{"id":221,"date":"2017-04-15T03:19:44","date_gmt":"2017-04-15T03:19:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/conceptstest1\/chapter\/assessing-the-fit-of-a-line-4-of-4\/"},"modified":"2017-05-28T05:22:19","modified_gmt":"2017-05-28T05:22:19","slug":"assessing-the-fit-of-a-line-4-of-4","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/chapter\/assessing-the-fit-of-a-line-4-of-4\/","title":{"raw":"Assessing the Fit of a Line (4 of 4)","rendered":"Assessing the Fit of a Line (4 of 4)"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use residuals, standard error, and <em>r<\/em><sup>2<\/sup> to assess the fit of a linear model.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Introduction<\/h3>\r\nOur final investigation into assessing the fit of the regression line focuses on typical error in the predictions.\r\n\r\nPreviously, we calculated the error in a single prediction by calculating\r\n<p style=\"text-align: center\">Residual = Observed value \u2212 Predicted value<\/p>\r\nBut we use the regression line to make predictions even when we do not have an observed value, so we need a method for using all of the residuals to compute a typical amount of error.\r\n\r\nWe ask the question, <em>How do we measure the typical amount of error for predictions from the regression line?<\/em>\r\n\r\nThe most common measure of the size of the typical error is the <strong>standard error of the regression<\/strong>, which is represented by <em>s<sub>e<\/sub><\/em>. It is calculated using the following formula:\r\n<p style=\"text-align: center\">[latex]{s}_{e}=\\sqrt{\\frac{\\text{SSE}}{n-2}}[\/latex]<\/p>\r\nwhere <em>SSE<\/em> stands for the sum of the squared errors.\r\n\r\nFinding the standard error of the regression is similar to finding the standard deviation of a distribution of data points from a single quantitative variable. In <em>Summarizing Data Graphically and Numerically<\/em>, we learned that the <em>standard deviation is roughly a measure of average distance about the mean<\/em>. Here the <em>standard error is roughly a measure of the average distance of the points about the regression line<\/em>.\r\n\r\nLet\u2019s return to our example where age is used to predict the maximum distance for reading highway signs.\r\n\r\nThe residual plot for the highway sign data set is shown below. We can visualize the SSE in the formula as simply the sum of the squares of all of the vertical (residual) line segments. After dividing by <em>n<\/em> \u2212 2, we have the average <em>squared<\/em> residual. Taking the square root then gives us a measure of the average size of the residuals.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031940\/m3_examining_relationships_topic_3_3_asses_fit_line_assessingfitline4.gif\" alt=\"Residual plot for highway sign data set\" width=\"364\" height=\"209\" \/>&nbsp;\r\n\r\nIn the case of the highway sign data, the value of <em>s<\/em><sub>e<\/sub> is 51.35. In the figure below, we added horizontal lines at <em>y<\/em> = 51.35 and <em>y<\/em> = \u221251.35, so the red line represents the typical size of the error.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031943\/m3_examining_relationships_topic_3_3_asses_fit_line_assessingfitline5.gif\" alt=\"Residual plot with red lines representing typical error size\" width=\"364\" height=\"209\" \/>&nbsp;\r\n\r\n<strong>Comment:<\/strong> When we mark the <em>s<sub>e<\/sub><\/em> on this residual plot, errors that fall outside of this range are larger than average. We see again that most of the errors that exceed \u00b151.35 are on the right. This illustrates that predictions of maximum reading distance for older drivers have larger error.\r\n\r\n<strong>Note:<\/strong> Most statistics software computes <em>r<\/em> and <em>r<\/em><sup>2<\/sup> and <em>s<sub>e<\/sub><\/em>. Therefore, our focus is not on calculating but on understanding and interpreting.\r\n\r\nNow let\u2019s apply the standard error of the regression as a measurement of typical error.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\n<h2>Highway Sign Visibility<\/h2>\r\nLet\u2019s take another look at the prediction we made earlier using the regression line equation:\r\n<p style=\"text-align: center\">Distance = 576 + (\u22123 * Age)<\/p>\r\nIn a previous example, we predicted the maximum distance that a 60-year-old driver can read a highway sign. We plugged Age = 60 into the equation and found that\r\n<p style=\"text-align: center\">Predicted distance = 576 + (\u22123 * 60) = 396<\/p>\r\nThe question we now ask is, How good is this prediction?\r\n\r\nUnfortunately, there is no 60-year-old driver in the original data set of 30 drivers, so we cannot calculate the residual. Instead, we use the <em>s<sub>e<\/sub><\/em> as a measurement of typical error.\r\n\r\nTechnology gives <em>s<sub>e<\/sub><\/em> = 51.35.\r\n\r\nSo how good is the prediction for the 60-year-old driver? Based on the <em>s<sub>e<\/sub><\/em> for this data, we estimate that our prediction of 396 feet is off by \u00b151 feet.\r\n\r\n<\/div>\r\n<table>\r\n<tbody>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\"><\/td>\r\n<td align=\"center\">Intro grade(%)<\/td>\r\n<td align=\"center\">Upper grade(%)<\/td>\r\n<td align=\"center\">Predictions<\/td>\r\n<td align=\"center\">Error (Residual)<\/td>\r\n<td align=\"center\">Error Squared<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 1<\/td>\r\n<td align=\"center\">65<\/td>\r\n<td align=\"center\">58<\/td>\r\n<td align=\"center\">59.1<\/td>\r\n<td align=\"center\">\u22121.1<\/td>\r\n<td align=\"center\">1.21<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 2<\/td>\r\n<td align=\"center\">71<\/td>\r\n<td align=\"center\">63<\/td>\r\n<td align=\"center\">65.4<\/td>\r\n<td align=\"center\">\u22122.4<\/td>\r\n<td align=\"center\">5.76<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 3<\/td>\r\n<td align=\"center\">72<\/td>\r\n<td align=\"center\">67<\/td>\r\n<td align=\"center\">66.4<\/td>\r\n<td align=\"center\">0.6<\/td>\r\n<td align=\"center\">0.36<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 4<\/td>\r\n<td align=\"center\">72<\/td>\r\n<td align=\"center\">77<\/td>\r\n<td align=\"center\">66.4<\/td>\r\n<td align=\"center\">10.6<\/td>\r\n<td align=\"center\">112.36<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 5<\/td>\r\n<td align=\"center\">75<\/td>\r\n<td align=\"center\">63<\/td>\r\n<td align=\"center\">69.6<\/td>\r\n<td align=\"center\">\u22126.6<\/td>\r\n<td align=\"center\">43.56<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 6<\/td>\r\n<td align=\"center\">83<\/td>\r\n<td align=\"center\">72<\/td>\r\n<td align=\"center\">77.9<\/td>\r\n<td align=\"center\">\u22125.9<\/td>\r\n<td align=\"center\">34.81<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 7<\/td>\r\n<td align=\"center\">85<\/td>\r\n<td align=\"center\">84<\/td>\r\n<td align=\"center\">80<\/td>\r\n<td align=\"center\">4<\/td>\r\n<td align=\"center\">16<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 8<\/td>\r\n<td align=\"center\">88<\/td>\r\n<td align=\"center\">83<\/td>\r\n<td align=\"center\">83.2<\/td>\r\n<td align=\"center\">\u22120.2<\/td>\r\n<td align=\"center\">0.04<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 9<\/td>\r\n<td align=\"center\">94<\/td>\r\n<td align=\"center\">89<\/td>\r\n<td align=\"center\">89.5<\/td>\r\n<td align=\"center\">\u22120.5<\/td>\r\n<td align=\"center\">0.25<\/td>\r\n<\/tr>\r\n<tr class=\"oli_table\">\r\n<td align=\"center\">Student 10<\/td>\r\n<td align=\"center\">96<\/td>\r\n<td align=\"center\">93<\/td>\r\n<td align=\"center\">91.5<\/td>\r\n<td align=\"center\">1.5<\/td>\r\n<td align=\"center\">2.25<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3515\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3516\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3517\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Learn By Doing<\/h3>\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3869\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<h3><strong>Let\u2019s Summarize<\/strong><\/h3>\r\n<ul>\r\n \t<li>When we use a regression line to make predictions, there is error in the prediction. We calculate this error as <strong>Observed data value \u2212 Predicted value<\/strong>. A residual is another name for the prediction error.<\/li>\r\n \t<li>We use residual plots to determine whether a linear model is a good summary of the relationship between the explanatory and response variables. In particular, we look for any <em>unexpected patterns<\/em> in the residuals that may suggest the data is not linear in form.<\/li>\r\n \t<li>We have two numeric measures to help us judge how well the regression line models the data.\r\n<ul>\r\n \t<li>The square of the correlation coefficient, <em>r<\/em><sup>2<\/sup>, is the proportion of the variation in the response variable that is explained by the least-squares regression line.<\/li>\r\n \t<li>The standard error of the regression, <em>s<sub>e<\/sub><\/em>, gives a typical prediction error based on all of the data. It roughly measures the average distance of the data from the regression line. In this way, it is similar to the standard deviation, which roughly measures average distance from the mean.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h3><\/h3>","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use residuals, standard error, and <em>r<\/em><sup>2<\/sup> to assess the fit of a linear model.<\/li>\n<\/ul>\n<\/div>\n<h3>Introduction<\/h3>\n<p>Our final investigation into assessing the fit of the regression line focuses on typical error in the predictions.<\/p>\n<p>Previously, we calculated the error in a single prediction by calculating<\/p>\n<p style=\"text-align: center\">Residual = Observed value \u2212 Predicted value<\/p>\n<p>But we use the regression line to make predictions even when we do not have an observed value, so we need a method for using all of the residuals to compute a typical amount of error.<\/p>\n<p>We ask the question, <em>How do we measure the typical amount of error for predictions from the regression line?<\/em><\/p>\n<p>The most common measure of the size of the typical error is the <strong>standard error of the regression<\/strong>, which is represented by <em>s<sub>e<\/sub><\/em>. It is calculated using the following formula:<\/p>\n<p style=\"text-align: center\">[latex]{s}_{e}=\\sqrt{\\frac{\\text{SSE}}{n-2}}[\/latex]<\/p>\n<p>where <em>SSE<\/em> stands for the sum of the squared errors.<\/p>\n<p>Finding the standard error of the regression is similar to finding the standard deviation of a distribution of data points from a single quantitative variable. In <em>Summarizing Data Graphically and Numerically<\/em>, we learned that the <em>standard deviation is roughly a measure of average distance about the mean<\/em>. Here the <em>standard error is roughly a measure of the average distance of the points about the regression line<\/em>.<\/p>\n<p>Let\u2019s return to our example where age is used to predict the maximum distance for reading highway signs.<\/p>\n<p>The residual plot for the highway sign data set is shown below. We can visualize the SSE in the formula as simply the sum of the squares of all of the vertical (residual) line segments. After dividing by <em>n<\/em> \u2212 2, we have the average <em>squared<\/em> residual. Taking the square root then gives us a measure of the average size of the residuals.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031940\/m3_examining_relationships_topic_3_3_asses_fit_line_assessingfitline4.gif\" alt=\"Residual plot for highway sign data set\" width=\"364\" height=\"209\" \/>&nbsp;<\/p>\n<p>In the case of the highway sign data, the value of <em>s<\/em><sub>e<\/sub> is 51.35. In the figure below, we added horizontal lines at <em>y<\/em> = 51.35 and <em>y<\/em> = \u221251.35, so the red line represents the typical size of the error.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1729\/2017\/04\/15031943\/m3_examining_relationships_topic_3_3_asses_fit_line_assessingfitline5.gif\" alt=\"Residual plot with red lines representing typical error size\" width=\"364\" height=\"209\" \/>&nbsp;<\/p>\n<p><strong>Comment:<\/strong> When we mark the <em>s<sub>e<\/sub><\/em> on this residual plot, errors that fall outside of this range are larger than average. We see again that most of the errors that exceed \u00b151.35 are on the right. This illustrates that predictions of maximum reading distance for older drivers have larger error.<\/p>\n<p><strong>Note:<\/strong> Most statistics software computes <em>r<\/em> and <em>r<\/em><sup>2<\/sup> and <em>s<sub>e<\/sub><\/em>. Therefore, our focus is not on calculating but on understanding and interpreting.<\/p>\n<p>Now let\u2019s apply the standard error of the regression as a measurement of typical error.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<h2>Highway Sign Visibility<\/h2>\n<p>Let\u2019s take another look at the prediction we made earlier using the regression line equation:<\/p>\n<p style=\"text-align: center\">Distance = 576 + (\u22123 * Age)<\/p>\n<p>In a previous example, we predicted the maximum distance that a 60-year-old driver can read a highway sign. We plugged Age = 60 into the equation and found that<\/p>\n<p style=\"text-align: center\">Predicted distance = 576 + (\u22123 * 60) = 396<\/p>\n<p>The question we now ask is, How good is this prediction?<\/p>\n<p>Unfortunately, there is no 60-year-old driver in the original data set of 30 drivers, so we cannot calculate the residual. Instead, we use the <em>s<sub>e<\/sub><\/em> as a measurement of typical error.<\/p>\n<p>Technology gives <em>s<sub>e<\/sub><\/em> = 51.35.<\/p>\n<p>So how good is the prediction for the 60-year-old driver? Based on the <em>s<sub>e<\/sub><\/em> for this data, we estimate that our prediction of 396 feet is off by \u00b151 feet.<\/p>\n<\/div>\n<table>\n<tbody>\n<tr class=\"oli_table\">\n<td align=\"center\"><\/td>\n<td align=\"center\">Intro grade(%)<\/td>\n<td align=\"center\">Upper grade(%)<\/td>\n<td align=\"center\">Predictions<\/td>\n<td align=\"center\">Error (Residual)<\/td>\n<td align=\"center\">Error Squared<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 1<\/td>\n<td align=\"center\">65<\/td>\n<td align=\"center\">58<\/td>\n<td align=\"center\">59.1<\/td>\n<td align=\"center\">\u22121.1<\/td>\n<td align=\"center\">1.21<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 2<\/td>\n<td align=\"center\">71<\/td>\n<td align=\"center\">63<\/td>\n<td align=\"center\">65.4<\/td>\n<td align=\"center\">\u22122.4<\/td>\n<td align=\"center\">5.76<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 3<\/td>\n<td align=\"center\">72<\/td>\n<td align=\"center\">67<\/td>\n<td align=\"center\">66.4<\/td>\n<td align=\"center\">0.6<\/td>\n<td align=\"center\">0.36<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 4<\/td>\n<td align=\"center\">72<\/td>\n<td align=\"center\">77<\/td>\n<td align=\"center\">66.4<\/td>\n<td align=\"center\">10.6<\/td>\n<td align=\"center\">112.36<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 5<\/td>\n<td align=\"center\">75<\/td>\n<td align=\"center\">63<\/td>\n<td align=\"center\">69.6<\/td>\n<td align=\"center\">\u22126.6<\/td>\n<td align=\"center\">43.56<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 6<\/td>\n<td align=\"center\">83<\/td>\n<td align=\"center\">72<\/td>\n<td align=\"center\">77.9<\/td>\n<td align=\"center\">\u22125.9<\/td>\n<td align=\"center\">34.81<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 7<\/td>\n<td align=\"center\">85<\/td>\n<td align=\"center\">84<\/td>\n<td align=\"center\">80<\/td>\n<td align=\"center\">4<\/td>\n<td align=\"center\">16<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 8<\/td>\n<td align=\"center\">88<\/td>\n<td align=\"center\">83<\/td>\n<td align=\"center\">83.2<\/td>\n<td align=\"center\">\u22120.2<\/td>\n<td align=\"center\">0.04<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 9<\/td>\n<td align=\"center\">94<\/td>\n<td align=\"center\">89<\/td>\n<td align=\"center\">89.5<\/td>\n<td align=\"center\">\u22120.5<\/td>\n<td align=\"center\">0.25<\/td>\n<\/tr>\n<tr class=\"oli_table\">\n<td align=\"center\">Student 10<\/td>\n<td align=\"center\">96<\/td>\n<td align=\"center\">93<\/td>\n<td align=\"center\">91.5<\/td>\n<td align=\"center\">1.5<\/td>\n<td align=\"center\">2.25<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3515\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3515&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3515\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3516\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3516&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3516\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<p>\t<iframe id=\"lumen_assessment_3517\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3517&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3517\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Learn By Doing<\/h3>\n<p>\t<iframe id=\"lumen_assessment_3869\" class=\"resizable\" src=\"https:\/\/assessments.lumenlearning.com\/assessments\/load?assessment_id=3869&#38;embed=1&#38;external_user_id=&#38;external_context_id=&#38;iframe_resize_id=lumen_assessment_3869\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><br \/>\n\t<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<h3><strong>Let\u2019s Summarize<\/strong><\/h3>\n<ul>\n<li>When we use a regression line to make predictions, there is error in the prediction. We calculate this error as <strong>Observed data value \u2212 Predicted value<\/strong>. A residual is another name for the prediction error.<\/li>\n<li>We use residual plots to determine whether a linear model is a good summary of the relationship between the explanatory and response variables. In particular, we look for any <em>unexpected patterns<\/em> in the residuals that may suggest the data is not linear in form.<\/li>\n<li>We have two numeric measures to help us judge how well the regression line models the data.\n<ul>\n<li>The square of the correlation coefficient, <em>r<\/em><sup>2<\/sup>, is the proportion of the variation in the response variable that is explained by the least-squares regression line.<\/li>\n<li>The standard error of the regression, <em>s<sub>e<\/sub><\/em>, gives a typical prediction error based on all of the data. It roughly measures the average distance of the data from the regression line. In this way, it is similar to the standard deviation, which roughly measures average distance from the mean.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3><\/h3>\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-221\">\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>Concepts in Statistics. <strong>Provided by<\/strong>: Open Learning Initiative. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/oli.cmu.edu\">http:\/\/oli.cmu.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":163,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Concepts in Statistics\",\"author\":\"\",\"organization\":\"Open Learning Initiative\",\"url\":\"http:\/\/oli.cmu.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"1bbe606f-ad84-4a6e-ad4e-3bef550b5ba7","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-221","chapter","type-chapter","status-publish","hentry"],"part":140,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/221","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/users\/163"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/221\/revisions"}],"predecessor-version":[{"id":1366,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/221\/revisions\/1366"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/parts\/140"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapters\/221\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/media?parent=221"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/pressbooks\/v2\/chapter-type?post=221"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/contributor?post=221"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/atd-herkimer-statisticssocsci\/wp-json\/wp\/v2\/license?post=221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}