{"id":213095,"date":"2025-05-09T10:55:00","date_gmt":"2025-05-09T10:55:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=213095"},"modified":"2025-05-09T10:55:02","modified_gmt":"2025-05-09T10:55:02","slug":"the-prices-of-a-certain-model-of-new-homes-are-normally-distributed-with-a-mean-of-150000-and-a-standard-deviation-of-900-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/09\/the-prices-of-a-certain-model-of-new-homes-are-normally-distributed-with-a-mean-of-150000-and-a-standard-deviation-of-900-2\/","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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To find the percentage of homes that <strong>exceed \\$152,700<\/strong>, we use the <strong>normal distribution<\/strong> with the given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mean (\u03bc)<\/strong> = \\$150,000<\/li>\n\n\n\n<li><strong>Standard deviation (\u03c3)<\/strong> = \\$900<\/li>\n\n\n\n<li><strong>X<\/strong> = \\$152,700<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 1: Calculate the Z-score<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The Z-score tells us how many standard deviations a value is from the mean. The formula is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>Z = \\frac{X &#8211; \\mu}{\\sigma}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>Z = \\frac{152,700 &#8211; 150,000}{900} = \\frac{2,700}{900} = 3<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 2: Find the area to the left of Z = 3<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using a standard normal distribution table or calculator:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>P(Z &lt; 3) = 0.9987<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This means that 99.87% of the homes cost <strong>less than or equal to<\/strong> \\$152,700.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Step 3: Find the area to the right (homes that exceed \\$152,700)<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>P(Z &gt; 3) = 1 &#8211; P(Z &lt; 3) = 1 &#8211; 0.9987 = 0.0013<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\text{Percentage} = 0.0013 \\times 100 = \\boxed{0.13\\%}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 <strong>Correct Answer: 0.13%<\/strong><\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (300+ words)<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In statistics, when values are <strong>normally distributed<\/strong>, they form a bell-shaped curve where most data points cluster around the mean. In this case, the home prices are centered around <strong>\\$150,000<\/strong>, with a <strong>standard deviation of \\$900<\/strong>, which measures how spread out the prices are.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The question asks for the <strong>percentage of homes priced above \\$152,700<\/strong>. To answer this, we use the concept of <strong>Z-scores<\/strong>, which convert any normal distribution into the <strong>standard normal distribution<\/strong> (mean = 0, standard deviation = 1), allowing us to use standard Z-tables.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By calculating the <strong>Z-score of 3<\/strong>, we found that \\$152,700 is 3 standard deviations above the mean. Statistically, values that lie this far above the mean are <strong>extremely rare<\/strong>. According to the standard normal distribution table, <strong>99.87%<\/strong> of values fall below this point. That leaves <strong>0.13%<\/strong> of the values above \\$152,700.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This small percentage confirms that it\u2019s <strong>highly unusual<\/strong> for a home of this model to exceed that price. In real estate, this could help buyers or sellers understand whether a price is typical or unusually high.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Understanding such probabilities helps in <strong>decision-making<\/strong>, <strong>price evaluation<\/strong>, and <strong>market analysis<\/strong>. For example, a buyer may recognize that a home priced above \\$152,700 is statistically rare and may need further justification (like better location or upgrades). For developers, this insight helps them set realistic pricing strategies.?<\/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 find the percentage of homes that exceed \\$152,700, we use the normal distribution with the given: Step 1: Calculate [&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-213095","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213095","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=213095"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/213095\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=213095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=213095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=213095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}