{"id":188952,"date":"2025-02-08T09:13:27","date_gmt":"2025-02-08T09:13:27","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188952"},"modified":"2025-02-08T09:13:31","modified_gmt":"2025-02-08T09:13:31","slug":"choose-an-american-household-at-random-and-let-the-random-variable-x-be-the-number-of-cars-including-suvs-and-light-trucks-they-own","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/08\/choose-an-american-household-at-random-and-let-the-random-variable-x-be-the-number-of-cars-including-suvs-and-light-trucks-they-own\/","title":{"rendered":"Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own"},"content":{"rendered":"\n

Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The standard deviation of X is sX = 1.08. If many households were selected at random, which of the following would be the best interpretation of the value 1.08?<\/p>\n\n\n\n

(a) The mean number of cars would be about 1.08.<\/p>\n\n\n\n

(b) The number of cars would typically be about 1.08 from the mean.<\/p>\n\n\n\n

(c) The number of cars would be at most 1.08 from the mean.<\/p>\n\n\n\n

(d) The number of cars would be within 1.08 from the mean about 68% of the time.<\/p>\n\n\n\n

(e) The mean number of cars would be about 1.08 from the expected value.<\/p>\n\n\n\n

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

The correct answer is:<\/p>\n\n\n\n

(d) The number of cars would be within 1.08 from the mean about 68% of the time.<\/strong><\/p>\n\n\n\n

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

The standard deviation, ( s_X = 1.08 ), measures how spread out the values of the random variable ( X ) (the number of cars owned by a household) are around the mean. This is a key concept in probability and statistics.<\/p>\n\n\n\n

The standard deviation is a measure of variability or dispersion, which tells us how much individual values deviate from the mean. In the context of a random variable, this means that most of the observed values of ( X ) (the number of cars) are likely to fall within a certain range of the mean. For a normal distribution (or approximately normal distribution, if we assume that the distribution of ( X ) is roughly symmetric), about 68% of the values of ( X ) will fall within one standard deviation from the mean. This is a well-known rule from statistics called the 68-95-99.7 rule<\/strong> (also called the empirical rule), which states:<\/p>\n\n\n\n

    \n
  • 68% of the data<\/strong> will fall within one standard deviation from the mean.<\/li>\n\n\n\n
  • 95% of the data<\/strong> will fall within two standard deviations from the mean.<\/li>\n\n\n\n
  • 99.7% of the data<\/strong> will fall within three standard deviations from the mean.<\/li>\n<\/ul>\n\n\n\n

    In this case, the standard deviation is 1.08, so about 68% of households would own between ( \\mu – 1.08 ) and ( \\mu + 1.08 ) cars, where ( \\mu ) represents the mean number of cars.<\/p>\n\n\n\n

    The other options do not accurately describe the interpretation of the standard deviation:<\/p>\n\n\n\n

      \n
    • (a)<\/strong> is incorrect because the mean number of cars is not necessarily 1.08.<\/li>\n\n\n\n
    • (b)<\/strong> is a vague interpretation and doesn\u2019t correctly describe the concept of standard deviation.<\/li>\n\n\n\n
    • (c)<\/strong> is misleading since the standard deviation doesn’t guarantee that all values are within 1.08 of the mean.<\/li>\n\n\n\n
    • (e)<\/strong> is incorrect because the mean number of cars is not described as being “about 1.08 from the expected value,” and the expected value is the mean.<\/li>\n<\/ul>\n\n\n\n

      Thus, option (d)<\/strong> correctly captures the meaning of the standard deviation in terms of the range within which most data points (households) will fall.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars: The standard deviation of X is sX = 1.08. If many households were selected […]<\/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-188952","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188952","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=188952"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188952\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}