{"id":221563,"date":"2025-05-29T19:27:33","date_gmt":"2025-05-29T19:27:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221563"},"modified":"2025-05-29T19:27:35","modified_gmt":"2025-05-29T19:27:35","slug":"the-quality-control-manager-at-a-light-bulb-factory-needs-to-determine-whether-the-mean-life-of-a-large-shipment-of-light-bulbs-is-equal-to-375-hours","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/29\/the-quality-control-manager-at-a-light-bulb-factory-needs-to-determine-whether-the-mean-life-of-a-large-shipment-of-light-bulbs-is-equal-to-375-hours\/","title":{"rendered":"The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours"},"content":{"rendered":"\n<p>The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 350 hours.a. At the 0.05 level of significance, is there evidence that the mean life is different from 375 hours?b. Compute the p-value and interpret its meaning.c. Construct a 95% confidence interval estimate of the population mean life of the light bulbs.d. Compare the results of (a) and (c). What conclusions do you reach?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Population standard deviation (\u03c3)<\/strong> = 100 hours<\/li>\n\n\n\n<li><strong>Sample size (n)<\/strong> = 64<\/li>\n\n\n\n<li><strong>Sample mean (x\u0304)<\/strong> = 350 hours<\/li>\n\n\n\n<li><strong>Hypothesized mean (\u03bc\u2080)<\/strong> = 375 hours<\/li>\n\n\n\n<li><strong>Significance level (\u03b1)<\/strong> = 0.05<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">a. Hypothesis Test<\/h3>\n\n\n\n<p><strong>Null Hypothesis (H\u2080):<\/strong> \u03bc = 375<br><strong>Alternative Hypothesis (H\u2081):<\/strong> \u03bc \u2260 375<\/p>\n\n\n\n<p>This is a two-tailed z-test.<\/p>\n\n\n\n<p>Calculate the test statistic: z=x\u02c9\u2212\u03bc0\u03c3\/n=350\u2212375100\/64=\u22122512.5=\u22122.0z = \\frac{\\bar{x} &#8211; \\mu_0}{\\sigma \/ \\sqrt{n}} = \\frac{350 &#8211; 375}{100 \/ \\sqrt{64}} = \\frac{-25}{12.5} = -2.0<\/p>\n\n\n\n<p>At a 0.05 significance level for a two-tailed test, the critical z-values are \u00b11.96.<br>Since -2.0 &lt; -1.96, we <strong>reject the null hypothesis<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">b. Compute the p-value<\/h3>\n\n\n\n<p>Using the standard normal distribution: p-value=2\u00d7P(Z&lt;\u22122.0)=2\u00d70.0228=0.0456p\\text{-value} = 2 \\times P(Z &lt; -2.0) = 2 \\times 0.0228 = 0.0456<\/p>\n\n\n\n<p><strong>Interpretation:<\/strong> The probability of observing a sample mean of 350 or more extreme, assuming the population mean is 375, is about 4.56%. Since this is less than the significance level (0.05), it supports rejecting the null hypothesis.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">c. 95% Confidence Interval<\/h3>\n\n\n\n<p>CI=x\u02c9\u00b1z\u03b1\/2\u00d7\u03c3n=350\u00b11.96\u00d710064=350\u00b124.5CI = \\bar{x} \\pm z_{\\alpha\/2} \\times \\frac{\\sigma}{\\sqrt{n}} = 350 \\pm 1.96 \\times \\frac{100}{\\sqrt{64}} = 350 \\pm 24.5 \u21d2(325.5,374.5)\\Rightarrow (325.5, 374.5)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">d. Comparison and Conclusion<\/h3>\n\n\n\n<p>From (a), we rejected the null hypothesis, indicating evidence that the mean is different from 375 hours. The confidence interval (325.5 to 374.5) from (c) does <strong>not<\/strong> include 375, which supports this conclusion. Both methods point to the same result: the true mean life of the light bulbs is <strong>significantly different<\/strong> from 375 hours. The company should investigate possible causes for the reduced average lifespan.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner6-143.jpeg\" alt=\"\" class=\"wp-image-221564\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 350 hours.a. At the 0.05 level of significance, [&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-221563","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221563","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=221563"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221563\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221563"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221563"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}