{"id":1948,"date":"2021-09-16T17:30:05","date_gmt":"2021-09-16T17:30:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1948"},"modified":"2023-12-05T09:27:45","modified_gmt":"2023-12-05T09:27:45","slug":"a-single-population-mean-using-the-normal-distribution-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-single-population-mean-using-the-normal-distribution-3\/","title":{"raw":"Working Backwards to Find the Error Bound or Sample Mean","rendered":"Working Backwards to Find the Error Bound or Sample Mean"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list12315\">\r\n \t<li>Given a confidence interval for a population mean, find the sample mean and error bound<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Working Backwards to Find the Error Bound or Sample Mean<\/h2>\r\nWhen we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.\r\n<h3>Finding the Error Bound<\/h3>\r\n<ul>\r\n \t<li>From the upper value for the interval, subtract the sample mean,<\/li>\r\n \t<li>OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.<\/li>\r\n<\/ul>\r\n<h3>Finding the Sample Mean<\/h3>\r\n<ul>\r\n \t<li>Subtract the error bound from the upper value of the confidence interval,<\/li>\r\n \t<li>OR, average the upper and lower endpoints of the confidence interval.<\/li>\r\n<\/ul>\r\n<p id=\"eip-117\" class=\" \">Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example 6<\/h3>\r\nSuppose we know that a confidence interval is <strong>(67.18, 68.82)<\/strong> and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.\r\n\r\n<strong>Calculate the Error Bound:<\/strong>\r\n<ul>\r\n \t<li>If we know that the sample mean is 68: <em data-redactor-tag=\"em\">EBM<\/em> = 68.82 \u2013 68 = 0.82<\/li>\r\n \t<li>If we don't know the sample mean:\u00a0<em>EBM\u00a0<\/em> = [latex]\\dfrac{(68.82-67.18)}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<strong>Calculate the Sample Mean:<\/strong>\r\n<ul>\r\n \t<li>If we know the error bound:\u00a0<span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]\u00a0<\/span>= 68.82 \u2013 0.82 = 68<\/li>\r\n \t<li>If we don't know the error bound:\u00a0<span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex] = [latex]\\dfrac{(67.18+68.82)}{2}[\/latex] = 68<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it 6<\/h3>\r\nSuppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.\r\n[reveal-answer q=\"113291\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"113291\"]\r\n\r\nSample mean is 45. Error bound is 2.88.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list12315\">\n<li>Given a confidence interval for a population mean, find the sample mean and error bound<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Working Backwards to Find the Error Bound or Sample Mean<\/h2>\n<p>When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. However, sometimes when we read statistical studies, the study may state the confidence interval only. If we know the confidence interval, we can work backwards to find both the error bound and the sample mean.<\/p>\n<h3>Finding the Error Bound<\/h3>\n<ul>\n<li>From the upper value for the interval, subtract the sample mean,<\/li>\n<li>OR, from the upper value for the interval, subtract the lower value. Then divide the difference by two.<\/li>\n<\/ul>\n<h3>Finding the Sample Mean<\/h3>\n<ul>\n<li>Subtract the error bound from the upper value of the confidence interval,<\/li>\n<li>OR, average the upper and lower endpoints of the confidence interval.<\/li>\n<\/ul>\n<p id=\"eip-117\" class=\"\">Notice that there are two methods to perform each calculation. You can choose the method that is easier to use with the information you know.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 6<\/h3>\n<p>Suppose we know that a confidence interval is <strong>(67.18, 68.82)<\/strong> and we want to find the error bound. We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean.<\/p>\n<p><strong>Calculate the Error Bound:<\/strong><\/p>\n<ul>\n<li>If we know that the sample mean is 68: <em data-redactor-tag=\"em\">EBM<\/em> = 68.82 \u2013 68 = 0.82<\/li>\n<li>If we don&#8217;t know the sample mean:\u00a0<em>EBM\u00a0<\/em> = [latex]\\dfrac{(68.82-67.18)}{2}[\/latex]<\/li>\n<\/ul>\n<p><strong>Calculate the Sample Mean:<\/strong><\/p>\n<ul>\n<li>If we know the error bound:\u00a0<span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex]\u00a0<\/span>= 68.82 \u2013 0.82 = 68<\/li>\n<li>If we don&#8217;t know the error bound:\u00a0<span style=\"white-space: nowrap;\">[latex]\\overline{x}[\/latex] = [latex]\\dfrac{(67.18+68.82)}{2}[\/latex] = 68<\/span><\/li>\n<\/ul>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try it 6<\/h3>\n<p>Suppose we know that a confidence interval is (42.12, 47.88). Find the error bound and the sample mean.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q113291\">Show Answer<\/span><\/p>\n<div id=\"q113291\" class=\"hidden-answer\" style=\"display: none\">\n<p>Sample mean is 45. Error bound is 2.88.<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1948\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>A Single Population Mean using the Normal Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution<\/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><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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"A Single Population Mean using the Normal Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-1-a-single-population-mean-using-the-normal-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"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-1948","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1948","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\/1948\/revisions"}],"predecessor-version":[{"id":3761,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1948\/revisions\/3761"}],"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\/1948\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1948"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1948"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1948"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}