{"id":2027,"date":"2021-09-23T19:30:27","date_gmt":"2021-09-23T19:30:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2027"},"modified":"2023-12-05T09:30:21","modified_gmt":"2023-12-05T09:30:21","slug":"summary-a-population-proportion","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-a-population-proportion\/","title":{"raw":"Summary: A Population Proportion","rendered":"Summary: A Population Proportion"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The distribution of the sample proportion follows a normal distribution with a mean of [latex]\\mu = np[\/latex]\u00a0and a standard deviation of [latex]\\sigma = \\sqrt{np(1-p)}[\/latex].\u00a0This is because the distribution of a sample proportion is based on the binomial probability distribution.<\/li>\r\n \t<li>[latex]P\u2019[\/latex] (read \u201cp prime) is the sample proportion (point estimate) for a confidence interval for a population proportion.<\/li>\r\n \t<li>The error bound formula is [latex]z (\\sqrt{\\frac{p' q'}{n}})[\/latex], where [latex]q'=1-p'[\/latex].<\/li>\r\n \t<li>The \u201cplus-four\u201d method of adding two additional successes and two additional failures can be used to produce more accurate confidence intervals.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>binomial probability distribution:\u00a0<\/strong>a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, <em>n<\/em>, of independent trials. The notation is: [latex]X \\sim B(n, p)[\/latex]. The mean is [latex]\\mu =np[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{np(1-p)}[\/latex].\r\n\r\n<strong>error bound for a population proportion (EBP):\u00a0<\/strong>the margin of error; depends on the confidence level, the sample size, and the estimated (from the sample) proportion of successes","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>The distribution of the sample proportion follows a normal distribution with a mean of [latex]\\mu = np[\/latex]\u00a0and a standard deviation of [latex]\\sigma = \\sqrt{np(1-p)}[\/latex].\u00a0This is because the distribution of a sample proportion is based on the binomial probability distribution.<\/li>\n<li>[latex]P\u2019[\/latex] (read \u201cp prime) is the sample proportion (point estimate) for a confidence interval for a population proportion.<\/li>\n<li>The error bound formula is [latex]z (\\sqrt{\\frac{p' q'}{n}})[\/latex], where [latex]q'=1-p'[\/latex].<\/li>\n<li>The \u201cplus-four\u201d method of adding two additional successes and two additional failures can be used to produce more accurate confidence intervals.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>binomial probability distribution:\u00a0<\/strong>a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, <em>n<\/em>, of independent trials. The notation is: [latex]X \\sim B(n, p)[\/latex]. The mean is [latex]\\mu =np[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{np(1-p)}[\/latex].<\/p>\n<p><strong>error bound for a population proportion (EBP):\u00a0<\/strong>the margin of error; depends on the confidence level, the sample size, and the estimated (from the sample) proportion of successes<\/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-2027\">\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\/8-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-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":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-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-2027","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2027","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":7,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2027\/revisions"}],"predecessor-version":[{"id":3784,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2027\/revisions\/3784"}],"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\/2027\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2027"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2027"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2027"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}