{"id":206560,"date":"2025-04-01T11:44:35","date_gmt":"2025-04-01T11:44:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=206560"},"modified":"2025-04-01T11:44:37","modified_gmt":"2025-04-01T11:44:37","slug":"the-mean-height-obtained-from-a-random-sample-of-size-100-is-64-inches","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/01\/the-mean-height-obtained-from-a-random-sample-of-size-100-is-64-inches\/","title":{"rendered":"The mean height obtained from a random sample of size 100 is 64 inches"},"content":{"rendered":"\n<p>The mean height obtained from a random sample of size 100 is 64 inches. The standard deviation of the height distribution of the population is known to be 3 inches. Test the statement that the mean height of the population is 67 inches at 5% level of significance. Also set up 99% limits of the mean height of the population.<\/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<h3 class=\"wp-block-heading\">Step 1: State the Hypotheses<\/h3>\n\n\n\n<p>We are testing whether the mean height of the population is 67 inches. This is a hypothesis test for a population mean.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Null hypothesis (H\u2080): The mean height of the population is 67 inches.<br>[<br>H\u2080: \\mu = 67<br>]<\/li>\n\n\n\n<li>Alternative hypothesis (H\u2081): The mean height of the population is not 67 inches.<br>[<br>H\u2081: \\mu \\neq 67<br>]<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Set Up the Test Statistic<\/h3>\n\n\n\n<p>Since we know the population standard deviation (\u03c3 = 3 inches), we can use the <strong>Z-test<\/strong> to test the hypothesis. The test statistic formula is:<\/p>\n\n\n\n<p>[<br>Z = \\frac{\\bar{X} &#8211; \\mu_0}{\\frac{\\sigma}{\\sqrt{n}}}<br>]<br>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(\\bar{X} = 64) (sample mean),<\/li>\n\n\n\n<li>(\\mu_0 = 67) (hypothesized population mean),<\/li>\n\n\n\n<li>(\\sigma = 3) (population standard deviation),<\/li>\n\n\n\n<li>(n = 100) (sample size).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the Z-Statistic<\/h3>\n\n\n\n<p>Substitute the known values into the Z-formula:<\/p>\n\n\n\n<p>[<br>Z = \\frac{64 &#8211; 67}{\\frac{3}{\\sqrt{100}}} = \\frac{-3}{0.3} = -10<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Determine the Critical Value<\/h3>\n\n\n\n<p>The test is two-tailed at the 5% significance level, so the critical values for the Z-test are \u00b11.96 (from standard Z-tables).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Compare the Test Statistic with the Critical Value<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If (|Z|) is greater than 1.96, we reject the null hypothesis.<\/li>\n\n\n\n<li>In this case, (|Z| = 10), which is much greater than 1.96.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 6: Conclusion<\/h3>\n\n\n\n<p>Since the calculated Z-value of -10 falls outside the acceptance region (-1.96 to +1.96), we <strong>reject<\/strong> the null hypothesis. There is enough evidence to conclude that the mean height of the population is not 67 inches at the 5% level of significance.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 7: Set Up the 99% Confidence Interval for the Population Mean<\/h3>\n\n\n\n<p>The formula for a confidence interval for the population mean when the population standard deviation is known is:<\/p>\n\n\n\n<p>[<br>CI = \\bar{X} \\pm Z_{\\alpha\/2} \\times \\frac{\\sigma}{\\sqrt{n}}<br>]<\/p>\n\n\n\n<p>For a 99% confidence level, the critical value (Z_{\\alpha\/2} = 2.576) (from standard Z-tables). Now, substitute the values:<\/p>\n\n\n\n<p>[<br>CI = 64 \\pm 2.576 \\times \\frac{3}{\\sqrt{100}} = 64 \\pm 2.576 \\times 0.3<br>]<br>[<br>CI = 64 \\pm 0.7728<br>]<br>So, the 99% confidence interval is:<\/p>\n\n\n\n<p>[<br>(63.2272, 64.7728)<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The null hypothesis is rejected, and we conclude that the mean height of the population is not 67 inches.<\/li>\n\n\n\n<li>The 99% confidence interval for the population mean is approximately <strong>(63.23 inches, 64.77 inches)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/04\/image-11.png\" alt=\"\" class=\"wp-image-206562\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The mean height obtained from a random sample of size 100 is 64 inches. The standard deviation of the height distribution of the population is known to be 3 inches. Test the statement that the mean height of the population is 67 inches at 5% level of significance. Also set up 99% limits of the [&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-206560","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/206560","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=206560"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/206560\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=206560"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=206560"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=206560"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}