{"id":242454,"date":"2025-07-04T07:37:14","date_gmt":"2025-07-04T07:37:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=242454"},"modified":"2025-07-04T07:37:16","modified_gmt":"2025-07-04T07:37:16","slug":"test-the-hypothesis-using-the-p-value-approach","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/test-the-hypothesis-using-the-p-value-approach\/","title":{"rendered":"Test the hypothesis using the P-value approach."},"content":{"rendered":"\n<p>Test the hypothesis using the P-value approach. Be sure to verify the requirements of the test. Ho: p = 0.93 versus H?: p? 0.93 n = 500, x = 450, ? = 0.01 Is npo (1-Po) ? 10? Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. No, because npo (1-Po) = B. Yes, because npo (1-Po) = 32.55 Now find p. ^p = 0.9 (Type an integer or a decimal. Do not round.) Find the test statistic Zo. Zo = -2.63 (Round to two decimal places as needed.) Find the P-value. P-value = 0.009 (Round to three decimal places as needed.) State the conclusion of the hypothesis test. Reject the null hypothesis, because the P-value is less than<\/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 test the hypothesis using the P-value approach, we need to follow these steps:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. Verify the requirements of the test<\/h3>\n\n\n\n<p>The hypothesis test is about the population proportion ppp. We want to test:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Null hypothesis (H0H_0H0\u200b):<\/strong> p=0.93p = 0.93p=0.93<\/li>\n\n\n\n<li><strong>Alternative hypothesis (HaH_aHa\u200b):<\/strong> p\u22600.93p \\neq 0.93p\ue020=0.93<\/li>\n\n\n\n<li><strong>Sample size (nnn):<\/strong> 500<\/li>\n\n\n\n<li><strong>Sample successes (xxx):<\/strong> 450<\/li>\n\n\n\n<li><strong>Significance level (\u03b1\\alpha\u03b1):<\/strong> 0.01<\/li>\n<\/ul>\n\n\n\n<p>For a z-test for proportions, the requirements are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>np0(1\u2212p0)\u226510n p_0 (1 &#8211; p_0) \\geq 10np0\u200b(1\u2212p0\u200b)\u226510, where p0p_0p0\u200b is the hypothesized population proportion.<\/li>\n\n\n\n<li>nnn should be large enough to ensure normality, which is often the case when both np0n p_0np0\u200b and n(1\u2212p0)n (1 &#8211; p_0)n(1\u2212p0\u200b) are greater than or equal to 10.<\/li>\n<\/ul>\n\n\n\n<p>Let\u2019s check if np0(1\u2212p0)\u226510n p_0 (1 &#8211; p_0) \\geq 10np0\u200b(1\u2212p0\u200b)\u226510:np0(1\u2212p0)=500\u00d70.93\u00d7(1\u22120.93)=500\u00d70.93\u00d70.07=32.55n p_0 (1 &#8211; p_0) = 500 \\times 0.93 \\times (1 &#8211; 0.93) = 500 \\times 0.93 \\times 0.07 = 32.55np0\u200b(1\u2212p0\u200b)=500\u00d70.93\u00d7(1\u22120.93)=500\u00d70.93\u00d70.07=32.55<\/p>\n\n\n\n<p>So, the answer is <strong>B. Yes, because np0(1\u2212p0)=32.55n p_0 (1 &#8211; p_0) = 32.55np0\u200b(1\u2212p0\u200b)=32.55<\/strong>. This satisfies the requirement for normality.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. Find the sample proportion<\/h3>\n\n\n\n<p>The sample proportion p^\\hat{p}p^\u200b is calculated as:p^=xn=450500=0.9\\hat{p} = \\frac{x}{n} = \\frac{450}{500} = 0.9p^\u200b=nx\u200b=500450\u200b=0.9<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. Calculate the test statistic Z0Z_0Z0\u200b<\/h3>\n\n\n\n<p>The test statistic for a hypothesis test on a population proportion is given by:Z0=p^\u2212p0p0(1\u2212p0)nZ_0 = \\frac{\\hat{p} &#8211; p_0}{\\sqrt{\\frac{p_0(1 &#8211; p_0)}{n}}}Z0\u200b=np0\u200b(1\u2212p0\u200b)\u200b\u200bp^\u200b\u2212p0\u200b\u200b<\/p>\n\n\n\n<p>Substitute the known values:Z0=0.9\u22120.930.93\u00d7(1\u22120.93)500Z_0 = \\frac{0.9 &#8211; 0.93}{\\sqrt{\\frac{0.93 \\times (1 &#8211; 0.93)}{500}}}Z0\u200b=5000.93\u00d7(1\u22120.93)\u200b\u200b0.9\u22120.93\u200bZ0=0.9\u22120.930.93\u00d70.07500Z_0 = \\frac{0.9 &#8211; 0.93}{\\sqrt{\\frac{0.93 \\times 0.07}{500}}}Z0\u200b=5000.93\u00d70.07\u200b\u200b0.9\u22120.93\u200bZ0=\u22120.030.0651500Z_0 = \\frac{-0.03}{\\sqrt{\\frac{0.0651}{500}}}Z0\u200b=5000.0651\u200b\u200b\u22120.03\u200bZ0=\u22120.030.0001302=\u22120.030.01141=\u22122.63Z_0 = \\frac{-0.03}{\\sqrt{0.0001302}} = \\frac{-0.03}{0.01141} = -2.63Z0\u200b=0.0001302\u200b\u22120.03\u200b=0.01141\u22120.03\u200b=\u22122.63<\/p>\n\n\n\n<p>So, the test statistic Z0=\u22122.63Z_0 = -2.63Z0\u200b=\u22122.63.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. Find the P-value<\/h3>\n\n\n\n<p>The P-value is the probability of observing a value as extreme as or more extreme than the test statistic under the null hypothesis. Since the test is two-tailed (because the alternative hypothesis is p\u22600.93p \\neq 0.93p\ue020=0.93), we calculate the two-tailed P-value as:P=2\u00d7P(Z\u2264\u22122.63)P = 2 \\times P(Z \\leq -2.63)P=2\u00d7P(Z\u2264\u22122.63)<\/p>\n\n\n\n<p>Using a standard normal distribution table or a calculator:P(Z\u2264\u22122.63)\u22480.0043P(Z \\leq -2.63) \\approx 0.0043P(Z\u2264\u22122.63)\u22480.0043<\/p>\n\n\n\n<p>Thus, the two-tailed P-value is:P=2\u00d70.0043=0.0086P = 2 \\times 0.0043 = 0.0086P=2\u00d70.0043=0.0086<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5. State the conclusion of the hypothesis test<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The P-value 0.00860.00860.0086 is less than the significance level \u03b1=0.01\\alpha = 0.01\u03b1=0.01.<\/li>\n\n\n\n<li>Since the P-value is less than \u03b1\\alpha\u03b1, we <strong>reject the null hypothesis<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p><strong>Reject the null hypothesis<\/strong>, because the P-value is less than the significance level of 0.01.<\/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-banner8-118.jpeg\" alt=\"\" class=\"wp-image-242456\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Test the hypothesis using the P-value approach. Be sure to verify the requirements of the test. Ho: p = 0.93 versus H?: p? 0.93 n = 500, x = 450, ? = 0.01 Is npo (1-Po) ? 10? Select the correct choice below and fill in the answer box to complete your choice. (Type an [&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-242454","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242454","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=242454"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242454\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=242454"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=242454"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=242454"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}