{"id":188090,"date":"2025-02-06T10:27:08","date_gmt":"2025-02-06T10:27:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188090"},"modified":"2025-02-06T10:27:10","modified_gmt":"2025-02-06T10:27:10","slug":"medical-researchers-have-developed-a-new-artificial-heart-constructed-primarily-of-titanium-and-plastic","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/06\/medical-researchers-have-developed-a-new-artificial-heart-constructed-primarily-of-titanium-and-plastic\/","title":{"rendered":"Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic"},"content":{"rendered":"\n<p>Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient&#8217;s body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation o= 0.2 hour. Is there evidence to support the claim that mean battery life exceeds 4 hours? Use a= 0.05. <\/p>\n\n\n\n<p>What is the test statistic for the test? Answer rounded-off to 2 decimal places<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine if there&#8217;s evidence that the mean battery life exceeds 4 hours, we perform a one-sample z-test for the population mean. Here&#8217;s how we proceed:<\/p>\n\n\n\n<p><strong>Given Data:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sample mean (( \\bar{x} )) = 4.05 hours- Population mean (( \\mu_0 )) = 4 hours- Population standard deviation (( \\sigma )) = 0.2 hours- Sample size (( n )) = 50- Significance level (( \\alpha )) = 0.05<br><strong>Step 1: State the Hypotheses<\/strong><\/li>\n\n\n\n<li>Null hypothesis (( H_0 )): ( \\mu = 4 ) hours (the mean battery life is 4 hours)- Alternative hypothesis (( H_1 )): ( \\mu > 4 ) hours (the mean battery life exceeds 4 hours)<br><strong>Step 2: Calculate the Test Statistic<\/strong><br>The test statistic for a z-test is calculated using the formula:<\/li>\n<\/ul>\n\n\n\n<p>[ z = \\frac{\\bar{x} &#8211; \\mu_0}{\\frac{\\sigma}{\\sqrt{n}}} ]<\/p>\n\n\n\n<p>Substituting the given values:<\/p>\n\n\n\n<p>[ z = \\frac{4.05 &#8211; 4}{\\frac{0.2}{\\sqrt{50}} ]<\/p>\n\n\n\n<p>Calculating the denominator:<\/p>\n\n\n\n<p>[ \\frac{0.2}{\\sqrt{50}} \\approx 0.028284 ]<\/p>\n\n\n\n<p>Now, calculating the z-value:<\/p>\n\n\n\n<p>[ z = \\frac{0.05}{0.028284} \\approx 1.77 ]<\/p>\n\n\n\n<p><strong>Step 3: Determine the Critical Value<\/strong><br>For a one-tailed test at ( \\alpha = 0.05 ), the critical z-value (( z_{\\alpha} )) is approximately 1.645. This value corresponds to the point beyond which we would reject the null hypothesis.<\/p>\n\n\n\n<p><strong>Step 4: Make a Decision<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the calculated z-value is greater than the critical z-value, we reject the null hypothesis.- If the calculated z-value is less than or equal to the critical z-value, we fail to reject the null hypothesis.<br>In this case, the calculated z-value is 1.77, which is greater than the critical z-value of 1.645. Therefore, we reject the null hypothesis.<\/li>\n<\/ul>\n\n\n\n<p><strong>Conclusion:<\/strong><br>There is sufficient evidence at the 0.05 significance level to support the claim that the mean battery life exceeds 4 hours.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient&#8217;s body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a [&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-188090","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188090","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=188090"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188090\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188090"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188090"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}