{"id":207651,"date":"2025-04-25T06:21:39","date_gmt":"2025-04-25T06:21:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=207651"},"modified":"2025-04-25T06:21:41","modified_gmt":"2025-04-25T06:21:41","slug":"a-certain-standardized-tests-math-scores-have-a-bell-shaped-distribution-with-a-mean-of-525-and-a-standard-deviation-of-105","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/25\/a-certain-standardized-tests-math-scores-have-a-bell-shaped-distribution-with-a-mean-of-525-and-a-standard-deviation-of-105\/","title":{"rendered":"A certain standardized test&#8217;s math scores have a bell-shaped distribution with a mean of 525 and a standard deviation of 105"},"content":{"rendered":"\n<p>A certain standardized test&#8217;s math scores have a bell-shaped distribution with a mean of 525 and a standard deviation of 105. Complete parts <\/p>\n\n\n\n<p>(a) through (c). (a) What percentage of standardized test scores is between 210 and 840? <\/p>\n\n\n\n<p>99.7% (Round to one decimal place as needed.)<\/p>\n\n\n\n<p> (b) What percentage of standardized test scores is less than 210 or greater than 340? <\/p>\n\n\n\n<p>(Round to one decimal place as needed.)<\/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\">(a) What percentage of standardized test scores is between 210 and 840?<\/h3>\n\n\n\n<p>To determine the percentage of standardized test scores between 210 and 840, we can use the properties of a <strong>bell-shaped (normal) distribution<\/strong>. The mean is 525, and the standard deviation is 105.<\/p>\n\n\n\n<p>We can use the <strong>Empirical Rule<\/strong>, also known as the 68-95-99.7 rule, which states:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>68%<\/strong> of the data falls within 1 standard deviation of the mean.<\/li>\n\n\n\n<li><strong>95%<\/strong> of the data falls within 2 standard deviations of the mean.<\/li>\n\n\n\n<li><strong>99.7%<\/strong> of the data falls within 3 standard deviations of the mean.<\/li>\n<\/ul>\n\n\n\n<p>To find how many standard deviations away 210 and 840 are from the mean:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For 210:<br>[<br>Z = \\frac{210 &#8211; 525}{105} = \\frac{-315}{105} = -3<br>]<\/li>\n\n\n\n<li>For 840:<br>[<br>Z = \\frac{840 &#8211; 525}{105} = \\frac{315}{105} = 3<br>]<\/li>\n<\/ul>\n\n\n\n<p>Thus, 210 is <strong>3 standard deviations below the mean<\/strong>, and 840 is <strong>3 standard deviations above the mean<\/strong>.<\/p>\n\n\n\n<p>According to the <strong>Empirical Rule<\/strong>, <strong>99.7%<\/strong> of the data lies between 3 standard deviations below and 3 standard deviations above the mean. Therefore, the percentage of standardized test scores between 210 and 840 is <strong>99.7%<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(b) What percentage of standardized test scores is less than 210 or greater than 840?<\/h3>\n\n\n\n<p>To find the percentage of scores less than 210 or greater than 840, we need to calculate the area outside the range of 210 and 840.<\/p>\n\n\n\n<p>Since <strong>99.7%<\/strong> of the data is between 210 and 840 (from part (a)), the remaining <strong>0.3%<\/strong> of the data is outside this range. This remaining percentage is split equally into two tails of the distribution.<\/p>\n\n\n\n<p>Thus, each tail has:<br>[<br>\\frac{0.3\\%}{2} = 0.15\\%<br>]<br>This means <strong>0.15%<\/strong> of the data is less than 210, and <strong>0.15%<\/strong> of the data is greater than 840.<\/p>\n\n\n\n<p>Therefore, the total percentage of scores less than 210 or greater than 840 is:<br>[<br>0.15\\% + 0.15\\% = 0.3\\%<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>In a normal distribution, the data is symmetrically distributed around the mean. The <strong>Empirical Rule<\/strong> provides a quick way to estimate the percentage of data within a certain number of standard deviations from the mean. By converting the raw scores (210 and 840) into <strong>z-scores<\/strong> and using the rule, we can easily determine the percentage of data that falls within or outside a specified range.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A certain standardized test&#8217;s math scores have a bell-shaped distribution with a mean of 525 and a standard deviation of 105. Complete parts (a) through (c). (a) What percentage of standardized test scores is between 210 and 840? 99.7% (Round to one decimal place as needed.) (b) What percentage of standardized test scores is less [&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-207651","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207651","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=207651"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207651\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=207651"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=207651"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=207651"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}