{"id":2138,"date":"2021-10-01T18:27:21","date_gmt":"2021-10-01T18:27:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2138"},"modified":"2023-12-05T09:38:16","modified_gmt":"2023-12-05T09:38:16","slug":"two-population-means-with-unknown-standard-deviations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/two-population-means-with-unknown-standard-deviations-2\/","title":{"raw":"Cohen's Standards for Small, Medium, and Large Effect Sizes","rendered":"Cohen&#8217;s Standards for Small, Medium, and Large Effect Sizes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul>\r\n \t<li>Calculate and interpret the measure of effect size, Cohen's <em>d<\/em><\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Cohen's Standards for Small, Medium, and Large Effect Sizes<\/h2>\r\n<strong>Cohen's <\/strong><em><strong data-redactor-tag=\"strong\">d<\/strong><\/em> is a measure of effect size based on the differences between two means. Cohen's <em>d<\/em>, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen's standards of small, medium, and large effect sizes.\r\n<h3>Cohen's Standard Effect Sizes<\/h3>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Size of Effect<\/th>\r\n<th><em>d<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Small<\/td>\r\n<td>0.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Medium<\/td>\r\n<td>0.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Large<\/td>\r\n<td>0.8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCohen's <em>d<\/em> is the measure of the difference between two means divided by the pooled standard deviation:\r\n\r\n[latex]\\displaystyle{d}=\\dfrac{{\\overline{{x}}_{{1}}-\\overline{{x}}_{{2}}}}{{{s}_{{\\text{pooled}}}}} \\text{ where } {s}_{{\\text{pooled}}}=\\sqrt{{\\dfrac{{{({n}_{{1}}-{1})}{{s}_{{1}}^{{2}}}+{({n}_{{2}}-{1})}{{s}_{{2}}^{{2}}}}}{{{n}_{{1}}+{n}_{{2}}-{2}}}}} [\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example 4<\/h3>\r\nCalculate Cohen's <em>d<\/em> for Example 2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.\r\n\r\n[reveal-answer q=\"49892\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"49892\"]\r\n<p id=\"fs-idp179952160\"><em data-effect=\"italics\">\u03bc<\/em><sub>1<\/sub> = 4 <em data-effect=\"italics\">s<\/em><sub>1<\/sub> = 1.5 <em data-effect=\"italics\">n<\/em><sub>1<\/sub> = 11<\/p>\r\n<em data-effect=\"italics\">\u03bc<\/em><sub>2<\/sub> = 3.5 <em data-effect=\"italics\">s<\/em><sub>2<\/sub> = 1 <em data-effect=\"italics\">n<\/em><sub>2<\/sub> = 9\r\n\r\n<em data-effect=\"italics\">d<\/em> = 0.384\r\n\r\nThe effect is small because 0.384 is between Cohen's value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences in the means for the two colleges is small indicating that there is not a significant difference between them.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 5<\/h3>\r\nCalculate Cohen's <em>d<\/em> for Example 3. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.\r\n\r\n[reveal-answer q=\"998861\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"998861\"]\r\n\r\n<em>d<\/em> = 0.834; The effect is large because 0.834 is greater than Cohen's 0.8 for a large effect size. The size of the differences between the means of the Final Exam scores of online students and students in a face-to-face class is large indicating a significant difference.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It 3<\/h3>\r\nWeighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013, are listed in the two tables below.\r\n\r\nNortheast\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>94.2<\/td>\r\n<td>75.2<\/td>\r\n<td>69.6<\/td>\r\n<td>52.0<\/td>\r\n<td>48.0<\/td>\r\n<td>41.9<\/td>\r\n<td>36.4<\/td>\r\n<td>33.4<\/td>\r\n<td>31.5<\/td>\r\n<td>27.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>77.3<\/td>\r\n<td>71.9<\/td>\r\n<td>67.5<\/td>\r\n<td>50.6<\/td>\r\n<td>46.2<\/td>\r\n<td>38.4<\/td>\r\n<td>35.2<\/td>\r\n<td>33.0<\/td>\r\n<td>28.7<\/td>\r\n<td>26.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>76.3<\/td>\r\n<td>71.7<\/td>\r\n<td>56.3<\/td>\r\n<td>48.7<\/td>\r\n<td>43.2<\/td>\r\n<td>37.6<\/td>\r\n<td>33.7<\/td>\r\n<td>31.8<\/td>\r\n<td>28.5<\/td>\r\n<td>26.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWest\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>126.0<\/td>\r\n<td>70.6<\/td>\r\n<td>65.2<\/td>\r\n<td>51.4<\/td>\r\n<td>45.5<\/td>\r\n<td>37.0<\/td>\r\n<td>33.0<\/td>\r\n<td>29.6<\/td>\r\n<td>23.7<\/td>\r\n<td>22.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>116.1<\/td>\r\n<td>70.6<\/td>\r\n<td>58.2<\/td>\r\n<td>51.2<\/td>\r\n<td>43.2<\/td>\r\n<td>36.0<\/td>\r\n<td>31.4<\/td>\r\n<td>28.7<\/td>\r\n<td>23.5<\/td>\r\n<td>21.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>78.2<\/td>\r\n<td>68.2<\/td>\r\n<td>55.6<\/td>\r\n<td>50.3<\/td>\r\n<td>39.0<\/td>\r\n<td>34.1<\/td>\r\n<td>31.0<\/td>\r\n<td>25.3<\/td>\r\n<td>23.4<\/td>\r\n<td>21.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIs there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:\r\n<ol>\r\n \t<li>Is this a test of two means or two proportions?<\/li>\r\n \t<li>Are the population standard deviations known or unknown?<\/li>\r\n \t<li>Which distribution do you use to perform the test?<\/li>\r\n \t<li>What is the random variable?<\/li>\r\n \t<li>What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.<\/li>\r\n \t<li>Is this test right-, left-, or two-tailed?<\/li>\r\n \t<li>What is the <em>p<\/em>-value?<\/li>\r\n \t<li>Do you reject or not reject the null hypothesis?<\/li>\r\n \t<li>At the ___ level of significance, from the sample data, there ______ (is\/is not) sufficient evidence to conclude that ______.<\/li>\r\n \t<li>Calculate Cohen's <em>d<\/em> and interpret it.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"713749\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"713749\"]\r\n<ol>\r\n \t<li>two means<\/li>\r\n \t<li>unknown<\/li>\r\n \t<li>Student's-t<\/li>\r\n \t<li>[latex]\\displaystyle\\overline{{X}}_{{{1}}}-\\overline{{X}}_{{{2}}} [\/latex]\r\n<ol>\r\n \t<li><em>H<\/em><sub>0<\/sub>: <em>\u03bc<\/em><sub>1<\/sub> = <em>\u03bc<\/em><sub>2<\/sub> Null hypothesis: the means of the weighted alphas are equal.<\/li>\r\n \t<li><em>H<\/em><sub>a<\/sub>: <em>\u03bc<\/em><sub>1<\/sub> \u2260 <em>\u03bc<\/em><sub>2<\/sub> Alternative hypothesis: the means of the weighted alphas are not equal.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>two-tailed<\/li>\r\n \t<li><em>p<\/em>-value = 0.8787<\/li>\r\n \t<li>do not reject the null hypothesis<\/li>\r\n \t<li>This indicates that the trends in stocks are about the same in the top 30 banks in each region.<img class=\"alignnone\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/n2nx-dj15s37i#fixme#fixme#fixme\" alt=\"This is a normal distribution curve with mean equal to zero. Both the right and left tails of the curve are shaded. Each tail represents 1\/2(p-value) = 0.4394.\" \/><\/li>\r\n \t<li>At the 5%<span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> level of significance, from the sample data, there <\/span><span style=\"text-decoration: underline;\">is not<\/span><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> sufficient evidence to conclude that <\/span><span style=\"text-decoration: underline;\">the mean weighted alphas for the banks in the northeast and the west are different.<\/span><\/li>\r\n \t<li>d=0.040, very small, because 0.040 is less than Cohen's value of 0.2 for small effect size.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul>\n<li>Calculate and interpret the measure of effect size, Cohen&#8217;s <em>d<\/em><\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Cohen&#8217;s Standards for Small, Medium, and Large Effect Sizes<\/h2>\n<p><strong>Cohen&#8217;s <\/strong><em><strong data-redactor-tag=\"strong\">d<\/strong><\/em> is a measure of effect size based on the differences between two means. Cohen&#8217;s <em>d<\/em>, named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen&#8217;s standards of small, medium, and large effect sizes.<\/p>\n<h3>Cohen&#8217;s Standard Effect Sizes<\/h3>\n<table>\n<thead>\n<tr>\n<th>Size of Effect<\/th>\n<th><em>d<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Small<\/td>\n<td>0.2<\/td>\n<\/tr>\n<tr>\n<td>Medium<\/td>\n<td>0.5<\/td>\n<\/tr>\n<tr>\n<td>Large<\/td>\n<td>0.8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Cohen&#8217;s <em>d<\/em> is the measure of the difference between two means divided by the pooled standard deviation:<\/p>\n<p>[latex]\\displaystyle{d}=\\dfrac{{\\overline{{x}}_{{1}}-\\overline{{x}}_{{2}}}}{{{s}_{{\\text{pooled}}}}} \\text{ where } {s}_{{\\text{pooled}}}=\\sqrt{{\\dfrac{{{({n}_{{1}}-{1})}{{s}_{{1}}^{{2}}}+{({n}_{{2}}-{1})}{{s}_{{2}}^{{2}}}}}{{{n}_{{1}}+{n}_{{2}}-{2}}}}}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 4<\/h3>\n<p>Calculate Cohen&#8217;s <em>d<\/em> for Example 2. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q49892\">Show Answer<\/span><\/p>\n<div id=\"q49892\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-idp179952160\"><em data-effect=\"italics\">\u03bc<\/em><sub>1<\/sub> = 4 <em data-effect=\"italics\">s<\/em><sub>1<\/sub> = 1.5 <em data-effect=\"italics\">n<\/em><sub>1<\/sub> = 11<\/p>\n<p><em data-effect=\"italics\">\u03bc<\/em><sub>2<\/sub> = 3.5 <em data-effect=\"italics\">s<\/em><sub>2<\/sub> = 1 <em data-effect=\"italics\">n<\/em><sub>2<\/sub> = 9<\/p>\n<p><em data-effect=\"italics\">d<\/em> = 0.384<\/p>\n<p>The effect is small because 0.384 is between Cohen&#8217;s value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences in the means for the two colleges is small indicating that there is not a significant difference between them.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 5<\/h3>\n<p>Calculate Cohen&#8217;s <em>d<\/em> for Example 3. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q998861\">Show Answer<\/span><\/p>\n<div id=\"q998861\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>d<\/em> = 0.834; The effect is large because 0.834 is greater than Cohen&#8217;s 0.8 for a large effect size. The size of the differences between the means of the Final Exam scores of online students and students in a face-to-face class is large indicating a significant difference.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It 3<\/h3>\n<p>Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013, are listed in the two tables below.<\/p>\n<p>Northeast<\/p>\n<table>\n<tbody>\n<tr>\n<td>94.2<\/td>\n<td>75.2<\/td>\n<td>69.6<\/td>\n<td>52.0<\/td>\n<td>48.0<\/td>\n<td>41.9<\/td>\n<td>36.4<\/td>\n<td>33.4<\/td>\n<td>31.5<\/td>\n<td>27.6<\/td>\n<\/tr>\n<tr>\n<td>77.3<\/td>\n<td>71.9<\/td>\n<td>67.5<\/td>\n<td>50.6<\/td>\n<td>46.2<\/td>\n<td>38.4<\/td>\n<td>35.2<\/td>\n<td>33.0<\/td>\n<td>28.7<\/td>\n<td>26.5<\/td>\n<\/tr>\n<tr>\n<td>76.3<\/td>\n<td>71.7<\/td>\n<td>56.3<\/td>\n<td>48.7<\/td>\n<td>43.2<\/td>\n<td>37.6<\/td>\n<td>33.7<\/td>\n<td>31.8<\/td>\n<td>28.5<\/td>\n<td>26.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>West<\/p>\n<table>\n<tbody>\n<tr>\n<td>126.0<\/td>\n<td>70.6<\/td>\n<td>65.2<\/td>\n<td>51.4<\/td>\n<td>45.5<\/td>\n<td>37.0<\/td>\n<td>33.0<\/td>\n<td>29.6<\/td>\n<td>23.7<\/td>\n<td>22.6<\/td>\n<\/tr>\n<tr>\n<td>116.1<\/td>\n<td>70.6<\/td>\n<td>58.2<\/td>\n<td>51.2<\/td>\n<td>43.2<\/td>\n<td>36.0<\/td>\n<td>31.4<\/td>\n<td>28.7<\/td>\n<td>23.5<\/td>\n<td>21.6<\/td>\n<\/tr>\n<tr>\n<td>78.2<\/td>\n<td>68.2<\/td>\n<td>55.6<\/td>\n<td>50.3<\/td>\n<td>39.0<\/td>\n<td>34.1<\/td>\n<td>31.0<\/td>\n<td>25.3<\/td>\n<td>23.4<\/td>\n<td>21.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Is there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:<\/p>\n<ol>\n<li>Is this a test of two means or two proportions?<\/li>\n<li>Are the population standard deviations known or unknown?<\/li>\n<li>Which distribution do you use to perform the test?<\/li>\n<li>What is the random variable?<\/li>\n<li>What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.<\/li>\n<li>Is this test right-, left-, or two-tailed?<\/li>\n<li>What is the <em>p<\/em>-value?<\/li>\n<li>Do you reject or not reject the null hypothesis?<\/li>\n<li>At the ___ level of significance, from the sample data, there ______ (is\/is not) sufficient evidence to conclude that ______.<\/li>\n<li>Calculate Cohen&#8217;s <em>d<\/em> and interpret it.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713749\">Show Answer<\/span><\/p>\n<div id=\"q713749\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>two means<\/li>\n<li>unknown<\/li>\n<li>Student&#8217;s-t<\/li>\n<li>[latex]\\displaystyle\\overline{{X}}_{{{1}}}-\\overline{{X}}_{{{2}}}[\/latex]\n<ol>\n<li><em>H<\/em><sub>0<\/sub>: <em>\u03bc<\/em><sub>1<\/sub> = <em>\u03bc<\/em><sub>2<\/sub> Null hypothesis: the means of the weighted alphas are equal.<\/li>\n<li><em>H<\/em><sub>a<\/sub>: <em>\u03bc<\/em><sub>1<\/sub> \u2260 <em>\u03bc<\/em><sub>2<\/sub> Alternative hypothesis: the means of the weighted alphas are not equal.<\/li>\n<\/ol>\n<\/li>\n<li>two-tailed<\/li>\n<li><em>p<\/em>-value = 0.8787<\/li>\n<li>do not reject the null hypothesis<\/li>\n<li>This indicates that the trends in stocks are about the same in the top 30 banks in each region.<img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/n2nx-dj15s37i#fixme#fixme#fixme\" alt=\"This is a normal distribution curve with mean equal to zero. Both the right and left tails of the curve are shaded. Each tail represents 1\/2(p-value) = 0.4394.\" \/><\/li>\n<li>At the 5%<span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> level of significance, from the sample data, there <\/span><span style=\"text-decoration: underline;\">is not<\/span><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\"> sufficient evidence to conclude that <\/span><span style=\"text-decoration: underline;\">the mean weighted alphas for the banks in the northeast and the west are different.<\/span><\/li>\n<li>d=0.040, very small, because 0.040 is less than Cohen&#8217;s value of 0.2 for small effect size.<\/li>\n<\/ol>\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-2138\">\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>Two Population Means with Unknown Standard Deviations. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/10-1-two-population-means-with-unknown-standard-deviations\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/10-1-two-population-means-with-unknown-standard-deviations<\/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>: OpenStax. <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":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Two Population Means with Unknown Standard Deviations\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/10-1-two-population-means-with-unknown-standard-deviations\",\"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\":\"OpenStax\",\"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-2138","chapter","type-chapter","status-publish","hentry"],"part":285,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2138","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":8,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2138\/revisions"}],"predecessor-version":[{"id":4056,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2138\/revisions\/4056"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/285"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2138\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2138"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2138"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}