{"id":196513,"date":"2025-03-05T05:18:04","date_gmt":"2025-03-05T05:18:04","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196513"},"modified":"2025-03-05T05:18:07","modified_gmt":"2025-03-05T05:18:07","slug":"the-length-of-human-pregnancies-is-approximately-normal-with-mean-mu-266-days-and-standard-deviation-sigma-16-days","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/05\/the-length-of-human-pregnancies-is-approximately-normal-with-mean-mu-266-days-and-standard-deviation-sigma-16-days\/","title":{"rendered":"The length of human pregnancies is approximately normal with mean mu = 266 days and standard deviation sigma = 16 days"},"content":{"rendered":"\n<p>The length of human pregnancies is approximately normal with mean mu = 266 days and standard deviation sigma = 16 days. Complete parts (a) through (f).<br>(a) What is the probability that a randomly selected pregnancy lasts less than 261 days?<br>The probability that a randomly selected pregnancy lasts less than 261 days is approximately<br>(Round to four decimal places as needed.)<\/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 randomly selected pregnancy lasts less than 261 days, given that human pregnancy lengths follow a normal distribution with a mean (\u03bc) of 266 days and a standard deviation (\u03c3) of 16 days, we can follow these steps:<\/p>\n\n\n\n<p><strong>1. Standardize the Value to a Z-Score:<\/strong><\/p>\n\n\n\n<p>First, we convert the raw score (261 days) to a standard normal variable (z-score) using the formula:\ue206<\/p>\n\n\n\n<p>[ z = \\frac{X &#8211; \\mu}{\\sigma} ]\ue206<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( X = 261 ) days\ue206<\/li>\n\n\n\n<li>( \\mu = 266 ) days\ue206<\/li>\n\n\n\n<li>( \\sigma = 16 ) days\ue206<\/li>\n<\/ul>\n\n\n\n<p>Calculating the z-score:<\/p>\n\n\n\n<p>[ z = \\frac{261 &#8211; 266}{16} = \\frac{-5}{16} \\approx -0.3125 ]\ue206<\/p>\n\n\n\n<p><strong>2. Find the Corresponding Probability:<\/strong><\/p>\n\n\n\n<p>The z-score of approximately -0.3125 indicates how many standard deviations the value of 261 days is below the mean. To determine the probability associated with this z-score, we can use standard normal distribution tables or computational tools.\ue206<\/p>\n\n\n\n<p>Using a standard normal distribution calculator:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mean (\u03bc):<\/strong> 266 days\ue206<\/li>\n\n\n\n<li><strong>Standard Deviation (\u03c3):<\/strong> 16 days\ue206<\/li>\n\n\n\n<li><strong>Value (X):<\/strong> 261 days\ue206<\/li>\n<\/ul>\n\n\n\n<p>Inputting these values, the calculator provides a cumulative probability of approximately 0.3770.\ue206<\/p>\n\n\n\n<p><strong>3. Interpret the Result:<\/strong><\/p>\n\n\n\n<p>A cumulative probability of 0.3770 means that there is a 37.70% chance that a randomly selected pregnancy will last less than 261 days.\ue206<\/p>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>In a normal distribution, the mean represents the central tendency of the data, and the standard deviation measures the spread or variability around the mean. Approximately 68% of values lie within one standard deviation of the mean (between \u03bc &#8211; \u03c3 and \u03bc + \u03c3), 95% within two standard deviations, and 99.7% within three standard deviations\u2014a concept known as the empirical rule. In this context, a pregnancy length of 261 days is slightly below the mean, falling within the first standard deviation but closer to the lower end.\ue206<\/p>\n\n\n\n<p><strong>Visual Representation:<\/strong><\/p>\n\n\n\n<p>To visualize this, consider a normal distribution curve centered at 266 days. A value of 261 days lies to the left of the mean, and the area under the curve to the left of this value represents the cumulative probability of approximately 37.70%.\ue206<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8c\/Standard_deviation_diagram.svg\/1200px-Standard_deviation_diagram.svg.png\" alt=\"Normal Distribution Curve\"\/><\/figure>\n\n\n\n<p><strong>Conclusion:<\/strong><\/p>\n\n\n\n<p>By standardizing the value and referring to the standard normal distribution, we determined that there is a 37.70% probability that a randomly selected pregnancy will last less than 261 days. This method leverages the properties of the normal distribution to assess probabilities associated with specific values.\ue206<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The length of human pregnancies is approximately normal with mean mu = 266 days and standard deviation sigma = 16 days. Complete parts (a) through (f).(a) What is the probability that a randomly selected pregnancy lasts less than 261 days?The probability that a randomly selected pregnancy lasts less than 261 days is approximately(Round to four [&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-196513","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196513","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=196513"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196513\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}