{"id":211329,"date":"2025-05-03T16:52:40","date_gmt":"2025-05-03T16:52:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=211329"},"modified":"2025-05-03T16:52:42","modified_gmt":"2025-05-03T16:52:42","slug":"the-prices-of-a-certain-model-of-new-homes-are-normally-distributed-with-a-mean-of-150000-and-a-standard-deviation-of-900","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/03\/the-prices-of-a-certain-model-of-new-homes-are-normally-distributed-with-a-mean-of-150000-and-a-standard-deviation-of-900\/","title":{"rendered":"The prices of a certain model of new homes are normally distributed with a mean of $150,000 and a standard deviation of $900"},"content":{"rendered":"\n<p>The prices of a certain model of new homes are normally distributed with a mean of $150,000 and a standard deviation of $900. What percentage of homes exceed $152,700?<\/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 what percentage of homes exceed \\$152,700, we need to first standardize the value and then use the standard normal distribution to find the corresponding percentage.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Standardize the Value<\/h3>\n\n\n\n<p>We can standardize the value of \\$152,700 using the formula for the z-score:<\/p>\n\n\n\n<p>$$<br>Z = \\frac{X &#8211; \\mu}{\\sigma}<br>$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$X = 152,700$ (the value we are interested in),<\/li>\n\n\n\n<li>$\\mu = 150,000$ (the mean price of homes),<\/li>\n\n\n\n<li>$\\sigma = 900$ (the standard deviation of home prices).<\/li>\n<\/ul>\n\n\n\n<p>Plugging the values into the formula:<\/p>\n\n\n\n<p>$$<br>Z = \\frac{152,700 &#8211; 150,000}{900} = \\frac{2,700}{900} = 3<br>$$<\/p>\n\n\n\n<p>So, the z-score for \\$152,700 is 3.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Find the Percentage from the Standard Normal Distribution<\/h3>\n\n\n\n<p>Now that we have the z-score, we can use a standard normal distribution table or a calculator to determine the cumulative probability for $Z = 3$. The cumulative probability for a z-score of 3 is approximately 0.99865, which means that about 99.865% of the home prices fall below \\$152,700.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the Percentage Above \\$152,700<\/h3>\n\n\n\n<p>Since the cumulative probability tells us the percentage of homes that cost <strong>less than<\/strong> \\$152,700, the percentage of homes that cost <strong>more<\/strong> than \\$152,700 is the complement of this value:<\/p>\n\n\n\n<p>$$<br>1 &#8211; 0.99865 = 0.00135<br>$$<\/p>\n\n\n\n<p>Thus, the percentage of homes that exceed \\$152,700 is approximately 0.135%.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p><strong>0.135% of homes exceed \\$152,700.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This result shows that the distribution of home prices is tightly clustered around the mean, with most homes priced close to \\$150,000. Only a very small percentage of homes have prices significantly higher than this value, such as those exceeding \\$152,700. This illustrates the properties of a normal distribution, where extreme values (far from the mean) are rare.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The prices of a certain model of new homes are normally distributed with a mean of $150,000 and a standard deviation of $900. What percentage of homes exceed $152,700? The correct answer and explanation is : To determine what percentage of homes exceed \\$152,700, we need to first standardize the value and then use the [&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-211329","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/211329","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=211329"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/211329\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=211329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=211329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=211329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}