{"id":197529,"date":"2025-03-07T10:31:18","date_gmt":"2025-03-07T10:31:18","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=197529"},"modified":"2025-03-07T10:31:20","modified_gmt":"2025-03-07T10:31:20","slug":"a-population-of-values-has-a-normal-distribution-with-%c2%b5-15-7-and-s-1-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/07\/a-population-of-values-has-a-normal-distribution-with-%c2%b5-15-7-and-s-1-5\/","title":{"rendered":"A population of values has a normal distribution with \u00b5 = 15.7 and s = 1.5"},"content":{"rendered":"\n<p>A population of values has a normal distribution with \u00b5 = 15.7 and s = 1.5. You intend to draw a random sample of size n = 18. First calculate z, round it to two (2) decimal places, then use the rounded z-score to determine the required probability accurate to four (4) decimal places. Find the probability that a single randomly selected value is less than 16.7. P(x &lt; 16.7) = ?<\/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 determine the probability that a single randomly selected value from a normally distributed population with a mean (\u00b5) of 15.7 and a standard deviation (\u03c3) of 1.5 is less than 16.7, we can follow these steps:<\/p>\n\n\n\n<p><strong>1. Calculate the Z-score:<\/strong><\/p>\n\n\n\n<p>The Z-score represents how many standard deviations a particular value (X) is away from the mean. It is calculated using the formula:<\/p>\n\n\n\n<p>[ Z = \\frac{X &#8211; \\mu}{\\sigma} ]<\/p>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( X = 16.7 )<\/li>\n\n\n\n<li>( \\mu = 15.7 )<\/li>\n\n\n\n<li>( \\sigma = 1.5 )<\/li>\n<\/ul>\n\n\n\n<p>Substitute these values into the formula:<\/p>\n\n\n\n<p>[ Z = \\frac{16.7 &#8211; 15.7}{1.5} = \\frac{1}{1.5} \\approx 0.6667 ]\ue206<\/p>\n\n\n\n<p>Rounded to two decimal places, the Z-score is:<\/p>\n\n\n\n<p>[ Z \\approx 0.67 ]<\/p>\n\n\n\n<p><strong>2. Determine the probability corresponding to the Z-score:<\/strong><\/p>\n\n\n\n<p>The Z-score of 0.67 indicates that the value 16.7 is 0.67 standard deviations above the mean. To find the probability that a randomly selected value is less than 16.7, we need to determine the cumulative probability associated with a Z-score of 0.67.<\/p>\n\n\n\n<p>Using a standard normal distribution table or a Z-score calculator, we find that the cumulative probability for a Z-score of 0.67 is approximately 0.7486. <\/p>\n\n\n\n<p><strong>3. Interpret the result:<\/strong><\/p>\n\n\n\n<p>A cumulative probability of 0.7486 means that there is a 74.86% chance that a randomly selected value from this population will be less than 16.7.<\/p>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric around the mean. The Z-score standardizes a value, allowing us to determine its position relative to the mean in terms of standard deviations. By converting the raw score to a Z-score and consulting the standard normal distribution, we can find the probability of observing a value less than or equal to the given score.<\/p>\n\n\n\n<p>The shaded area under the curve to the left of Z = 0.67 represents the cumulative probability of approximately 74.86%.<\/p>\n\n\n\n<p><strong>Conclusion:<\/strong><\/p>\n\n\n\n<p>By calculating the Z-score for the value 16.7 and determining the corresponding cumulative probability, we find that there is a 74.86% probability that a single randomly selected value from this normal distribution will be less than 16.7.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A population of values has a normal distribution with \u00b5 = 15.7 and s = 1.5. You intend to draw a random sample of size n = 18. First calculate z, round it to two (2) decimal places, then use the rounded z-score to determine the required probability accurate to four (4) decimal places. Find [&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-197529","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197529","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=197529"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197529\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197529"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197529"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}