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 < 16.7) = ?<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n 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 1. Calculate the z-score:<\/strong><\/p>\n\n\n\n The z-score represents how many standard deviations a particular value (x) is from the mean and is calculated using the formula:<\/p>\n\n\n\n [ z = \\frac{x – \\mu}{\\sigma} ]<\/p>\n\n\n\n Applying the given values:<\/p>\n\n\n\n [ z = \\frac{16.7 – 15.7}{1.5} ]<\/p>\n\n\n\n This calculation yields a z-score of approximately 0.67.<\/p>\n\n\n\n 2. Determine the cumulative probability:<\/strong><\/p>\n\n\n\n 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 corresponding to this z-score.<\/p>\n\n\n\n Using standard normal distribution tables or a calculator, a z-score of 0.67 corresponds to a cumulative probability of approximately 0.7486. This means there’s a 74.86% chance that a randomly selected value from this distribution is less than 16.7.<\/p>\n\n\n\n Explanation:<\/strong><\/p>\n\n\n\n In a normal distribution, the mean divides the data into two equal halves. The standard deviation measures the average distance of data points from the mean. A z-score transforms a raw score into a standard score, indicating how many standard deviations the raw score is from the mean. By calculating the z-score for 16.7, we standardize it, allowing us to use standard normal distribution tables to find the corresponding cumulative probability.<\/p>\n\n\n\n Understanding z-scores and cumulative probabilities is crucial in statistics for assessing how typical or atypical a particular value is within a given distribution.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 […]<\/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-197719","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197719","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=197719"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197719\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197719"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197719"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}