{"id":1911,"date":"2021-09-15T14:50:54","date_gmt":"2021-09-15T14:50:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1911"},"modified":"2023-12-05T09:23:59","modified_gmt":"2023-12-05T09:23:59","slug":"summary-using-the-central-limit-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-using-the-central-limit-theorem\/","title":{"raw":"Summary: Using the Central Limit Theorem","rendered":"Summary: Using the Central Limit Theorem"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">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=\"1\">As the sample size gets larger, the mean of the sample means approaches the population mean. This is due to the law of large numbers.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (CLT) is not for calculating probabilities involving an individual value.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>central limit theorem (for means and sums):\u00a0<\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, we are sampling with size <em>n<\/em>, and we are interested in two new RVs: the sample mean, [latex]\\overline{X}[\/latex] and the sample sum [latex]\\sum x[\/latex]. If the size (<em>n<\/em>) of the sample is sufficiently large, then [latex]\\overline{X} \\sim N (M, \\frac{\\sigma}{\\sqrt{n}})[\/latex] and [latex]\\sum X \\sim N (n \\mu, \\sqrt{n} \\sigma)[\/latex].\u00a0If the size (<em>n<\/em>) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal <em>n<\/em> times the population mean. The standard deviation of the distribution of the sample means, [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex], is called the standard error of the mean.\r\n\r\n<strong>exponential distribution:\u00a0<\/strong>a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is [latex]X \\sim Exp(m)[\/latex]. The mean is [latex]\\mu = \\frac{1}{m}[\/latex] and the standard deviation is [latex]\\sigma = \\frac{1}{m}[\/latex]. The probability density function is [latex]f(x)=me^{(-mx)}, x \\geq 0[\/latex] and the cumulative distribution function is [latex]P(X \\leq x) = 1-e^{(-mx)}[\/latex].\r\n\r\n<strong>uniform distribution:\u00a0<\/strong>a continuous random variable (RV) that has equally likely outcomes over the domain, [latex]a&lt;x&lt;b[\/latex]. Notation: [latex]X \\sim U(a,b)[\/latex]. The mean is [latex]\\mu = \\frac{a+b}{2}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{(b-a)^2}{12}}[\/latex]. The probability density function is [latex]f(x)=\\frac{1}{b-a}[\/latex] for [latex]a&lt;x&lt;b[\/latex] or [latex]a \\leq x \\leq b[\/latex]. The cumulative distribution is [latex]P(X \\leq x) = \\frac{x-a}{b-a}[\/latex].","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">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=\"1\">As the sample size gets larger, the mean of the sample means approaches the population mean. This is due to the law of large numbers.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (CLT) is not for calculating probabilities involving an individual value.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>central limit theorem (for means and sums):\u00a0<\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, we are sampling with size <em>n<\/em>, and we are interested in two new RVs: the sample mean, [latex]\\overline{X}[\/latex] and the sample sum [latex]\\sum x[\/latex]. If the size (<em>n<\/em>) of the sample is sufficiently large, then [latex]\\overline{X} \\sim N (M, \\frac{\\sigma}{\\sqrt{n}})[\/latex] and [latex]\\sum X \\sim N (n \\mu, \\sqrt{n} \\sigma)[\/latex].\u00a0If the size (<em>n<\/em>) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal <em>n<\/em> times the population mean. The standard deviation of the distribution of the sample means, [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex], is called the standard error of the mean.<\/p>\n<p><strong>exponential distribution:\u00a0<\/strong>a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is [latex]X \\sim Exp(m)[\/latex]. The mean is [latex]\\mu = \\frac{1}{m}[\/latex] and the standard deviation is [latex]\\sigma = \\frac{1}{m}[\/latex]. The probability density function is [latex]f(x)=me^{(-mx)}, x \\geq 0[\/latex] and the cumulative distribution function is [latex]P(X \\leq x) = 1-e^{(-mx)}[\/latex].<\/p>\n<p><strong>uniform distribution:\u00a0<\/strong>a continuous random variable (RV) that has equally likely outcomes over the domain, [latex]a<x<b[\/latex]. Notation: [latex]X \\sim U(a,b)[\/latex]. The mean is [latex]\\mu = \\frac{a+b}{2}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{(b-a)^2}{12}}[\/latex]. The probability density function is [latex]f(x)=\\frac{1}{b-a}[\/latex] for [latex]a<x<b[\/latex] or [latex]a \\leq x \\leq b[\/latex]. The cumulative distribution is [latex]P(X \\leq x) = \\frac{x-a}{b-a}[\/latex].\n<\/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-1911\">\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-1911","chapter","type-chapter","status-publish","hentry"],"part":262,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1911","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":9,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1911\/revisions"}],"predecessor-version":[{"id":3724,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1911\/revisions\/3724"}],"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\/1911\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1911"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1911"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1911"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}