{"id":270161,"date":"2025-07-25T05:12:04","date_gmt":"2025-07-25T05:12:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=270161"},"modified":"2025-07-25T05:12:06","modified_gmt":"2025-07-25T05:12:06","slug":"a-study-of-12000-able-bodied-male-students-at-the-university-of-illinois-found-that-their-times-for-the-mile-run-were-approximately-normal-with-mean-7-11-minutes-and-standard-deviation-0-74-minute","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/25\/a-study-of-12000-able-bodied-male-students-at-the-university-of-illinois-found-that-their-times-for-the-mile-run-were-approximately-normal-with-mean-7-11-minutes-and-standard-deviation-0-74-minute\/","title":{"rendered":"A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute"},"content":{"rendered":"\n

A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute. Choose a student at random from this group and call his time for the mile Y (a) Write the event “the student could run a mile in less than 6.18 minutes” in terms of the value of the random variable Y Use the symbols “< or “> as appropriate to indicate the bounds on Y (b) What is the probability of the event from part (a)?<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

a) Event in terms of the random variable Y:<\/h3>\n\n\n\n

The random variable YYY represents the time, in minutes, it takes for a randomly selected student to run a mile. The problem is asking for the event where a student’s mile time is less than 6.18 minutes<\/strong>. This event can be written in terms of the random variable YYY as:Y<6.18Y < 6.18Y<6.18<\/p>\n\n\n\n

This means that we are interested in the probability that the time for a randomly selected student is less than 6.18 minutes.<\/p>\n\n\n\n

(b) Probability of the event from part (a):<\/h3>\n\n\n\n

To calculate the probability, we need to use the fact that the distribution of the times is approximately Normal<\/strong> with a mean \u03bc=7.11\\mu = 7.11\u03bc=7.11 minutes and a standard deviation \u03c3=0.74\\sigma = 0.74\u03c3=0.74 minutes.<\/p>\n\n\n\n

To find the probability that Y<6.18Y < 6.18Y<6.18, we first need to standardize the value using the Z-score formula<\/strong>:Z=Y\u2212\u03bc\u03c3Z = \\frac{Y – \\mu}{\\sigma}Z=\u03c3Y\u2212\u03bc\u200b<\/p>\n\n\n\n

Substitute the known values into this formula:Z=6.18\u22127.110.74=\u22120.930.74\u2248\u22121.26Z = \\frac{6.18 – 7.11}{0.74} = \\frac{-0.93}{0.74} \\approx -1.26Z=0.746.18\u22127.11\u200b=0.74\u22120.93\u200b\u2248\u22121.26<\/p>\n\n\n\n

Next, we need to find the probability that Z<\u22121.26Z < -1.26Z<\u22121.26. To do this, we can use the standard normal distribution table (or a calculator with the normal distribution function) to find the cumulative probability associated with a Z-score of -1.26.<\/p>\n\n\n\n

Looking up Z=\u22121.26Z = -1.26Z=\u22121.26 in the standard normal table, we find that the cumulative probability is approximately 0.103<\/strong>.<\/p>\n\n\n\n

Thus, the probability that the student could run a mile in less than 6.18 minutes is:P(Y<6.18)\u22480.103P(Y < 6.18) \\approx 0.103P(Y<6.18)\u22480.103<\/p>\n\n\n\n

Conclusion:<\/h3>\n\n\n\n

The probability that a randomly chosen student at the University of Illinois could run a mile in less than 6.18 minutes is approximately 0.103<\/strong>, or about 10.3%<\/strong>. This means there is about a 10.3% chance that a randomly selected student from this group would run the mile in under 6.18 minutes.<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

A study of 12,000 able-bodied male students at the University of Illinois found that their times for the mile run were approximately Normal with mean 7.11 minutes and standard deviation 0.74 minute. Choose a student at random from this group and call his time for the mile Y (a) Write the event “the student could […]<\/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-270161","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270161","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=270161"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270161\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270161"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270161"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}