{"id":1311,"date":"2023-06-22T02:17:24","date_gmt":"2023-06-22T02:17:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-13-cheat-sheet\/"},"modified":"2025-02-11T03:57:08","modified_gmt":"2025-02-11T03:57:08","slug":"module-13-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/module-13-cheat-sheet\/","title":{"raw":"Module 12: Cheat Sheet","rendered":"Module 12: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+12_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>One-Sample Hypothesis Test for Means<\/strong><\/p>\r\n<ul>\r\n\t<li>\r\n<p>The<strong> null hypothesis ([latex]H_0[\/latex]) <\/strong>is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.<\/p>\r\n<ul>\r\n\t<li>\r\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]H_0: \\mu=\\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/span><\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The<strong> alternative hypothesis ([latex]H_A[\/latex])<\/strong>\u00a0is a claim about the population that is contradictory to [latex]H_0[\/latex] and what we conclude when we reject [latex]H_0[\/latex].<br \/>\r\n<ul>\r\n\t<li>[latex]H_A: \\mu \\lt \\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\r\n\t<li>[latex]H_A: \\mu&gt;\\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\r\n\t<li>[latex]H_A: \\mu\\ne \\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li><strong>Conditions for a One-Sample [latex]t[\/latex]-Test<\/strong><\/li>\r\n<\/ul>\r\n<ol>\r\n\t<li style=\"list-style-type: none;\">\r\n<ol>\r\n\t<li>The sample is a <strong>random sample<\/strong> from the population of interest or it is reasonable to regard the sample as if it is random. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in a sample that is representative of the population.<\/li>\r\n\t<li>For each population, the distribution of the variable that was measured is <strong>approximately normal, or the sample size for the sample from that population is large<\/strong>. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n\t<li>Test statistics: <strong style=\"font-size: 0.9em; word-spacing: normal;\">[latex]t[\/latex]-statistic:<\/strong><span style=\"background-color: initial; font-size: 0.9em; word-spacing: normal;\"> [latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<br \/>\r\n<\/span><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">The distribution of [latex]t[\/latex]-scores depends on the degrees of freedom, that is, [latex]df = n \u2013 1[\/latex].<\/span><\/li>\r\n<\/ul>\r\n<p><strong>Two-Sample Hypothesis Test for Means<\/strong><\/p>\r\n<ul>\r\n\t<li><strong>Independent samples<\/strong> are two samples where the individuals selected for the first sample do not influence the individuals selected for the second sample.<\/li>\r\n\t<li>A hypothesis test for comparing two population means is often referred to as a <strong>two-sample t-test<\/strong>.<\/li>\r\n\t<li>The <strong>null hypothesis ([latex]H_0[\/latex])<\/strong> is a statement about the population that is either believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">Null hypothesis: [latex]H_0: \\mu_1=\\mu_2[\/latex] or [latex]H_0: \\mu_1-\\mu_2=0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>The <strong>alternative hypothesis ([latex]H_A[\/latex])<\/strong> is a claim about the population that is contradictory to [latex]H_0[\/latex] and what we conclude when we reject [latex]H_0[\/latex].\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">Alternative hypothesis:\r\n\r\n<ul>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1\\lt \\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2\\lt 0[\/latex]<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1&gt;\\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2&gt;0[\/latex]<\/li>\r\n\t<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1\\ne \\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2\\ne0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Conditions for a t-test<\/li>\r\n<\/ul>\r\n<ol>\r\n\t<li style=\"list-style-type: none;\">\r\n<ol>\r\n\t<li style=\"font-weight: 400;\">The sample should be randomly selected or reasonably representative of the population.<\/li>\r\n\t<li style=\"font-weight: 400;\">For each population, the distribution of the variable that was measured is approximately normal, or the sample size for the sample from that population is large. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n\t<li>standard error of [latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/li>\r\n\t<li>The test statistic to compare two population means is calculated using the following formula: [latex]t = \\dfrac{\\text{estimate of parameter - null hypothesis value}}{\\text{standard error}} = \\dfrac{(\\bar{x}_1-\\bar{x}_2)-(\\mu_1-\\mu_2)}{\\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}}[\/latex]<\/li>\r\n<\/ul>\r\n<ul>\r\n\t<li><strong>Paired samples<\/strong> or <strong>Dependent samples<\/strong> are samples that are chosen in a way that results in the observations in one sample being paired with the observations in the other sample.<\/li>\r\n\t<li>The mean of the differences is equal to the difference in means: [latex]\\mu_d = \\mu_\\text{after}-\\mu_\\text{before}[\/latex]<\/li>\r\n\t<li>In summary, where [latex]k[\/latex] is the value of the null hypothesis, we have:<\/li>\r\n<\/ul>\r\n<div style=\"display: grid; grid-template-columns: repeat(1, minmax(0, 1fr)); overflow: auto; white-space: normal;\" tabindex=\"0\" align=\"left\">\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Alternative Hypothesis for Independent Samples<\/th>\r\n<th>Alternative Hypothesis for Dependent Samples<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]H_A: \\mu_1-\\mu_2&gt;k[\/latex]<\/td>\r\n<td>[latex]H_A: \\mu_d&gt;k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]H_A: \\mu_1-\\mu_2 \\lt k[\/latex]<\/td>\r\n<td>[latex]H_A: \\mu_d \\lt k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]H_A: \\mu_1-\\mu_2 \\ne k[\/latex]<\/td>\r\n<td>[latex]H_A: \\mu_d \\ne k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<ul>\r\n\t<li>The notations for the summary statistics used to compare paired populations or samples are shown in the following table. We will use [latex]d[\/latex] to represent the difference variable.<\/li>\r\n<\/ul>\r\n<div style=\"display: grid; grid-template-columns: repeat(1, minmax(0, 1fr)); overflow: auto; white-space: normal;\" tabindex=\"0\" align=\"left\">\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Summary Statistics<\/th>\r\n<th>Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Population mean of difference<\/td>\r\n<td>[latex]\\mu_d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sample mean of difference<\/td>\r\n<td>[latex]\\bar{d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Population standard deviation of difference<\/td>\r\n<td>[latex]\\sigma_d[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sample standard deviation of difference<\/td>\r\n<td>[latex]s_d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<ul>\r\n\t<li>The test statistic for the dependent (paired) t-test is calculated using the following formulas: [latex]\\text{standard error of the difference}=\\dfrac{s_d}{\\sqrt{n}}[\/latex]<br \/>\r\n[latex]\\text{test statistic }(t)=\\dfrac{\\text{estimator - null value}}{\\text{standard error of estimator}}=\\dfrac{\\bar{d}-\\text{null value}}{\\text{standard error of difference}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<p><strong>[latex]t[\/latex]-statistic<\/strong><\/p>\r\n<p>[latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\r\n<p><strong>standard error of difference of means<\/strong><\/p>\r\n<p>[latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/p>\r\n<p><strong>standard deviation of difference of means<\/strong><\/p>\r\n<p>[latex]\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}[\/latex]<\/p>\r\n<h2>Glossary<\/h2>\r\n<p><b>independent sample<\/b><\/p>\r\n<p style=\"padding-left: 40px;\">random samples from each population<\/p>\r\n<p><b>paired samples, dependent samples<\/b><\/p>\r\n<p style=\"padding-left: 40px;\">samples that are chosen in a way that results in the observations in one sample being paired with the observations in the other sample<\/p>","rendered":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Statistics+Exemplar\/Cheat+Sheets\/Module+12_+Cheat+Sheet.pdf\" target=\"_blank\" rel=\"noopener\">Download a PDF of this page here.<\/a><\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>One-Sample Hypothesis Test for Means<\/strong><\/p>\n<ul>\n<li>\n<p>The<strong> null hypothesis ([latex]H_0[\/latex]) <\/strong>is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.<\/p>\n<ul>\n<li>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]H_0: \\mu=\\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li>The<strong> alternative hypothesis ([latex]H_A[\/latex])<\/strong>\u00a0is a claim about the population that is contradictory to [latex]H_0[\/latex] and what we conclude when we reject [latex]H_0[\/latex].\n<ul>\n<li>[latex]H_A: \\mu \\lt \\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\n<li>[latex]H_A: \\mu>\\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\n<li>[latex]H_A: \\mu\\ne \\mu_0[\/latex], [latex]\\mu_0[\/latex] is the null value.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li><strong>Conditions for a One-Sample [latex]t[\/latex]-Test<\/strong><\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>The sample is a <strong>random sample<\/strong> from the population of interest or it is reasonable to regard the sample as if it is random. It is reasonable to regard the sample as a random sample if it was selected in a way that should result in a sample that is representative of the population.<\/li>\n<li>For each population, the distribution of the variable that was measured is <strong>approximately normal, or the sample size for the sample from that population is large<\/strong>. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ul>\n<li>Test statistics: <strong style=\"font-size: 0.9em; word-spacing: normal;\">[latex]t[\/latex]-statistic:<\/strong><span style=\"background-color: initial; font-size: 0.9em; word-spacing: normal;\"> [latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<br \/>\n<\/span><span style=\"font-size: 1rem; orphans: 1; text-align: initial; background-color: initial; word-spacing: normal;\">The distribution of [latex]t[\/latex]-scores depends on the degrees of freedom, that is, [latex]df = n \u2013 1[\/latex].<\/span><\/li>\n<\/ul>\n<p><strong>Two-Sample Hypothesis Test for Means<\/strong><\/p>\n<ul>\n<li><strong>Independent samples<\/strong> are two samples where the individuals selected for the first sample do not influence the individuals selected for the second sample.<\/li>\n<li>A hypothesis test for comparing two population means is often referred to as a <strong>two-sample t-test<\/strong>.<\/li>\n<li>The <strong>null hypothesis ([latex]H_0[\/latex])<\/strong> is a statement about the population that is either believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.\n<ul>\n<li style=\"font-weight: 400;\">Null hypothesis: [latex]H_0: \\mu_1=\\mu_2[\/latex] or [latex]H_0: \\mu_1-\\mu_2=0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The <strong>alternative hypothesis ([latex]H_A[\/latex])<\/strong> is a claim about the population that is contradictory to [latex]H_0[\/latex] and what we conclude when we reject [latex]H_0[\/latex].\n<ul>\n<li style=\"font-weight: 400;\">Alternative hypothesis:\n<ul>\n<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1\\lt \\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2\\lt 0[\/latex]<\/li>\n<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1>\\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2>0[\/latex]<\/li>\n<li style=\"font-weight: 400;\">[latex]H_A: \\mu_1\\ne \\mu_2[\/latex] or [latex]H_A: \\mu_1-\\mu_2\\ne0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Conditions for a t-test<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"font-weight: 400;\">The sample should be randomly selected or reasonably representative of the population.<\/li>\n<li style=\"font-weight: 400;\">For each population, the distribution of the variable that was measured is approximately normal, or the sample size for the sample from that population is large. Usually, a sample of size [latex]30[\/latex] or more is considered to be \u201clarge.\u201d If a sample size is less than [latex]30[\/latex], you should look at a plot of the data from that sample (a dotplot, a boxplot, or, if the sample size isn\u2019t really small, a histogram) to make sure that the distribution looks approximately symmetric and that there are no outliers.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ul>\n<li>standard error of [latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/li>\n<li>The test statistic to compare two population means is calculated using the following formula: [latex]t = \\dfrac{\\text{estimate of parameter - null hypothesis value}}{\\text{standard error}} = \\dfrac{(\\bar{x}_1-\\bar{x}_2)-(\\mu_1-\\mu_2)}{\\sqrt{\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}}}[\/latex]<\/li>\n<\/ul>\n<ul>\n<li><strong>Paired samples<\/strong> or <strong>Dependent samples<\/strong> are samples that are chosen in a way that results in the observations in one sample being paired with the observations in the other sample.<\/li>\n<li>The mean of the differences is equal to the difference in means: [latex]\\mu_d = \\mu_\\text{after}-\\mu_\\text{before}[\/latex]<\/li>\n<li>In summary, where [latex]k[\/latex] is the value of the null hypothesis, we have:<\/li>\n<\/ul>\n<div style=\"display: grid; grid-template-columns: repeat(1, minmax(0, 1fr)); overflow: auto; white-space: normal; text-align: left;\" tabindex=\"0\">\n<table>\n<thead>\n<tr>\n<th>Alternative Hypothesis for Independent Samples<\/th>\n<th>Alternative Hypothesis for Dependent Samples<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]H_A: \\mu_1-\\mu_2>k[\/latex]<\/td>\n<td>[latex]H_A: \\mu_d>k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]H_A: \\mu_1-\\mu_2 \\lt k[\/latex]<\/td>\n<td>[latex]H_A: \\mu_d \\lt k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]H_A: \\mu_1-\\mu_2 \\ne k[\/latex]<\/td>\n<td>[latex]H_A: \\mu_d \\ne k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ul>\n<li>The notations for the summary statistics used to compare paired populations or samples are shown in the following table. We will use [latex]d[\/latex] to represent the difference variable.<\/li>\n<\/ul>\n<div style=\"display: grid; grid-template-columns: repeat(1, minmax(0, 1fr)); overflow: auto; white-space: normal; text-align: left;\" tabindex=\"0\">\n<table>\n<thead>\n<tr>\n<th>Summary Statistics<\/th>\n<th>Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Population mean of difference<\/td>\n<td>[latex]\\mu_d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sample mean of difference<\/td>\n<td>[latex]\\bar{d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Population standard deviation of difference<\/td>\n<td>[latex]\\sigma_d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sample standard deviation of difference<\/td>\n<td>[latex]s_d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<ul>\n<li>The test statistic for the dependent (paired) t-test is calculated using the following formulas: [latex]\\text{standard error of the difference}=\\dfrac{s_d}{\\sqrt{n}}[\/latex]<br \/>\n[latex]\\text{test statistic }(t)=\\dfrac{\\text{estimator - null value}}{\\text{standard error of estimator}}=\\dfrac{\\bar{d}-\\text{null value}}{\\text{standard error of difference}}[\/latex]<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<p><strong>[latex]t[\/latex]-statistic<\/strong><\/p>\n<p>[latex]t=\\dfrac{\\stackrel{\u00af}{x}-\u03bc}{\\frac{s}{\\sqrt{n}}}[\/latex]<\/p>\n<p><strong>standard error of difference of means<\/strong><\/p>\n<p>[latex]\\bar{x}_1-\\bar{x}_2[\/latex]: [latex]\\sqrt{\\dfrac{s_1^2}{n_1}+\\dfrac{s_2^2}{n_2}}[\/latex]<\/p>\n<p><strong>standard deviation of difference of means<\/strong><\/p>\n<p>[latex]\\sqrt{\\frac{\\sigma_1^2}{n_1}+\\frac{\\sigma_2^2}{n_2}}[\/latex]<\/p>\n<h2>Glossary<\/h2>\n<p><b>independent sample<\/b><\/p>\n<p style=\"padding-left: 40px;\">random samples from each population<\/p>\n<p><b>paired samples, dependent samples<\/b><\/p>\n<p style=\"padding-left: 40px;\">samples that are chosen in a way that results in the observations in one sample being paired with the observations in the other sample<\/p>\n","protected":false},"author":8,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1309,"module-header":"cheat_sheet","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\/1311"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1311\/revisions"}],"predecessor-version":[{"id":6260,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1311\/revisions\/6260"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1309"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1311\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1311"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1311"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1311"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}