{"id":2019,"date":"2021-09-23T19:13:32","date_gmt":"2021-09-23T19:13:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2019"},"modified":"2023-12-05T09:29:13","modified_gmt":"2023-12-05T09:29:13","slug":"summary-a-single-population-mean-using-the-student-t-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-a-single-population-mean-using-the-student-t-distribution\/","title":{"raw":"Summary: A Single Population Mean using the Student's t-Distribution","rendered":"Summary: A Single Population Mean using the Student&#8217;s t-Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The procedure for forming a confidence interval for a mean when the sample size is small and the population standard deviation is not known is similar to forming a confidence interval for a large sample size with a known population standard deviation.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Instead of basing the error bound on the normal distribution, the t-distribution is used with degrees of freedom (df) equal to the sample size minus 1.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The error bound formula is [latex]\\large t(\\frac{s}{\\sqrt{n}})[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>degrees of freedom (df):\u00a0<\/strong>the number of objects in a sample that are free to vary\r\n\r\n<strong>Student's t-distribution:\u00a0<\/strong>investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">It is continuous and assumes any real values.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">It approaches the standard normal distribution as <em>n<\/em> get larger.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There is a \"family\" of t-distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The procedure for forming a confidence interval for a mean when the sample size is small and the population standard deviation is not known is similar to forming a confidence interval for a large sample size with a known population standard deviation.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Instead of basing the error bound on the normal distribution, the t-distribution is used with degrees of freedom (df) equal to the sample size minus 1.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The error bound formula is [latex]\\large t(\\frac{s}{\\sqrt{n}})[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>degrees of freedom (df):\u00a0<\/strong>the number of objects in a sample that are free to vary<\/p>\n<p><strong>Student&#8217;s t-distribution:\u00a0<\/strong>investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">It is continuous and assumes any real values.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">It approaches the standard normal distribution as <em>n<\/em> get larger.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There is a &#8220;family&#8221; of t-distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.<\/li>\n<\/ul>\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-2019\">\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":15,"template":"","meta":{"_candela_citation":"[{\"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\"},{\"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-2019","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2019","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\/2019\/revisions"}],"predecessor-version":[{"id":3773,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2019\/revisions\/3773"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/269"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2019\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2019"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2019"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2019"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}