{"id":2716,"date":"2021-11-12T13:51:06","date_gmt":"2021-11-12T13:51:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2716"},"modified":"2023-12-05T09:43:58","modified_gmt":"2023-12-05T09:43:58","slug":"putting-it-together-the-chi-square-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/putting-it-together-the-chi-square-distribution\/","title":{"raw":"Putting It Together: The Chi-Square Distribution","rendered":"Putting It Together: The Chi-Square Distribution"},"content":{"raw":"<h2>Let\u2019s Summarize<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply to all of the chi-square hypothesis tests in this module:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square distribution is a distribution that is skewed to the right.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The variability (or spread) of the chi-square distribution depends on the degrees of freedom of the distribution.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The test statistic for a chi-square distribution is always greater than or equal to zero.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square goodness-of-fit test:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that the distribution fits the hypothesized proportions. The alternative hypothesis is that the distribution does not fit the hypothesized proportions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Expected counts are found by taking the total count and multiplying by each of the hypothesized proportions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom is [latex]k \u2013 1[\/latex], where [latex]k[\/latex] is the number of categories.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each category.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square test of independence:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that there is no association between the two categorical variables. The alternative hypothesis is that there is an association between the two categorical variables.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom is [latex](r \u2013 1)(c \u2013 1)[\/latex], where r is the number of rows in the contingency table and [latex]c[\/latex] is the number of columns in the contingency table.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each cell in the contingency table.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square test of homogeneity:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that the distribution of the two populations is the same. The alternative hypothesis is that the distribution of the two populations is not the same.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom for a chi-square test of homogeneity for two populations is [latex]k \u2013 1[\/latex], where [latex]k[\/latex] is the number of response values.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each category.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nTo determine which type of chi-square test is being done, consider the number of samples and the general research question that is being answered.\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Type of Chi-Square Test<\/strong><\/td>\r\n<td><strong>Number of Samples<\/strong><\/td>\r\n<td><strong>Question<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Goodness-of-Fit<\/td>\r\n<td>One Sample<\/td>\r\n<td>Does the population fit the given distribution?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Test of Independence<\/td>\r\n<td>One Sample<\/td>\r\n<td>Is there an association between the two categorical variables?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Test of Homogeneity<\/td>\r\n<td>Two Independent Samples<\/td>\r\n<td>Do the two populations follow the same distribution?<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"<h2>Let\u2019s Summarize<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply to all of the chi-square hypothesis tests in this module:\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square distribution is a distribution that is skewed to the right.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The variability (or spread) of the chi-square distribution depends on the degrees of freedom of the distribution.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The test statistic for a chi-square distribution is always greater than or equal to zero.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square goodness-of-fit test:\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that the distribution fits the hypothesized proportions. The alternative hypothesis is that the distribution does not fit the hypothesized proportions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Expected counts are found by taking the total count and multiplying by each of the hypothesized proportions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom is [latex]k \u2013 1[\/latex], where [latex]k[\/latex] is the number of categories.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each category.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square test of independence:\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that there is no association between the two categorical variables. The alternative hypothesis is that there is an association between the two categorical variables.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom is [latex](r \u2013 1)(c \u2013 1)[\/latex], where r is the number of rows in the contingency table and [latex]c[\/latex] is the number of columns in the contingency table.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each cell in the contingency table.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The following concepts apply for a chi-square test of homogeneity:\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The null hypothesis is that the distribution of the two populations is the same. The alternative hypothesis is that the distribution of the two populations is not the same.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The degrees of freedom for a chi-square test of homogeneity for two populations is [latex]k \u2013 1[\/latex], where [latex]k[\/latex] is the number of response values.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The chi-square test statistic is the sum of [latex]\\frac{(\\mathrm{Observed} \\ \u2013 \\ \\mathrm{Expected})^2}{\\mathrm{Expected}}[\/latex] for each category.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>To determine which type of chi-square test is being done, consider the number of samples and the general research question that is being answered.<\/p>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td><strong>Type of Chi-Square Test<\/strong><\/td>\n<td><strong>Number of Samples<\/strong><\/td>\n<td><strong>Question<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Goodness-of-Fit<\/td>\n<td>One Sample<\/td>\n<td>Does the population fit the given distribution?<\/td>\n<\/tr>\n<tr>\n<td>Test of Independence<\/td>\n<td>One Sample<\/td>\n<td>Is there an association between the two categorical variables?<\/td>\n<\/tr>\n<tr>\n<td>Test of Homogeneity<\/td>\n<td>Two Independent Samples<\/td>\n<td>Do the two populations follow the same distribution?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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-2716\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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-2716","chapter","type-chapter","status-publish","hentry"],"part":293,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2716","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":4,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2716\/revisions"}],"predecessor-version":[{"id":3951,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2716\/revisions\/3951"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/293"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}