{"id":197039,"date":"2025-03-06T13:19:21","date_gmt":"2025-03-06T13:19:21","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=197039"},"modified":"2025-03-06T13:19:24","modified_gmt":"2025-03-06T13:19:24","slug":"a-14-yr-old-boy-is-66-inches-tall","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/06\/a-14-yr-old-boy-is-66-inches-tall\/","title":{"rendered":"A 14 yr old boy is 66 inches tall"},"content":{"rendered":"\n<p>A 14 yr old boy is 66 inches tall. His 6 yr old brother is 47 inches tall. Assume the heights of boys are normally distributed.<\/p>\n\n\n\n<p>Age: 6 Mean Height: 45.8 inches Standard Deviation: 1.4<\/p>\n\n\n\n<p>Age: 14 Mean Height: 64.3 inches Standard Deviation: 2.8<\/p>\n\n\n\n<p>a) Which boy is taller for his age? Explain.<\/p>\n\n\n\n<p>b) If there are 11 other boys in the 6 yr old&#8217;s class at school, how many of them would you expect to be taller than he is?<\/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>\ue200i\ue202turn0image0\ue202turn0image2\ue202turn0image6\ue202turn0image11\ue201To assess the heights of the 14-year-old and 6-year-old boys relative to their age groups, we can calculate their z-scores. A z-score indicates how many standard deviations a particular value is from the mean. The formula for the z-score is:<\/p>\n\n\n\n<p>[ z = \\frac{X &#8211; \\mu}{\\sigma} ]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( X ) is the individual&#8217;s height,<\/li>\n\n\n\n<li>( \\mu ) is the mean height for that age group,<\/li>\n\n\n\n<li>( \\sigma ) is the standard deviation for that age group.<\/li>\n<\/ul>\n\n\n\n<p><strong>a) Which boy is taller for his age?<\/strong><\/p>\n\n\n\n<p><em>For the 14-year-old boy:<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Height (( X )) = 66 inches<\/li>\n\n\n\n<li>Mean height (( \\mu )) = 64.3 inches<\/li>\n\n\n\n<li>Standard deviation (( \\sigma )) = 2.8 inches<\/li>\n<\/ul>\n\n\n\n<p>Calculating the z-score:<\/p>\n\n\n\n<p>[ z = \\frac{66 &#8211; 64.3}{2.8} = \\frac{1.7}{2.8} \\approx 0.607 ]<\/p>\n\n\n\n<p>This z-score of approximately 0.607 means the 14-year-old boy&#8217;s height is about 0.607 standard deviations above the mean for his age group.<\/p>\n\n\n\n<p><em>For the 6-year-old boy:<\/em><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Height (( X )) = 47 inches<\/li>\n\n\n\n<li>Mean height (( \\mu )) = 45.8 inches<\/li>\n\n\n\n<li>Standard deviation (( \\sigma )) = 1.4 inches<\/li>\n<\/ul>\n\n\n\n<p>Calculating the z-score:<\/p>\n\n\n\n<p>[ z = \\frac{47 &#8211; 45.8}{1.4} = \\frac{1.2}{1.4} \\approx 0.857 ]<\/p>\n\n\n\n<p>This z-score of approximately 0.857 indicates the 6-year-old boy&#8217;s height is about 0.857 standard deviations above the mean for his age group.<\/p>\n\n\n\n<p><strong>Conclusion:<\/strong> The 6-year-old boy has a higher z-score (0.857) compared to the 14-year-old boy (0.607), indicating that, relative to their respective age groups, the 6-year-old is taller for his age.\ue206<\/p>\n\n\n\n<p><strong>b) If there are 11 other boys in the 6-year-old&#8217;s class at school, how many of them would you expect to be taller than he is?<\/strong><\/p>\n\n\n\n<p>A z-score of 0.857 corresponds to a percentile rank of approximately 80.5%. This means the 6-year-old boy is taller than about 80.5% of boys his age. Consequently, approximately 19.5% of boys his age are taller than he is.<\/p>\n\n\n\n<p>In a class with 11 other boys:<\/p>\n\n\n\n<p>[ \\text{Number of boys taller} = 11 \\times 0.195 \\approx 2.145 ]<\/p>\n\n\n\n<p>Rounding to the nearest whole number, we would expect about 2 of the 11 other boys to be taller than the 6-year-old boy.<\/p>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>The z-score is a statistical measurement that describes a value&#8217;s position relative to the mean of a group of values, measured in terms of standard deviations. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean. By converting individual heights into z-scores, we can compare how each boy&#8217;s height relates to his age group&#8217;s average height, even though the age groups have different means and standard deviations.<\/p>\n\n\n\n<p>Percentiles indicate the relative standing of a value within a dataset. For example, being in the 80.5th percentile means the individual is taller than 80.5% of the population. In this context, knowing the percentile helps estimate how many peers are taller or shorter than the individual.<\/p>\n\n\n\n<p>By applying these statistical tools, we can objectively assess and compare the boys&#8217; heights relative to their age groups and estimate how many peers might be taller or shorter.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A 14 yr old boy is 66 inches tall. His 6 yr old brother is 47 inches tall. Assume the heights of boys are normally distributed. Age: 6 Mean Height: 45.8 inches Standard Deviation: 1.4 Age: 14 Mean Height: 64.3 inches Standard Deviation: 2.8 a) Which boy is taller for his age? Explain. b) If [&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-197039","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197039","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=197039"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197039\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197039"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197039"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}