The heights of women follow a normal distribution with a mean of 64.5 inches and a standard deviation of 2.8 inches. (a) (2 points) What percent of women are taller than 72 inches (6 feet)? <\/p>\n\n\n\n
(b) (4 points) From a SRS of 50 women, what is the probability that their mean height is between 60 inches and 72 inches?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To solve these questions, we’ll use properties of the normal distribution.<\/p>\n\n\n\n We are given:<\/p>\n\n\n\n We need to find the probability that a randomly selected woman is taller than 72 inches. This can be done by calculating the Z-score for 72 inches and then finding the corresponding probability.<\/p>\n\n\n\n The Z-score formula is:<\/p>\n\n\n\n [ Where:<\/p>\n\n\n\n Substituting the given values:<\/p>\n\n\n\n [ Now, we need to find the probability that a woman is taller than 72 inches. Using a standard normal distribution table or a Z-table, we look up the cumulative probability for a Z-score of 2.68. The cumulative probability for a Z-score of 2.68 is approximately 0.9963. This represents the probability of being less than<\/strong> 72 inches.<\/p>\n\n\n\n To find the probability of being greater than<\/strong> 72 inches, we subtract this value from 1:<\/p>\n\n\n\n [ So, approximately 0.37%<\/strong> of women are taller than 72 inches.<\/p>\n\n\n\n When we are dealing with a sample mean, we need to use the standard error of the mean (SEM). The formula for SEM is:<\/p>\n\n\n\n [ Where:<\/p>\n\n\n\n Substituting the values:<\/p>\n\n\n\n [ Now, we want to find the probability that the sample mean height is between 60 and 72 inches. To do this, we first calculate the Z-scores for 60 and 72 inches using the formula:<\/p>\n\n\n\n [ For ( X = 60 ) inches:<\/p>\n\n\n\n [ For ( X = 72 ) inches:<\/p>\n\n\n\n [ Both Z-scores are extremely extreme. The probability corresponding to a Z-score of -11.36 is essentially 0, and the probability for a Z-score of 18.94 is essentially 1. Therefore, the probability that the sample mean height is between 60 and 72 inches is approximately 1<\/strong>.<\/p>\n\n\n\n The heights of women follow a normal distribution with a mean of 64.5 inches and a standard deviation of 2.8 inches. (a) (2 points) What percent of women are taller than 72 inches (6 feet)? (b) (4 points) From a SRS of 50 women, what is the probability that their mean height is between 60 […]<\/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-188940","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188940","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=188940"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188940\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188940"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188940"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}(a) What percent of women are taller than 72 inches (6 feet)?<\/h3>\n\n\n\n
\n
Z = \\frac{X – \\mu}{\\sigma}
]<\/p>\n\n\n\n\n
Z = \\frac{72 – 64.5}{2.8} = \\frac{7.5}{2.8} \\approx 2.68
]<\/p>\n\n\n\n
P(X > 72) = 1 – 0.9963 = 0.0037
]<\/p>\n\n\n\n(b) From a SRS (Simple Random Sample) of 50 women, what is the probability that their mean height is between 60 inches and 72 inches?<\/h3>\n\n\n\n
\\text{SEM} = \\frac{\\sigma}{\\sqrt{n}}
]<\/p>\n\n\n\n\n
\\text{SEM} = \\frac{2.8}{\\sqrt{50}} \\approx \\frac{2.8}{7.071} \\approx 0.396
]<\/p>\n\n\n\n
Z = \\frac{X – \\mu}{\\text{SEM}}
]<\/p>\n\n\n\n
Z_1 = \\frac{60 – 64.5}{0.396} = \\frac{-4.5}{0.396} \\approx -11.36
]<\/p>\n\n\n\n
Z_2 = \\frac{72 – 64.5}{0.396} = \\frac{7.5}{0.396} \\approx 18.94
]<\/p>\n\n\n\nSummary:<\/h3>\n\n\n\n
\n