=ohm1067&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>two-sample [latex]z[\/latex]-test of proportions<\/h3>\n<ol>\n<li>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q249099\">Write out the null and alternative hypotheses<\/button><\/p>\n<div id=\"q249099\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Null hypothesis: [latex]H_0: p_1=p_2[\/latex] or [latex]H_0: p_1-p_2=0[\/latex]<\/li>\n<li>Alternative hypothesis:\n<ul>\n<li>[latex]H_A: p_1\\lt p_2[\/latex] or [latex]H_A: p_1-p_2\\lt 0[\/latex]<\/li>\n<li>[latex]H_A: p_1>p_2[\/latex] or [latex]H_A: p_1-p_2>0[\/latex]<\/li>\n<li>[latex]H_A: p_1\\ne p_2[\/latex] or [latex]H_A: p_1-p_2\\ne0[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q454056\">Check the conditions for the hypothesis test.<\/button><\/p>\n<div id=\"q454056\" class=\"hidden-answer\" style=\"display: none\"> For testing a one-sample [latex]z[\/latex]-test for proportions, we require:<\/p>\n<ul>\n<li>Large Counts: For [latex]\\hat{p}_c = \\frac{x_1+x_2}{n_1+n_2}[\/latex], check that: [latex]n_1\\hat{p}_c \\ge 10[\/latex], [latex]n_2\\hat{p}_c \\ge 10[\/latex], [latex]n_1(1-\\hat{p}_c) \\ge 10[\/latex], and [latex]n_2(1-\\hat{p}_c) \\ge 10[\/latex].<\/li>\n<li>Random samples\/assignment: Check that the two samples are independent and random samples or that they come from randomly assigned groups in an experiment.<\/li>\n<li>10%:\u00a0Check that\u00a0[latex]n_1<0.10(N_1)[\/latex] and [latex]n_2<0.10(N_2)[\/latex].<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>Calculate a test statistic.<\/li>\n<li>Calculate a P-value.<\/li>\n<li>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q919782\">Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<\/button><\/p>\n<div id=\"q919782\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td><strong>Decision<\/strong><\/td>\n<td><strong>Conclusion<\/strong><\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\le\\alpha[\/latex], there is enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<tr>\n<td>If P-value [latex]\\gt\\alpha[\/latex], there is not enough evidence to reject the null hypothesis.<\/td>\n<td>At the [latex]\\alpha\\times[\/latex]100% significance level, the data do not provide convincing evidence in support of the alternative hypothesis.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/li>\n<li>Write a conclusion in context (e.g., we do\/do not have convincing evidence\u2026).<\/li>\n<\/ol>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" width=\"100%\" height=\"1200\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\"><\/span><\/iframe><br \/>\n[<a href=\"https:\/\/lumen-learning.shinyapps.io\/2sample_prop\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section class=\"textbox example\">Two types of medication for hives are being tested to determine if there is a\u00a0<strong>difference in the proportions of adult patient reactions<\/strong>.<\/p>\n<ul>\n<li>Twenty\u00a0out of a random\u00a0sample of [latex]200[\/latex]\u00a0adults given medication A still had hives [latex]30[\/latex] minutes after taking the medication.<\/li>\n<li>Twelve out of another random sample of [latex]200[\/latex] adults given medication B still had hives [latex]30[\/latex] minutes after taking the medication.<\/li>\n<\/ul>\n<p>Test at a [latex]1\\%[\/latex] level of significance.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q991590\">Show answer<\/button><\/p>\n<div id=\"q991590\" class=\"hidden-answer\" style=\"display: none\">\n<p>The problem asks for a difference in proportions, making it a test of two proportions.<\/p>\n<p>Let [latex]A[\/latex] and [latex]B[\/latex] be the subscripts for medication [latex]A[\/latex] and medication [latex]B[\/latex], respectively. Then [latex]p_A[\/latex] and [latex]p_B[\/latex] are the desired population proportions.<\/p>\n<p><strong>Step 1: Write out the hypotheses.<\/strong><\/p>\n<ul style=\"list-style-type: none;\">\n<li>[latex]H_0: p_A - p_B = 0[\/latex]<\/li>\n<li>[latex]H_A: p_A - p_B \\ne 0[\/latex]<\/li>\n<\/ul>\n<p><strong>Step 2: Check the conditions.<\/strong><\/p>\n<p>Our samples are random, so there is no problem there. Again, we want to determine whether the normal model is a good fit for the sampling distribution of sample proportions.<\/p>\n<p style=\"text-align: center;\">[latex]\\hat{p}_c = \\frac{20+12}{200+200} = 0.08[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n_1\\hat{p}_c = n_2\\hat{p}_c = 200(0.08) = 16 \\ge 10[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n_1(1-\\hat{p}_c) = n_2(1-\\hat{p}_c) = 200(1-0.08) = 200(0.92) = 184 \\ge 10[\/latex]<\/p>\n<p>Because these are all more than [latex]10[\/latex], we can use the normal model to find the P-value.<\/p>\n<p><strong>Step 3: Calculate the test statistic.<\/strong><\/p>\n<p>Since we can use the normal model, we need to calculate the [latex]z[\/latex]-test statistic for the difference in the sample proportion. Using the statistical tool, we get [latex]z= 1.47[\/latex].<\/p>\n<p>The observed sample difference is about [latex]1.47[\/latex] standard errors above\u00a0 the null hypothesis value of [latex]0[\/latex].<\/p>\n<p><strong>Step 4: Calculate the P-value.<\/strong><\/p>\n<p>Using our statistical tool, the P-value is [latex]0.1404[\/latex].<\/p>\n<p><strong>Step 5: Compare P-value with significance level.<\/strong><\/p>\n<p>Now we compare the P-value to the level of significance, [latex]\\alpha = 0.01[\/latex]. In this case, the P-value of [latex]0.1404[\/latex] is greater than [latex]0.01[\/latex], which means we do not have enough evidence to reject the null hypothesis.<\/p>\n<p><strong>Step 6: Write conclusion in context of the problem.<\/strong><\/p>\n<p>At a [latex]1\\%[\/latex] level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference in the proportions of adult patients who did not react after [latex]30[\/latex] minutes to medication [latex]A[\/latex] and medication [latex]B[\/latex].<\/p><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":8,"menu_order":39,"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":"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\/1257"}],"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\/1257\/revisions"}],"predecessor-version":[{"id":5094,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1257\/revisions\/5094"}],"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\/1257\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1257"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1257"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1257"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}