{"id":266866,"date":"2025-07-23T07:50:53","date_gmt":"2025-07-23T07:50:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266866"},"modified":"2025-07-23T07:50:56","modified_gmt":"2025-07-23T07:50:56","slug":"iq-scores-form-a-normal-distribution-with-a-mean-of-%ce%bc-100-and-a-standard-deviation-of-%cf%83-15","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/iq-scores-form-a-normal-distribution-with-a-mean-of-%ce%bc-100-and-a-standard-deviation-of-%cf%83-15\/","title":{"rendered":"IQ scores form a normal distribution with a mean of \u03bc = 100 and a standard deviation of \u03c3 = 15."},"content":{"rendered":"\n<p>IQ scores form a normal distribution with a mean of \u03bc = 100 and a standard deviation of \u03c3 = 15. What is the probability of obtaining a sample mean greater than M = 103? for a random sample of n = 9 people? for a random sample of n = 25 people? for a random sample of n = 100 people<\/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<p>To find the probability of obtaining a sample mean greater than M=103M = 103M=103 for different sample sizes, we can use the Z-score formula for sample means: Z=M\u2212\u03bc\u03c3nZ = \\frac{M &#8211; \\mu}{\\frac{\\sigma}{\\sqrt{n}}}Z=n\u200b\u03c3\u200bM\u2212\u03bc\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>MMM is the sample mean of interest (103),<\/li>\n\n\n\n<li>\u03bc\\mu\u03bc is the population mean (100),<\/li>\n\n\n\n<li>\u03c3\\sigma\u03c3 is the population standard deviation (15),<\/li>\n\n\n\n<li>nnn is the sample size.<\/li>\n<\/ul>\n\n\n\n<p>We will calculate the Z-score for each sample size and then use the standard normal distribution table to find the corresponding probability.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For n=9n = 9n=9:<\/h3>\n\n\n\n<p>First, calculate the standard error of the mean: SE=\u03c3n=159=153=5SE = \\frac{\\sigma}{\\sqrt{n}} = \\frac{15}{\\sqrt{9}} = \\frac{15}{3} = 5SE=n\u200b\u03c3\u200b=9\u200b15\u200b=315\u200b=5<\/p>\n\n\n\n<p>Now, calculate the Z-score: Z=103\u22121005=35=0.6Z = \\frac{103 &#8211; 100}{5} = \\frac{3}{5} = 0.6Z=5103\u2212100\u200b=53\u200b=0.6<\/p>\n\n\n\n<p>Using the Z-table, the probability corresponding to a Z-score of 0.6 is approximately 0.7257. This gives the probability of obtaining a sample mean less than 103. To find the probability of obtaining a sample mean greater than 103, subtract this value from 1: P(M&gt;103)=1\u22120.7257=0.2743P(M &gt; 103) = 1 &#8211; 0.7257 = 0.2743P(M&gt;103)=1\u22120.7257=0.2743<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For n=25n = 25n=25:<\/h3>\n\n\n\n<p>Calculate the standard error of the mean: SE=\u03c3n=1525=155=3SE = \\frac{\\sigma}{\\sqrt{n}} = \\frac{15}{\\sqrt{25}} = \\frac{15}{5} = 3SE=n\u200b\u03c3\u200b=25\u200b15\u200b=515\u200b=3<\/p>\n\n\n\n<p>Now, calculate the Z-score: Z=103\u22121003=33=1Z = \\frac{103 &#8211; 100}{3} = \\frac{3}{3} = 1Z=3103\u2212100\u200b=33\u200b=1<\/p>\n\n\n\n<p>From the Z-table, the probability corresponding to a Z-score of 1 is approximately 0.8413. To find the probability of obtaining a sample mean greater than 103: P(M&gt;103)=1\u22120.8413=0.1587P(M &gt; 103) = 1 &#8211; 0.8413 = 0.1587P(M&gt;103)=1\u22120.8413=0.1587<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For n=100n = 100n=100:<\/h3>\n\n\n\n<p>Calculate the standard error of the mean: SE=\u03c3n=15100=1510=1.5SE = \\frac{\\sigma}{\\sqrt{n}} = \\frac{15}{\\sqrt{100}} = \\frac{15}{10} = 1.5SE=n\u200b\u03c3\u200b=100\u200b15\u200b=1015\u200b=1.5<\/p>\n\n\n\n<p>Now, calculate the Z-score: Z=103\u22121001.5=31.5=2Z = \\frac{103 &#8211; 100}{1.5} = \\frac{3}{1.5} = 2Z=1.5103\u2212100\u200b=1.53\u200b=2<\/p>\n\n\n\n<p>From the Z-table, the probability corresponding to a Z-score of 2 is approximately 0.9772. To find the probability of obtaining a sample mean greater than 103: P(M&gt;103)=1\u22120.9772=0.0228P(M &gt; 103) = 1 &#8211; 0.9772 = 0.0228P(M&gt;103)=1\u22120.9772=0.0228<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Probabilities:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For n=9n = 9n=9, the probability is 0.2743.<\/li>\n\n\n\n<li>For n=25n = 25n=25, the probability is 0.1587.<\/li>\n\n\n\n<li>For n=100n = 100n=100, the probability is 0.0228.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The probability of obtaining a sample mean greater than 103 decreases as the sample size increases because larger sample sizes result in a smaller standard error. As a result, the sample means cluster more closely around the population mean, making extreme values (like 103) less likely.<\/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-banner6-1625.jpeg\" alt=\"\" class=\"wp-image-266867\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>IQ scores form a normal distribution with a mean of \u03bc = 100 and a standard deviation of \u03c3 = 15. What is the probability of obtaining a sample mean greater than M = 103? for a random sample of n = 9 people? for a random sample of n = 25 people? for 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-266866","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266866","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=266866"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266866\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266866"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266866"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266866"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}