{"id":234207,"date":"2025-06-14T02:18:55","date_gmt":"2025-06-14T02:18:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234207"},"modified":"2025-06-14T02:18:57","modified_gmt":"2025-06-14T02:18:57","slug":"iq-is-normally-distributed-with-a-mean-of-100-and-a-standard-deviation-of-15","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/iq-is-normally-distributed-with-a-mean-of-100-and-a-standard-deviation-of-15\/","title":{"rendered":"IQ is normally distributed with a mean of 100 and a standard deviation of 15"},"content":{"rendered":"\n<p>IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose an individual is randomly chosen. a) Find the probability that the person has an IQ greater than 125. b) Find the probability that the person has an IQ score between 105 and 118. c) What is the IQ score of a person whose percentile rank is at the 75th percentile, P75? d) Use the information from part (c) to fill in the blanks and circle the correct choice in the following statement. <strong>% of the individuals (persons) have an IQ score less than\/more than <em>__<\/em><\/strong> e) &#8220;MENSA&#8221; is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the &#8220;MENSA&#8221; organization.<\/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 solve the questions, we use the <strong>normal distribution<\/strong>, which is symmetric and bell-shaped. Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean (\u03bc) = 100<\/li>\n\n\n\n<li>Standard deviation (\u03c3) = 15<br>Let IQ be represented by the variable X\u223cN(100,152)X \\sim N(100, 15^2)X\u223cN(100,152).<\/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\"><strong>a) P(X &gt; 125)<\/strong><\/h3>\n\n\n\n<p><strong>Step 1: Convert to z-score<\/strong>z=X\u2212\u03bc\u03c3=125\u221210015=2515\u22481.67z = \\frac{X &#8211; \\mu}{\\sigma} = \\frac{125 &#8211; 100}{15} = \\frac{25}{15} \\approx 1.67z=\u03c3X\u2212\u03bc\u200b=15125\u2212100\u200b=1525\u200b\u22481.67<\/p>\n\n\n\n<p><strong>Step 2: Find P(Z &gt; 1.67)<\/strong><br>Using the standard normal distribution table:P(Z&gt;1.67)=1\u2212P(Z\u22641.67)=1\u22120.9525=0.0475P(Z &gt; 1.67) = 1 &#8211; P(Z \\leq 1.67) = 1 &#8211; 0.9525 = 0.0475P(Z&gt;1.67)=1\u2212P(Z\u22641.67)=1\u22120.9525=0.0475<\/p>\n\n\n\n<p><strong>Answer:<\/strong>P(X&gt;125)=0.0475P(X &gt; 125) = 0.0475P(X&gt;125)=0.0475<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>b) P(105 &lt; X &lt; 118)<\/strong><\/h3>\n\n\n\n<p><strong>Step 1: Convert both to z-scores<\/strong>z1=105\u221210015=515=0.33z_1 = \\frac{105 &#8211; 100}{15} = \\frac{5}{15} = 0.33z1\u200b=15105\u2212100\u200b=155\u200b=0.33z2=118\u221210015=1815=1.20z_2 = \\frac{118 &#8211; 100}{15} = \\frac{18}{15} = 1.20z2\u200b=15118\u2212100\u200b=1518\u200b=1.20<\/p>\n\n\n\n<p><strong>Step 2: Find the probabilities<\/strong>P(0.33&lt;Z&lt;1.20)=P(Z&lt;1.20)\u2212P(Z&lt;0.33)P(0.33 &lt; Z &lt; 1.20) = P(Z &lt; 1.20) &#8211; P(Z &lt; 0.33)P(0.33&lt;Z&lt;1.20)=P(Z&lt;1.20)\u2212P(Z&lt;0.33)<\/p>\n\n\n\n<p>From the z-table:P(Z&lt;1.20)=0.8849,P(Z&lt;0.33)=0.6293P(Z &lt; 1.20) = 0.8849,\\quad P(Z &lt; 0.33) = 0.6293P(Z&lt;1.20)=0.8849,P(Z&lt;0.33)=0.6293P(105&lt;X&lt;118)=0.8849\u22120.6293=0.2556P(105 &lt; X &lt; 118) = 0.8849 &#8211; 0.6293 = 0.2556P(105&lt;X&lt;118)=0.8849\u22120.6293=0.2556<\/p>\n\n\n\n<p><strong>Answer:<\/strong>P(105&lt;X&lt;118)=0.2556P(105 &lt; X &lt; 118) = 0.2556P(105&lt;X&lt;118)=0.2556<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>c) Find P\u2087\u2085 (75th percentile)<\/strong><\/h3>\n\n\n\n<p>Use z-tables to find the z-score corresponding to the 75th percentile:P(Z&lt;z)=0.75\u21d2z\u22480.674P(Z &lt; z) = 0.75 \\Rightarrow z \\approx 0.674P(Z&lt;z)=0.75\u21d2z\u22480.674<\/p>\n\n\n\n<p>Convert z to IQ score:X=\u03bc+z\u03c3=100+(0.674)(15)\u2248100+10.11=110.11X = \\mu + z\\sigma = 100 + (0.674)(15) \\approx 100 + 10.11 = 110.11X=\u03bc+z\u03c3=100+(0.674)(15)\u2248100+10.11=110.11<\/p>\n\n\n\n<p><strong>Answer:<\/strong>P75=110.11P_{75} = 110.11P75\u200b=110.11<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>d) Interpret the percentile<\/strong><\/h3>\n\n\n\n<p>From part (c):<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>&#8220;75% of individuals have an IQ score less than 110.11.&#8221;<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p><strong>Answer:<\/strong>75%&nbsp;of&nbsp;the&nbsp;individuals&nbsp;have&nbsp;an&nbsp;IQ&nbsp;score&nbsp;less&nbsp;than&nbsp;110.11\\boxed{75\\%} \\text{ of the individuals have an IQ score } \\boxed{\\text{less than 110.11}}75%\u200b&nbsp;of&nbsp;the&nbsp;individuals&nbsp;have&nbsp;an&nbsp;IQ&nbsp;score&nbsp;less&nbsp;than&nbsp;110.11\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>e) Minimum IQ for MENSA (Top 2%)<\/strong><\/h3>\n\n\n\n<p>Top 2% \u2192 98th percentile<br>Find the z-score for the 98th percentile:P(Z&lt;z)=0.98\u21d2z\u22482.054P(Z &lt; z) = 0.98 \\Rightarrow z \\approx 2.054P(Z&lt;z)=0.98\u21d2z\u22482.054<\/p>\n\n\n\n<p>Convert to IQ score:X=\u03bc+z\u03c3=100+(2.054)(15)\u2248100+30.81=130.81X = \\mu + z\\sigma = 100 + (2.054)(15) \\approx 100 + 30.81 = 130.81X=\u03bc+z\u03c3=100+(2.054)(15)\u2248100+30.81=130.81<\/p>\n\n\n\n<p><strong>Answer:<\/strong>Minimum&nbsp;IQ&nbsp;to&nbsp;qualify&nbsp;for&nbsp;MENSA=130.81\\text{Minimum IQ to qualify for MENSA} = 130.81Minimum&nbsp;IQ&nbsp;to&nbsp;qualify&nbsp;for&nbsp;MENSA=130.81<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Summary<\/strong><\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Part<\/th><th>Result<\/th><\/tr><\/thead><tbody><tr><td>a)<\/td><td>P(X &gt; 125) = 0.0475<\/td><\/tr><tr><td>b)<\/td><td>P(105 &lt; X &lt; 118) = 0.2556<\/td><\/tr><tr><td>c)<\/td><td>75th percentile = 110.11<\/td><\/tr><tr><td>d)<\/td><td>75% have IQ less than 110.11<\/td><\/tr><tr><td>e)<\/td><td>Minimum IQ for MENSA = 130.81<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>These calculations demonstrate how the normal distribution is used to model real-world data such as intelligence scores, allowing for meaningful interpretation of probabilities and percentiles.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-494.jpeg\" alt=\"\" class=\"wp-image-234208\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose an individual is randomly chosen. a) Find the probability that the person has an IQ greater than 125. b) Find the probability that the person has an IQ score between 105 and 118. c) What is the IQ score [&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-234207","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234207","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=234207"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234207\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}