{"id":1253,"date":"2023-06-22T02:09:39","date_gmt":"2023-06-22T02:09:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/two-sample-test-for-proportions-learn-it-3\/"},"modified":"2025-05-16T03:46:47","modified_gmt":"2025-05-16T03:46:47","slug":"two-sample-test-for-proportions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/two-sample-test-for-proportions-learn-it-3\/","title":{"raw":"Two-Sample Test for Proportions: Apply It 1","rendered":"Two-Sample Test for Proportions: Apply It 1"},"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>Two-Sample Hypothesis Test for Proportions<\/h2>\r\n<p>As mentioned before, hypothesis testing is part of inference. We previously stated that the purpose of a hypothesis test is to use sample data to test a claim about a population parameter. However, studies often compare two groups. Comparing two proportions is quite common. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.<\/p>\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>Write out the null and alternative hypotheses.\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]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Check the conditions for the hypothesis test. For testing a two-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].<\/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>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\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<\/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=\"1125\" frameborder=\"no\" data-mce-fragment=\"1\"><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\"><strong><strong><strong>Vaccine Efficiency[footnote]https:\/\/www.nytimes.com\/2020\/12\/13\/learning\/what-does-95-effective-mean-teaching-the-math-of-vaccine-efficacy.html[\/footnote]<\/strong><\/strong><\/strong>In 2020, after a trial showed that Pfizer\u2019s\u00a0coronavirus vaccine\u00a0had an\u00a0efficacy\u00a0rate of [latex]95\\%[\/latex], the Food and Drug Administration\u00a0approved the vaccine.<br \/>\r\n<span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"><br \/>\r\nThe [latex]43,660[\/latex] subjects were split evenly between the placebo and vaccine groups (about [latex]21,830[\/latex] subjects per group). In the placebo group \u2014 the group that got a \u201cfake\u201d vaccine \u2014 [latex]162[\/latex] became infected with the coronavirus and showed symptoms. In the vaccine group \u2014 the group that got the real vaccine \u2014 that number was only eight. <\/span>At the [latex]5\\%[\/latex] significance level, is the vaccine group\u2019s infection risk\u00a0<em class=\"css-2fg4z9 e1gzwzxm0\">that different<\/em>\u00a0from the placebo group\u2019s?[reveal-answer q=\"224900\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"224900\"]<br \/>\r\n<strong>Step 1: Write out the hypotheses.<\/strong>\r\n<ul style=\"list-style-type: none;\">\r\n\t<li>\u00a0<\/li>\r\n\t<li>[latex]H_0: p_p = p_v[\/latex] or [latex]H_0: p_p - p_v=0[\/latex]<\/li>\r\n\t<li>[latex]H_A: p_p \\ne p_v[\/latex] or [latex]H_A: p_p - p_v\\ne0[\/latex]<\/li>\r\n<\/ul>\r\n<p>where [latex]p_p[\/latex] is the proportion of the placebo subjects that became infected and [latex]p_v[\/latex] is the proportion of the vaccine subjects that became infected.<\/p>\r\n<p><strong>Step 2: Check the conditions.<\/strong><\/p>\r\n<p>The sample is random, so there is no problem there. Again, we want to determine whether the normal model is a good fit.<\/p>\r\n<p>[latex]\\hat{p}_c = \\frac{162+8}{21830+21830} = \\frac{170}{43660}[\/latex]<\/p>\r\n<ul>\r\n\t<li>[latex]n_p\\hat{p}_c = 21830(\\frac{170}{43660})= 85\\ge 10[\/latex],<\/li>\r\n\t<li>[latex]n_v\\hat{p}_c = 21830(\\frac{170}{43660})= 85\\ge 10[\/latex],<\/li>\r\n\t<li>[latex]n_p(1-\\hat{p}_c) = 21830(1-\\frac{170}{43660})= 21745\\ge 10[\/latex], and<\/li>\r\n\t<li>[latex]n_v(1-\\hat{p}_c) =21830(1-\\frac{170}{43660})= 21745\\ge 10[\/latex].<\/li>\r\n<\/ul>\r\n<p>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>[latex]p_p = \\frac{162}{21830} \\approx 0.74\\%[\/latex]<\/p>\r\n<p>[latex]p_v =\\frac{8}{21830} \\approx 0.04\\%[\/latex]<\/p>\r\n<p>We can use our statistical tool to calculate our test statistic.<\/p>\r\n<p><img class=\"aligncenter wp-image-428 size-medium\" style=\"font-size: 0.9em;\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/23211629\/Screen-Shot-2022-11-03-at-11.50.19-AM.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"259\" \/><\/p>\r\n<p style=\"text-align: left;\">[latex]z=11.8[\/latex]<\/p>\r\n<p>The difference in the proportion is about [latex]11.8[\/latex] standard errors above the null hypothesis value ([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[\/latex].<\/p>\r\n<p><strong>Step 5: Compare the P-value with the significance level.<\/strong><\/p>\r\n<p>Now we compare the P-value to the level of significance, [latex]\\alpha = 0.05[\/latex]. In this case, the P-value of [latex]0[\/latex] is less than [latex]0.05[\/latex], which means we do have enough evidence to reject the null hypothesis.<\/p>\r\n<p><strong>Step 6: Write the conclusion in the context of the problem.<\/strong><\/p>\r\n<p>The data from this study does provide convincing evidence that the vaccine group\u2019s infection risk is <em class=\"css-2fg4z9 e1gzwzxm0\">different<\/em> from the placebo group\u2019s.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Fun Fact: <\/strong>How did Pfizer get an efficacy rate of [latex]95\\%[\/latex]?<\/p>\r\n<p>Efficacy Rate = [latex]\\frac{p_p-p_v}{p_p} =\\frac{0.74\\%-0.04\\%}{0.74\\%}=\\frac{0.7\\%}{0.74\\%} \\approx 95\\%[\/latex]<\/p>\r\n<p>[\/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;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>Two-Sample Hypothesis Test for Proportions<\/h2>\n<p>As mentioned before, hypothesis testing is part of inference. We previously stated that the purpose of a hypothesis test is to use sample data to test a claim about a population parameter. However, studies often compare two groups. Comparing two proportions is quite common. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>two-sample [latex]z[\/latex]-test of proportions<\/h3>\n<ol>\n<li>Write out the null and alternative hypotheses.\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]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Check the conditions for the hypothesis test. For testing a two-sample [latex]z[\/latex]-test for proportions, we require:\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].<\/li>\n<\/ul>\n<\/li>\n<li>Calculate a test statistic.<\/li>\n<li>Calculate a P-value.<\/li>\n<li>Compare the P-value to the significance level, [latex]\\alpha[\/latex], to make a decision.<br \/>\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<\/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=\"1125\" frameborder=\"no\" data-mce-fragment=\"1\"><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\"><strong><strong><strong>Vaccine Efficiency<a class=\"footnote\" title=\"https:\/\/www.nytimes.com\/2020\/12\/13\/learning\/what-does-95-effective-mean-teaching-the-math-of-vaccine-efficacy.html\" id=\"return-footnote-1253-1\" href=\"#footnote-1253-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/strong><\/strong><\/strong>In 2020, after a trial showed that Pfizer\u2019s\u00a0coronavirus vaccine\u00a0had an\u00a0efficacy\u00a0rate of [latex]95\\%[\/latex], the Food and Drug Administration\u00a0approved the vaccine.<br \/>\n<span style=\"font-size: 1rem; text-align: initial; background-color: initial;\"><br \/>\nThe [latex]43,660[\/latex] subjects were split evenly between the placebo and vaccine groups (about [latex]21,830[\/latex] subjects per group). In the placebo group \u2014 the group that got a \u201cfake\u201d vaccine \u2014 [latex]162[\/latex] became infected with the coronavirus and showed symptoms. In the vaccine group \u2014 the group that got the real vaccine \u2014 that number was only eight. <\/span>At the [latex]5\\%[\/latex] significance level, is the vaccine group\u2019s infection risk\u00a0<em class=\"css-2fg4z9 e1gzwzxm0\">that different<\/em>\u00a0from the placebo group\u2019s?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q224900\">Show Solution<\/button><\/p>\n<div id=\"q224900\" class=\"hidden-answer\" style=\"display: none\">\n<strong>Step 1: Write out the hypotheses.<\/strong><\/p>\n<ul style=\"list-style-type: none;\">\n<li>\u00a0<\/li>\n<li>[latex]H_0: p_p = p_v[\/latex] or [latex]H_0: p_p - p_v=0[\/latex]<\/li>\n<li>[latex]H_A: p_p \\ne p_v[\/latex] or [latex]H_A: p_p - p_v\\ne0[\/latex]<\/li>\n<\/ul>\n<p>where [latex]p_p[\/latex] is the proportion of the placebo subjects that became infected and [latex]p_v[\/latex] is the proportion of the vaccine subjects that became infected.<\/p>\n<p><strong>Step 2: Check the conditions.<\/strong><\/p>\n<p>The sample is random, so there is no problem there. Again, we want to determine whether the normal model is a good fit.<\/p>\n<p>[latex]\\hat{p}_c = \\frac{162+8}{21830+21830} = \\frac{170}{43660}[\/latex]<\/p>\n<ul>\n<li>[latex]n_p\\hat{p}_c = 21830(\\frac{170}{43660})= 85\\ge 10[\/latex],<\/li>\n<li>[latex]n_v\\hat{p}_c = 21830(\\frac{170}{43660})= 85\\ge 10[\/latex],<\/li>\n<li>[latex]n_p(1-\\hat{p}_c) = 21830(1-\\frac{170}{43660})= 21745\\ge 10[\/latex], and<\/li>\n<li>[latex]n_v(1-\\hat{p}_c) =21830(1-\\frac{170}{43660})= 21745\\ge 10[\/latex].<\/li>\n<\/ul>\n<p>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>[latex]p_p = \\frac{162}{21830} \\approx 0.74\\%[\/latex]<br \/>\n[latex]p_v =\\frac{8}{21830} \\approx 0.04\\%[\/latex]<\/p>\n<p>We can use our statistical tool to calculate our test statistic.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-428 size-medium\" style=\"font-size: 0.9em;\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/23211629\/Screen-Shot-2022-11-03-at-11.50.19-AM.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"259\" \/><\/p>\n<p style=\"text-align: left;\">[latex]z=11.8[\/latex]<\/p>\n<p>The difference in the proportion is about [latex]11.8[\/latex] standard errors above the null hypothesis value ([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[\/latex].<\/p>\n<p><strong>Step 5: Compare the P-value with the significance level.<\/strong><\/p>\n<p>Now we compare the P-value to the level of significance, [latex]\\alpha = 0.05[\/latex]. In this case, the P-value of [latex]0[\/latex] is less than [latex]0.05[\/latex], which means we do have enough evidence to reject the null hypothesis.<\/p>\n<p><strong>Step 6: Write the conclusion in the context of the problem.<\/strong><\/p>\n<p>The data from this study does provide convincing evidence that the vaccine group\u2019s infection risk is <em class=\"css-2fg4z9 e1gzwzxm0\">different<\/em> from the placebo group\u2019s.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Fun Fact: <\/strong>How did Pfizer get an efficacy rate of [latex]95\\%[\/latex]?<\/p>\n<p>Efficacy Rate = [latex]\\frac{p_p-p_v}{p_p} =\\frac{0.74\\%-0.04\\%}{0.74\\%}=\\frac{0.7\\%}{0.74\\%} \\approx 95\\%[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-1253-1\">https:\/\/www.nytimes.com\/2020\/12\/13\/learning\/what-does-95-effective-mean-teaching-the-math-of-vaccine-efficacy.html <a href=\"#return-footnote-1253-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":37,"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":"apply_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\/1253"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1253\/revisions"}],"predecessor-version":[{"id":6775,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1253\/revisions\/6775"}],"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\/1253\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1253"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1253"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1253"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}