{"id":244353,"date":"2025-07-05T04:56:48","date_gmt":"2025-07-05T04:56:48","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=244353"},"modified":"2025-07-05T04:56:51","modified_gmt":"2025-07-05T04:56:51","slug":"construct-a-95-confidence-interval-for-the-difference-between-the-two-population-proportions-p1-p2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/05\/construct-a-95-confidence-interval-for-the-difference-between-the-two-population-proportions-p1-p2\/","title":{"rendered":"Construct a 95% confidence interval for the difference between the two population proportions P1-P2."},"content":{"rendered":"\n<p>Construct a 95% confidence interval for the difference between the two population proportions P1-P2. Population one has a sample size of 510 and a sample proportion of 25%. Population two has a sample size of 450 and a sample proportion of 19% (calculator).<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To construct a 95% confidence interval for the difference between two population proportions (P1 &#8211; P2), we can use the following formula:CI=(p1\u2212p2)\u00b1Z\u03b1\/2\u00d7p1(1\u2212p1)n1+p2(1\u2212p2)n2\\text{CI} = (p_1 &#8211; p_2) \\pm Z_{\\alpha\/2} \\times \\sqrt{\\frac{p_1(1 &#8211; p_1)}{n_1} + \\frac{p_2(1 &#8211; p_2)}{n_2}}CI=(p1\u200b\u2212p2\u200b)\u00b1Z\u03b1\/2\u200b\u00d7n1\u200bp1\u200b(1\u2212p1\u200b)\u200b+n2\u200bp2\u200b(1\u2212p2\u200b)\u200b\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>p1p_1p1\u200b and p2p_2p2\u200b are the sample proportions for population one and population two, respectively.<\/li>\n\n\n\n<li>n1n_1n1\u200b and n2n_2n2\u200b are the sample sizes for population one and population two, respectively.<\/li>\n\n\n\n<li>Z\u03b1\/2Z_{\\alpha\/2}Z\u03b1\/2\u200b is the critical value from the standard normal distribution for a 95% confidence level, which is approximately 1.96.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Identify the given values<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Population one: n1=510n_1 = 510n1\u200b=510, p1=0.25p_1 = 0.25p1\u200b=0.25<\/li>\n\n\n\n<li>Population two: n2=450n_2 = 450n2\u200b=450, p2=0.19p_2 = 0.19p2\u200b=0.19<\/li>\n\n\n\n<li>Z\u03b1\/2=1.96Z_{\\alpha\/2} = 1.96Z\u03b1\/2\u200b=1.96 for a 95% confidence level.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the standard error<\/h3>\n\n\n\n<p>We use the formula for the standard error of the difference between two proportions:SE=p1(1\u2212p1)n1+p2(1\u2212p2)n2SE = \\sqrt{\\frac{p_1(1 &#8211; p_1)}{n_1} + \\frac{p_2(1 &#8211; p_2)}{n_2}}SE=n1\u200bp1\u200b(1\u2212p1\u200b)\u200b+n2\u200bp2\u200b(1\u2212p2\u200b)\u200b\u200b<\/p>\n\n\n\n<p>Substituting the given values:SE=0.25(1\u22120.25)510+0.19(1\u22120.19)450=0.25\u00d70.75510+0.19\u00d70.81450SE = \\sqrt{\\frac{0.25(1 &#8211; 0.25)}{510} + \\frac{0.19(1 &#8211; 0.19)}{450}} = \\sqrt{\\frac{0.25 \\times 0.75}{510} + \\frac{0.19 \\times 0.81}{450}}SE=5100.25(1\u22120.25)\u200b+4500.19(1\u22120.19)\u200b\u200b=5100.25\u00d70.75\u200b+4500.19\u00d70.81\u200b\u200bSE=0.1875510+0.1539450=0.0003686+0.0003419=0.0007105\u22480.0267SE = \\sqrt{\\frac{0.1875}{510} + \\frac{0.1539}{450}} = \\sqrt{0.0003686 + 0.0003419} = \\sqrt{0.0007105} \\approx 0.0267SE=5100.1875\u200b+4500.1539\u200b\u200b=0.0003686+0.0003419\u200b=0.0007105\u200b\u22480.0267<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the confidence interval<\/h3>\n\n\n\n<p>Now, we can calculate the confidence interval:CI=(p1\u2212p2)\u00b1Z\u03b1\/2\u00d7SECI = (p_1 &#8211; p_2) \\pm Z_{\\alpha\/2} \\times SECI=(p1\u200b\u2212p2\u200b)\u00b1Z\u03b1\/2\u200b\u00d7SECI=(0.25\u22120.19)\u00b11.96\u00d70.0267CI = (0.25 &#8211; 0.19) \\pm 1.96 \\times 0.0267CI=(0.25\u22120.19)\u00b11.96\u00d70.0267CI=0.06\u00b11.96\u00d70.0267=0.06\u00b10.0523CI = 0.06 \\pm 1.96 \\times 0.0267 = 0.06 \\pm 0.0523CI=0.06\u00b11.96\u00d70.0267=0.06\u00b10.0523<\/p>\n\n\n\n<p>Thus, the confidence interval is:CI=(0.060\u22120.0523,0.060+0.0523)=(0.0077,0.1123)CI = (0.060 &#8211; 0.0523, 0.060 + 0.0523) = (0.0077, 0.1123)CI=(0.060\u22120.0523,0.060+0.0523)=(0.0077,0.1123)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>The 95% confidence interval for the difference between the two population proportions P1\u2212P2P_1 &#8211; P_2P1\u200b\u2212P2\u200b is <strong>(0.0077, 0.1123)<\/strong>. This means we are 95% confident that the true difference in proportions between population one and population two lies between 0.0077 and 0.1123.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-436.jpeg\" alt=\"\" class=\"wp-image-244354\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Construct a 95% confidence interval for the difference between the two population proportions P1-P2. Population one has a sample size of 510 and a sample proportion of 25%. Population two has a sample size of 450 and a sample proportion of 19% (calculator). The Correct Answer and Explanation is: To construct a 95% confidence interval [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-244353","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=244353"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/244353\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=244353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=244353"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=244353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}