{"id":2011,"date":"2021-09-23T19:07:38","date_gmt":"2021-09-23T19:07:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2011"},"modified":"2023-12-05T09:28:16","modified_gmt":"2023-12-05T09:28:16","slug":"summary-a-single-population-mean-using-the-normal-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-a-single-population-mean-using-the-normal-distribution\/","title":{"raw":"Summary: A Single Population Mean using the Normal Distribution","rendered":"Summary: A Single Population Mean using the Normal Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The sample mean is a single value (point estimate) used to estimate the value of the population mean.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">An error bound is added and subtracted to the sample mean to form a confidence interval.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The size of the error bound is based on the confidence level, standard deviation of the population and the sample size.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The <em>z<\/em>-score for a confidence interval for a mean is based on a normal distribution when the population standard deviation is known.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The confidence level refers to the long-run success rate of all possible confidence intervals from all possible random samples of a given sample size.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">For a given confidence level, intervals are narrower for larger sample sizes.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">For a given sample size, intervals are wider for larger confidence levels.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The sample size can be calculated for a specified level of confidence and error bound.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>Confidence Interval (CI):\u00a0<\/strong>an interval estimate for an unknown population parameter. This depends on:\r\n<ul>\r\n \t<li>the desired confidence level<\/li>\r\n \t<li>the information that is known about the distribution (for example, known standard deviation),<\/li>\r\n \t<li>and the sample and its size.<\/li>\r\n<\/ul>\r\n<strong>Confidence Level (CL):\u00a0<\/strong>the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.\r\n\r\n<strong>Error Bound for a Population Mean (EBM):\u00a0<\/strong>the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation.\r\n\r\n<strong>inferential statistics:\u00a0<\/strong>also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that four percent of the production is defective.\r\n\r\n<strong>normal distribution:<\/strong>\u00a0a continuous random variable (RV) with pdf [latex]\\large f(x)= \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{\\frac{-(x-\\mu)^2}{2 \\sigma ^2}}[\/latex] where \u03bc is the mean of the distribution and \u03c3 is the standard deviation; notation: [latex]X \\sim N(\u03bc, \u03c3)[\/latex]. If [latex]\u03bc = 0[\/latex] and [latex]\u03c3 = 1[\/latex], the RV is called <strong>the standard normal distribution<\/strong>.\r\n\r\n<strong>parameter:\u00a0\u00a0<\/strong>a numerical characteristic of a population\r\n\r\n<strong>point estimate:\u00a0<\/strong>a single number computed from a sample and used to estimate a population parameter\r\n\r\n<strong>standard deviation:\u00a0<\/strong>a number 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 \u03c3 for population standard deviation","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The sample mean is a single value (point estimate) used to estimate the value of the population mean.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">An error bound is added and subtracted to the sample mean to form a confidence interval.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The size of the error bound is based on the confidence level, standard deviation of the population and the sample size.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The <em>z<\/em>-score for a confidence interval for a mean is based on a normal distribution when the population standard deviation is known.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The confidence level refers to the long-run success rate of all possible confidence intervals from all possible random samples of a given sample size.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">For a given confidence level, intervals are narrower for larger sample sizes.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">For a given sample size, intervals are wider for larger confidence levels.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The sample size can be calculated for a specified level of confidence and error bound.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>Confidence Interval (CI):\u00a0<\/strong>an interval estimate for an unknown population parameter. This depends on:<\/p>\n<ul>\n<li>the desired confidence level<\/li>\n<li>the information that is known about the distribution (for example, known standard deviation),<\/li>\n<li>and the sample and its size.<\/li>\n<\/ul>\n<p><strong>Confidence Level (CL):\u00a0<\/strong>the percent expression for the probability that the confidence interval contains the true population parameter; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.<\/p>\n<p><strong>Error Bound for a Population Mean (EBM):\u00a0<\/strong>the margin of error; depends on the confidence level, sample size, and known or estimated population standard deviation.<\/p>\n<p><strong>inferential statistics:\u00a0<\/strong>also called statistical inference or inductive statistics; this facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that four percent of the production is defective.<\/p>\n<p><strong>normal distribution:<\/strong>\u00a0a continuous random variable (RV) with pdf [latex]\\large f(x)= \\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{\\frac{-(x-\\mu)^2}{2 \\sigma ^2}}[\/latex] where \u03bc is the mean of the distribution and \u03c3 is the standard deviation; notation: [latex]X \\sim N(\u03bc, \u03c3)[\/latex]. If [latex]\u03bc = 0[\/latex] and [latex]\u03c3 = 1[\/latex], the RV is called <strong>the standard normal distribution<\/strong>.<\/p>\n<p><strong>parameter:\u00a0\u00a0<\/strong>a numerical characteristic of a population<\/p>\n<p><strong>point estimate:\u00a0<\/strong>a single number computed from a sample and used to estimate a population parameter<\/p>\n<p><strong>standard deviation:\u00a0<\/strong>a number 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 \u03c3 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-2011\">\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":12,"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-2011","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2011","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":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2011\/revisions"}],"predecessor-version":[{"id":3765,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2011\/revisions\/3765"}],"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\/2011\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2011"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2011"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2011"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}