{"id":2408,"date":"2021-10-13T17:28:03","date_gmt":"2021-10-13T17:28:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2408"},"modified":"2023-12-05T09:38:31","modified_gmt":"2023-12-05T09:38:31","slug":"summary-two-population-means-with-unknown-standard-deviations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-two-population-means-with-unknown-standard-deviations\/","title":{"raw":"Summary: Two Population Means with Unknown Standard Deviations","rendered":"Summary: Two Population Means with Unknown Standard Deviations"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The steps for performing a hypothesis test for two population means with unknown standard deviation are generally the same as the steps for conducting a hypothesis test for one population mean with unknown standard deviation, using a <em>t<\/em>-distribution.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">It is recommended that calculations for the test statistic and degrees of freedom be done using technology.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Because the population standard deviations are not known, the sample standard deviations are used for calculations.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">When the sum of the sample sizes is more than 30, a normal distribution can be used to approximate the student <em>t<\/em>-distribution.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Cohen's <em>d<\/em> is a measure of effect size based on the differences between two means.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>Degrees of Freedom (<em>df<\/em>):\u00a0<\/strong>the number of objects in a sample that are free to vary.\r\n\r\n<strong>Standard Deviation: <\/strong>a\u00a0number that is equal to the square root of the variance and measures how far data values are from their mean; notation: <em>s<\/em> for sample standard deviation and <em>\u03c3<\/em> for population standard deviation.","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The steps for performing a hypothesis test for two population means with unknown standard deviation are generally the same as the steps for conducting a hypothesis test for one population mean with unknown standard deviation, using a <em>t<\/em>-distribution.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">It is recommended that calculations for the test statistic and degrees of freedom be done using technology.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Because the population standard deviations are not known, the sample standard deviations are used for calculations.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">When the sum of the sample sizes is more than 30, a normal distribution can be used to approximate the student <em>t<\/em>-distribution.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Cohen&#8217;s <em>d<\/em> is a measure of effect size based on the differences between two means.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>Degrees of Freedom (<em>df<\/em>):\u00a0<\/strong>the number of objects in a sample that are free to vary.<\/p>\n<p><strong>Standard Deviation: <\/strong>a\u00a0number that is equal to the square root of the variance and measures how far data values are from their mean; notation: <em>s<\/em> for sample standard deviation and <em>\u03c3<\/em> for population standard deviation.<\/p>\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-2408\">\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\/10-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/10-key-terms<\/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":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/10-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2408","chapter","type-chapter","status-publish","hentry"],"part":285,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2408","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\/2408\/revisions"}],"predecessor-version":[{"id":3900,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2408\/revisions\/3900"}],"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\/2408\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2408"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2408"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2408"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2408"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}