{"id":668,"date":"2021-08-13T15:54:20","date_gmt":"2021-08-13T15:54:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=668"},"modified":"2023-12-05T08:56:28","modified_gmt":"2023-12-05T08:56:28","slug":"measures-of-the-center-of-the-data-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/measures-of-the-center-of-the-data-2\/","title":{"raw":"The Law of Large Numbers and the Mean","rendered":"The Law of Large Numbers and the Mean"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Calculate a mean from a grouped frequency table<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Law of Large Numbers and the Mean<\/h2>\r\nThe Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean [latex]\\displaystyle\\overline{{x}}[\/latex] of the sample is very likely to get closer and closer to [latex]\u00b5[\/latex]. This is discussed in more detail later in the text.\r\n<h3>Sampling Distributions and Statistic of a Sampling Distribution<\/h3>\r\nYou can think of a sampling distribution as a relative frequency distribution with a great many samples. Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.\r\n<table style=\"height: 337px; width: 226px;\">\r\n<tbody>\r\n<tr>\r\n<td># of movies<\/td>\r\n<td>Relative Frequency<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\frac{{5}}{{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\frac{{15}}{{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\frac{{6}}{{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\frac{{3}}{{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]\\displaystyle\\frac{{1}}{{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.<\/strong>\r\n\r\nA <strong>statistic<\/strong> is a number calculated from a sample. Statistic examples include the <strong>mean<\/strong>, the <strong>median<\/strong>, and the <strong>mode<\/strong> as well as others. The sample mean [latex]\\displaystyle\\overline{{x}}[\/latex] is an example of a statistic which estimates the population mean [latex]\u03bc[\/latex].\r\n<h3>Calculating the Mean of Grouped Frequency Tables<\/h3>\r\nWhen only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:\r\n[latex]\\displaystyle\\text{mean}=\\frac{{\\text{data sum}}}{{\\text{number of data values}}}[\/latex]. We simply need to modify the definition to fit within the restrictions of a frequency table.\r\n\r\nSince we do not know the individual data values we can instead find the <strong>midpoint<\/strong> of each interval. The midpoint is [latex]\\displaystyle\\frac{{\\text{lower boundary } + \\text{ upper boundary}}}{{2}}[\/latex] We can now modify the mean definition to be [latex]\\displaystyle\\text{Mean of Frequency Table} = \\frac{\\sum\\nolimits{fm}}{\\sum\\nolimits{f}}[\/latex] where [latex]f[\/latex] = the frequency of the interval and [latex]m[\/latex] = the midpoint of the interval.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA frequency table displaying professor Blount's last statistic test is shown. Find the best estimate of the class mean.\r\n<table style=\"height: 261px; width: 458px;\">\r\n<tbody>\r\n<tr>\r\n<td>Grade Interval<\/td>\r\n<td>Number of Students<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]50\u201356.5[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"124082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124082\"]\r\n\r\nFind the midpoints for all intervals\r\n<table style=\"height: 261px; width: 459px;\">\r\n<tbody>\r\n<tr>\r\n<td>Grade Interval<\/td>\r\n<td>\u00a0Midpoint<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]50\u201356.5[\/latex]<\/td>\r\n<td>[latex]53.25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\r\n<td>[latex]65.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\r\n<td>[latex]71.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\r\n<td>[latex]77.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\r\n<td>[latex]83.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\r\n<td>[latex]95.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li>Calculate the sum of the product of each interval frequency and midpoint. [latex]\\sum\\nolimits{fm}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<center>[latex]53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.2[\/latex]<\/center>\r\n<ul>\r\n \t<li>[latex]\\displaystyle\\mu=\\frac{{\\sum{f}{m}}}{{\\sum{f}}} =\\frac{{1460.25}}{{19}}={76.86}[\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMaris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:\r\n<table style=\"height: 174px; width: 456px;\">\r\n<tbody>\r\n<tr>\r\n<td>Hours Teenagers Spend on Video Games<\/td>\r\n<td>Number of Teenagers<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0\u20133.5[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3.5\u20137.5[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7.5\u201311.5[\/latex]<\/td>\r\n<td>\u00a0[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11.5\u201315.5[\/latex]<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]15.5\u201319.5[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat is the best estimate for the mean number of hours spent playing video games?\r\n\r\n[reveal-answer q=\"124083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"124083\"]\r\n\r\nFind the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.\r\n\r\nThe midpoints are [latex]1.75[\/latex], [latex]5.5[\/latex], [latex]9.5[\/latex], [latex]13.5[\/latex], [latex]17.5[\/latex]. [latex]Mean = (1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9) = 409.75[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Calculate a mean from a grouped frequency table<\/li>\n<\/ul>\n<\/div>\n<h2>The Law of Large Numbers and the Mean<\/h2>\n<p>The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean [latex]\\displaystyle\\overline{{x}}[\/latex] of the sample is very likely to get closer and closer to [latex]\u00b5[\/latex]. This is discussed in more detail later in the text.<\/p>\n<h3>Sampling Distributions and Statistic of a Sampling Distribution<\/h3>\n<p>You can think of a sampling distribution as a relative frequency distribution with a great many samples. Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.<\/p>\n<table style=\"height: 337px; width: 226px;\">\n<tbody>\n<tr>\n<td># of movies<\/td>\n<td>Relative Frequency<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\displaystyle\\frac{{5}}{{30}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\displaystyle\\frac{{15}}{{30}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\displaystyle\\frac{{6}}{{30}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]\\displaystyle\\frac{{3}}{{30}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]\\displaystyle\\frac{{1}}{{30}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.<\/strong><\/p>\n<p>A <strong>statistic<\/strong> is a number calculated from a sample. Statistic examples include the <strong>mean<\/strong>, the <strong>median<\/strong>, and the <strong>mode<\/strong> as well as others. The sample mean [latex]\\displaystyle\\overline{{x}}[\/latex] is an example of a statistic which estimates the population mean [latex]\u03bc[\/latex].<\/p>\n<h3>Calculating the Mean of Grouped Frequency Tables<\/h3>\n<p>When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean:<br \/>\n[latex]\\displaystyle\\text{mean}=\\frac{{\\text{data sum}}}{{\\text{number of data values}}}[\/latex]. We simply need to modify the definition to fit within the restrictions of a frequency table.<\/p>\n<p>Since we do not know the individual data values we can instead find the <strong>midpoint<\/strong> of each interval. The midpoint is [latex]\\displaystyle\\frac{{\\text{lower boundary } + \\text{ upper boundary}}}{{2}}[\/latex] We can now modify the mean definition to be [latex]\\displaystyle\\text{Mean of Frequency Table} = \\frac{\\sum\\nolimits{fm}}{\\sum\\nolimits{f}}[\/latex] where [latex]f[\/latex] = the frequency of the interval and [latex]m[\/latex] = the midpoint of the interval.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A frequency table displaying professor Blount&#8217;s last statistic test is shown. Find the best estimate of the class mean.<\/p>\n<table style=\"height: 261px; width: 458px;\">\n<tbody>\n<tr>\n<td>Grade Interval<\/td>\n<td>Number of Students<\/td>\n<\/tr>\n<tr>\n<td>[latex]50\u201356.5[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124082\">Show Solution<\/span><\/p>\n<div id=\"q124082\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the midpoints for all intervals<\/p>\n<table style=\"height: 261px; width: 459px;\">\n<tbody>\n<tr>\n<td>Grade Interval<\/td>\n<td>\u00a0Midpoint<\/td>\n<\/tr>\n<tr>\n<td>[latex]50\u201356.5[\/latex]<\/td>\n<td>[latex]53.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.5\u201362.5[\/latex]<\/td>\n<td>[latex]59.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]62.5\u201368.5[\/latex]<\/td>\n<td>[latex]65.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]68.5\u201374.5[\/latex]<\/td>\n<td>[latex]71.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]74.5\u201380.5[\/latex]<\/td>\n<td>[latex]77.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]80.5\u201386.5[\/latex]<\/td>\n<td>[latex]83.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]86.5\u201392.5[\/latex]<\/td>\n<td>[latex]89.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]92.5\u201398.5[\/latex]<\/td>\n<td>[latex]95.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Calculate the sum of the product of each interval frequency and midpoint. [latex]\\sum\\nolimits{fm}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div style=\"text-align: center;\">[latex]53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.2[\/latex]<\/div>\n<ul>\n<li>[latex]\\displaystyle\\mu=\\frac{{\\sum{f}{m}}}{{\\sum{f}}} =\\frac{{1460.25}}{{19}}={76.86}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:<\/p>\n<table style=\"height: 174px; width: 456px;\">\n<tbody>\n<tr>\n<td>Hours Teenagers Spend on Video Games<\/td>\n<td>Number of Teenagers<\/td>\n<\/tr>\n<tr>\n<td>[latex]0\u20133.5[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3.5\u20137.5[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]7.5\u201311.5[\/latex]<\/td>\n<td>\u00a0[latex]12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]11.5\u201315.5[\/latex]<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]15.5\u201319.5[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What is the best estimate for the mean number of hours spent playing video games?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q124083\">Show Solution<\/span><\/p>\n<div id=\"q124083\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.<\/p>\n<p>The midpoints are [latex]1.75[\/latex], [latex]5.5[\/latex], [latex]9.5[\/latex], [latex]13.5[\/latex], [latex]17.5[\/latex]. [latex]Mean = (1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9) = 409.75[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-668\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax, Statistics, Measures of the Center of the Data. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Measures of the Center of the Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-5-measures-of-the-center-of-the-data\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-668","chapter","type-chapter","status-publish","hentry"],"part":31,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/668","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/668\/revisions"}],"predecessor-version":[{"id":3484,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/668\/revisions\/3484"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/31"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/668\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=668"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=668"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=668"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=668"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}