{"id":207648,"date":"2025-04-25T06:18:55","date_gmt":"2025-04-25T06:18:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=207648"},"modified":"2025-04-25T06:18:57","modified_gmt":"2025-04-25T06:18:57","slug":"a-certain-standardized-tests-math-scores-have-a-bell-shaped-distribution-with-a-mean-of-530-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-530-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 530 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 530 and a standard deviation of 105. Complete parts (a) through (c).<br>(a) What percentage of standardized test scores is between 425 and 635?<br>% (Round to one decimal place as needed.)<br>(b) What percentage of standardized test scores is less than 425 or greater than 635?<br>% (Round to one decimal place as needed.)<br>(c) What percentage of standardized test scores is greater than 740?<br>% (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<p>To answer these questions, we\u2019ll use the properties of the normal distribution. A bell-shaped distribution means the data follows a normal distribution, with the mean and standard deviation provided.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mean (\u03bc) = 530<\/strong><\/li>\n\n\n\n<li><strong>Standard deviation (\u03c3) = 105<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">(a) What percentage of standardized test scores is between 425 and 635?<\/h3>\n\n\n\n<p>First, we calculate how many standard deviations away 425 and 635 are from the mean using the Z-score formula:<\/p>\n\n\n\n<p>[<br>Z = \\frac{X &#8211; \\mu}{\\sigma}<br>]<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(X) is the raw score (either 425 or 635),<\/li>\n\n\n\n<li>(\\mu) is the mean (530),<\/li>\n\n\n\n<li>(\\sigma) is the standard deviation (105).<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">For 425:<\/h4>\n\n\n\n<p>[<br>Z_{425} = \\frac{425 &#8211; 530}{105} = \\frac{-105}{105} = -1<br>]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">For 635:<\/h4>\n\n\n\n<p>[<br>Z_{635} = \\frac{635 &#8211; 530}{105} = \\frac{105}{105} = 1<br>]<\/p>\n\n\n\n<p>Now, we need to find the percentage of scores between Z = -1 and Z = 1. This corresponds to the area under the normal curve between these two Z-scores. Using standard Z-tables or a calculator, we find that the area between Z = -1 and Z = 1 is approximately <strong>68%<\/strong>. Therefore, <strong>68%<\/strong> of the scores fall between 425 and 635.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(b) What percentage of standardized test scores is less than 425 or greater than 635?<\/h3>\n\n\n\n<p>This is asking for the percentage of scores outside the range between 425 and 635. From part (a), we know that 68% of the scores are between these values. Since the total percentage of data in a normal distribution is 100%, the percentage outside this range is:<\/p>\n\n\n\n<p>[<br>100\\% &#8211; 68\\% = 32\\%<br>]<\/p>\n\n\n\n<p>Thus, <strong>32%<\/strong> of the scores are either less than 425 or greater than 635.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(c) What percentage of standardized test scores is greater than 740?<\/h3>\n\n\n\n<p>Now, we need to calculate how many standard deviations 740 is above the mean:<\/p>\n\n\n\n<p>[<br>Z_{740} = \\frac{740 &#8211; 530}{105} = \\frac{210}{105} = 2<br>]<\/p>\n\n\n\n<p>The Z-score of 2 corresponds to the area to the left of Z = 2 in a standard normal distribution. From Z-tables or a calculator, the area to the left of Z = 2 is approximately <strong>0.9772<\/strong>, meaning that 97.72% of the data is less than 740. Therefore, the percentage greater than 740 is:<\/p>\n\n\n\n<p>[<br>100\\% &#8211; 97.72\\% = 2.28\\%<br>]<\/p>\n\n\n\n<p>So, <strong>2.3%<\/strong> of the standardized test scores are greater than 740.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a) <strong>68.0%<\/strong> of the scores are between 425 and 635.<\/li>\n\n\n\n<li>(b) <strong>32.0%<\/strong> of the scores are less than 425 or greater than 635.<\/li>\n\n\n\n<li>(c) <strong>2.3%<\/strong> of the scores are greater than 740.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>A certain standardized test&#8217;s math scores have a bell-shaped distribution with a mean of 530 and a standard deviation of 105. Complete parts (a) through (c).(a) What percentage of standardized test scores is between 425 and 635?% (Round to one decimal place as needed.)(b) What percentage of standardized test scores is less than 425 or [&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-207648","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207648","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=207648"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/207648\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=207648"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=207648"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=207648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}