{"id":196731,"date":"2025-03-05T19:25:44","date_gmt":"2025-03-05T19:25:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196731"},"modified":"2025-03-05T19:25:47","modified_gmt":"2025-03-05T19:25:47","slug":"the-amounts-of-time-per-workout-an-athlete-uses-a-stairclimber-are-normally-distributed-with-a-mean-of-25-minutes-and-a-standard-deviation-of-7-minutes","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/05\/the-amounts-of-time-per-workout-an-athlete-uses-a-stairclimber-are-normally-distributed-with-a-mean-of-25-minutes-and-a-standard-deviation-of-7-minutes\/","title":{"rendered":"The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 25 minutes and a standard deviation of 7 minutes"},"content":{"rendered":"\n

The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 25 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 21 minutes, (b) between 25 and 33 minutes, and (c) more than 40 minutes.<\/p>\n\n\n\n

(a) The probability that a randomly selected athlete uses a stairclimber for less than 21 minutes is (Round to four decimal places as needed.)<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Answers:<\/h3>\n\n\n\n

(a) The probability that a randomly selected athlete uses a stairclimber for less than 21 minutes is 0.2839<\/strong> (rounded to four decimal places).
(b) The probability that a randomly selected athlete uses a stairclimber for between 25 and 33 minutes is 0.3735<\/strong>.
(c) The probability that a randomly selected athlete uses a stairclimber for more than 40 minutes is 0.0161<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The problem states that the time an athlete spends on a stairclimber follows a normal distribution with a mean<\/strong> of 25 minutes<\/strong> and a standard deviation<\/strong> of 7 minutes<\/strong>. We use the z-score formula<\/strong> to determine probabilities: Z=X\u2212\u03bc\u03c3Z = \\frac{X – \\mu}{\\sigma}<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • XX is the specific value (e.g., 21, 33, 40 minutes),<\/li>\n\n\n\n
  • \u03bc\\mu (mean) is 25 minutes<\/strong>,<\/li>\n\n\n\n
  • \u03c3\\sigma (standard deviation) is 7 minutes<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    (a) Less than 21 minutes:<\/h4>\n\n\n\n

    Z=21\u2212257=\u221247\u2248\u22120.571Z = \\frac{21 – 25}{7} = \\frac{-4}{7} \\approx -0.571<\/p>\n\n\n\n

    Using the normal distribution table, the probability for Z=\u22120.571Z = -0.571 is 0.2839<\/strong>.<\/p>\n\n\n\n

    (b) Between 25 and 33 minutes:<\/h4>\n\n\n\n

    Z1=25\u2212257=0Z_1 = \\frac{25 – 25}{7} = 0 Z2=33\u2212257=87\u22481.143Z_2 = \\frac{33 – 25}{7} = \\frac{8}{7} \\approx 1.143<\/p>\n\n\n\n

    Using the normal table,
    P(Z1)=0.5000P(Z_1) = 0.5000,
    P(Z2)=0.8735P(Z_2) = 0.8735,
    So, the probability is 0.8735\u22120.5000=0.37350.8735 – 0.5000 = 0.3735.<\/p>\n\n\n\n

    (c) More than 40 minutes:<\/h4>\n\n\n\n

    Z=40\u2212257=157\u22482.143Z = \\frac{40 – 25}{7} = \\frac{15}{7} \\approx 2.143<\/p>\n\n\n\n

    Using the normal table,
    P(Z\u22642.143)=0.9839P(Z \\leq 2.143) = 0.9839,
    so P(X>40)=1\u22120.9839=0.0161P(X > 40) = 1 – 0.9839 = 0.0161.<\/p>\n\n\n\n

    Thus, the probabilities are:<\/p>\n\n\n\n

      \n
    • 28.39%<\/strong> chance of working out less than 21 minutes.<\/li>\n\n\n\n
    • 37.35%<\/strong> chance of working out between 25 and 33 minutes.<\/li>\n\n\n\n
    • 1.61%<\/strong> chance of working out more than 40 minutes.<\/li>\n<\/ul>\n\n\n\n

      The graph visually represents these probabilities using shaded areas under the normal curve.<\/p>\n","protected":false},"excerpt":{"rendered":"

      The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 25 minutes and a standard deviation of 7 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 21 minutes, (b) between 25 and 33 minutes, and (c) more than 40 […]<\/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-196731","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196731","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=196731"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196731\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196731"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196731"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196731"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}