{"id":2707,"date":"2021-11-12T13:38:11","date_gmt":"2021-11-12T13:38:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2707"},"modified":"2023-12-05T09:24:19","modified_gmt":"2023-12-05T09:24:19","slug":"putting-it-together-the-central-limit-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/putting-it-together-the-central-limit-theorem\/","title":{"raw":"Putting It Together: The Central Limit Theorem","rendered":"Putting It Together: The Central Limit Theorem"},"content":{"raw":"<h2>Let\u2019s Summarize<\/h2>\r\nIf we have a quantitative data set from a population with mean [latex]\u00b5[\/latex] and standard deviation [latex]\u03c3[\/latex], the model for the theoretical sampling distribution of means of all random samples of size [latex]n[\/latex] has the following properties:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of means is [latex]\u00b5[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of means is [latex]\u03c3[\/latex] divided by the square root of the sample size, [latex]n[\/latex]. This is also called the standard error of the mean.\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Notice that as [latex]n[\/latex] grows, the standard error of the sampling distribution of means shrinks. That means that larger samples give more accurate estimates of a population mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">For a large enough sample size, the sampling distribution of means is approximately normal (even if the population is not normal). This is called the central limit theorem.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Even if a distribution is non-normal, if the sample size is sufficiently large, a normal distribution can be used to calculate probabilities involving sample means and sample sums. This is even true for exponential distributions and uniform distributions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">The general rule is that if [latex]n[\/latex] is at least 30, then the sampling distribution of means will be approximately normal. However, if the population is already normal, then any sample size will produce a normal sampling distribution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The Central Limit Theorem is not for calculating probabilities involving an individual value.<\/li>\r\n<\/ul>","rendered":"<h2>Let\u2019s Summarize<\/h2>\n<p>If we have a quantitative data set from a population with mean [latex]\u00b5[\/latex] and standard deviation [latex]\u03c3[\/latex], the model for the theoretical sampling distribution of means of all random samples of size [latex]n[\/latex] has the following properties:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of means is [latex]\u00b5[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of means is [latex]\u03c3[\/latex] divided by the square root of the sample size, [latex]n[\/latex]. This is also called the standard error of the mean.\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Notice that as [latex]n[\/latex] grows, the standard error of the sampling distribution of means shrinks. That means that larger samples give more accurate estimates of a population mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">For a large enough sample size, the sampling distribution of means is approximately normal (even if the population is not normal). This is called the central limit theorem.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Even if a distribution is non-normal, if the sample size is sufficiently large, a normal distribution can be used to calculate probabilities involving sample means and sample sums. This is even true for exponential distributions and uniform distributions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">The general rule is that if [latex]n[\/latex] is at least 30, then the sampling distribution of means will be approximately normal. However, if the population is already normal, then any sample size will produce a normal sampling distribution.<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The Central Limit Theorem is not for calculating probabilities involving an individual value.<\/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-2707\">\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":16,"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-2707","chapter","type-chapter","status-publish","hentry"],"part":262,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2707","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":3,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2707\/revisions"}],"predecessor-version":[{"id":3726,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2707\/revisions\/3726"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/262"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2707\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2707"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2707"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2707"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}