{"id":1956,"date":"2021-09-16T17:45:31","date_gmt":"2021-09-16T17:45:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1956"},"modified":"2023-12-05T09:30:02","modified_gmt":"2023-12-05T09:30:02","slug":"a-population-proportion-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/a-population-proportion-3\/","title":{"raw":"Calculating the Sample Size n","rendered":"Calculating the Sample Size n"},"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>Calculate the sample size needed for a given error bound for a confidence interval for a population proportion<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<strong>Calculating the Sample Size n<\/strong>\r\nIf researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.\r\n\r\nThe error bound formula for a population proportion is\r\n<ul>\r\n \t<li>EBP = [latex]\\displaystyle({z}_{\\frac{{\\alpha}}{{2}}})(\\sqrt{\\frac{{p'q'}}{{n}}})[\/latex]<\/li>\r\n \t<li>Solving for <em>n<\/em> gives you an equation for the sample size.<\/li>\r\n \t<li>[latex]\\displaystyle{n}=\\frac{{{\\left({z}_{\\frac{{\\alpha}}{{2}}}\\right)}^{2}({p'}{q'})}}{{{EBP}^{2}}}[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 5<\/h3>\r\nSuppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones?\r\n[reveal-answer q=\"430478\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"430478\"]\r\n\r\nFrom the problem, we know that <strong>EBP = 0.03<\/strong> (3%=0.03) and [latex]{z_{\\frac{a}{2}}}[\/latex] z<sub>0.05<\/sub>\u00a0= 1.645 because the confidence level is 90%.\r\n\r\nHowever, in order to find <em>n<\/em>, we need to know the estimated (sample) proportion <em>p\u2032<\/em>. Remember that <em>q\u2032<\/em> = 1 \u2013 <em>p\u2032<\/em>. But, we do not know <em>p\u2032<\/em> yet. Since we multiply <em>p\u2032<\/em> and <em>q\u2032<\/em> together, we make them both equal to 0.5 because <em>p\u2032q\u2032<\/em> = (0.5)(0.5) = 0.25 results in the largest possible product. (Try other products: (0.6)(0.4) = 0.24; (0.3)(0.7) = 0.21; (0.2)(0.8) = 0.16 and so on). The largest possible product gives us the largest <em>n<\/em>. This gives us a large enough sample so that we can be 90% confident that we are within three percentage points of the true population proportion. To calculate the sample size <em>n<\/em>, use the formula and make the substitutions.\r\n\r\n<em>n<\/em> = [latex]{\\dfrac{z^2p'q'}{EBP^2}}[\/latex] gives\u00a0<em>n<\/em> = [latex]{\\dfrac{1.645^2(0.5)(0.5)}{0.03^2}}[\/latex] = 751.7\r\n\r\nRound the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 5<\/h3>\r\nSuppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90% confident that the estimated proportion is within five percentage points of the true population proportion of customers who click on ads on their smartphones?\r\n[reveal-answer q=\"189675\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"189675\"]\r\n\r\n271 customers should be surveyed.\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>Calculate the sample size needed for a given error bound for a confidence interval for a population proportion<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p><strong>Calculating the Sample Size n<\/strong><br \/>\nIf researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.<\/p>\n<p>The error bound formula for a population proportion is<\/p>\n<ul>\n<li>EBP = [latex]\\displaystyle({z}_{\\frac{{\\alpha}}{{2}}})(\\sqrt{\\frac{{p'q'}}{{n}}})[\/latex]<\/li>\n<li>Solving for <em>n<\/em> gives you an equation for the sample size.<\/li>\n<li>[latex]\\displaystyle{n}=\\frac{{{\\left({z}_{\\frac{{\\alpha}}{{2}}}\\right)}^{2}({p'}{q'})}}{{{EBP}^{2}}}[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example 5<\/h3>\n<p>Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q430478\">Show Answer<\/span><\/p>\n<div id=\"q430478\" class=\"hidden-answer\" style=\"display: none\">\n<p>From the problem, we know that <strong>EBP = 0.03<\/strong> (3%=0.03) and [latex]{z_{\\frac{a}{2}}}[\/latex] z<sub>0.05<\/sub>\u00a0= 1.645 because the confidence level is 90%.<\/p>\n<p>However, in order to find <em>n<\/em>, we need to know the estimated (sample) proportion <em>p\u2032<\/em>. Remember that <em>q\u2032<\/em> = 1 \u2013 <em>p\u2032<\/em>. But, we do not know <em>p\u2032<\/em> yet. Since we multiply <em>p\u2032<\/em> and <em>q\u2032<\/em> together, we make them both equal to 0.5 because <em>p\u2032q\u2032<\/em> = (0.5)(0.5) = 0.25 results in the largest possible product. (Try other products: (0.6)(0.4) = 0.24; (0.3)(0.7) = 0.21; (0.2)(0.8) = 0.16 and so on). The largest possible product gives us the largest <em>n<\/em>. This gives us a large enough sample so that we can be 90% confident that we are within three percentage points of the true population proportion. To calculate the sample size <em>n<\/em>, use the formula and make the substitutions.<\/p>\n<p><em>n<\/em> = [latex]{\\dfrac{z^2p'q'}{EBP^2}}[\/latex] gives\u00a0<em>n<\/em> = [latex]{\\dfrac{1.645^2(0.5)(0.5)}{0.03^2}}[\/latex] = 751.7<\/p>\n<p>Round the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 5<\/h3>\n<p>Suppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90% confident that the estimated proportion is within five percentage points of the true population proportion of customers who click on ads on their smartphones?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q189675\">Show Answer<\/span><\/p>\n<div id=\"q189675\" class=\"hidden-answer\" style=\"display: none\">\n<p>271 customers should be surveyed.<\/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-1956\">\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 Population Proportion. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-3-a-population-proportion\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-3-a-population-proportion<\/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":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"A Population Proportion\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/8-3-a-population-proportion\",\"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-1956","chapter","type-chapter","status-publish","hentry"],"part":269,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1956","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\/1956\/revisions"}],"predecessor-version":[{"id":3782,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1956\/revisions\/3782"}],"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\/1956\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1956"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1956"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1956"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}