{"id":149835,"date":"2024-10-07T20:00:28","date_gmt":"2024-10-07T20:00:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=149835"},"modified":"2024-10-07T20:00:32","modified_gmt":"2024-10-07T20:00:32","slug":"a-ferry-will-safely-accommodate-68-tons-of-passenger-cars","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/07\/a-ferry-will-safely-accommodate-68-tons-of-passenger-cars\/","title":{"rendered":"A ferry will safely accommodate 68 tons of passenger cars"},"content":{"rendered":"\n<p>A ferry will safely accommodate 68 tons of passenger cars. Assume that the mean weight of a passenger car is 1.8 tons with standard deviation 0.5 tons. If a random sample of 35 cars are loaded onto the ferry, what is the probability that the maximum safe weight will be exceeded?<\/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>The correct answer is:<mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-3-color\"><strong> P(X>68)=1\u22120.9545\u22480.0455<\/strong><\/mark><\/p>\n\n\n\n<p>To find the probability that the total weight of 35 randomly selected passenger cars exceeds the ferry&#8217;s capacity of 68 tons, we can use the Central Limit Theorem (CLT). The CLT states that, for large enough sample sizes, the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Define Parameters<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean weight of a passenger car (( \\mu )): 1.8 tons<\/li>\n\n\n\n<li>Standard deviation of the weight (( \\sigma )): 0.5 tons<\/li>\n\n\n\n<li>Sample size (( n )): 35 cars<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the Sampling Distribution<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Mean of the Sampling Distribution<\/strong>:<br>[<br>\\mu_{\\bar{x}} = \\mu \\times n = 1.8 \\text{ tons}<br>]<\/li>\n\n\n\n<li><strong>Standard Deviation of the Sampling Distribution<\/strong> (Standard Error):<br>[<br>\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}} = \\frac{0.5}{\\sqrt{35}} \\approx 0.0844 \\text{ tons}<br>]<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the Total Weight<\/h3>\n\n\n\n<p>To find the probability that the total weight exceeds 68 tons, we first convert this into the equivalent mean for the sample of 35 cars:<br>[<br>\\text{Total weight limit} = 68 \\text{ tons}<br>]<br>[<br>\\text{Mean weight limit per car} = \\frac{68}{35} \\approx 1.943 \\text{ tons}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Standardize<\/h3>\n\n\n\n<p>Next, we standardize this value to find the z-score:<br>[<br>z = \\frac{\\bar{x} &#8211; \\mu_{\\bar{x}}}{\\sigma_{\\bar{x}}} = \\frac{1.943 &#8211; 1.8}{0.0844} \\approx 1.693<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 5: Find the Probability<\/h3>\n\n\n\n<p>Using a z-table or a calculator, we can find the probability corresponding to ( z = 1.693 ). This gives us the area to the left of the z-score.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The area to the left of ( z = 1.693 ) is approximately ( 0.9545 ). Therefore, the probability that the total weight exceeds 68 tons is:<br>[<br>P(X > 68) = 1 &#8211; 0.9545 \\approx 0.0455<br>]<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Thus, the probability that the maximum safe weight of the ferry will be exceeded when loading 35 cars is approximately ( 0.0455 ) or ( 4.55\\% ). This indicates a relatively low likelihood of exceeding the weight limit, meaning the ferry can safely accommodate the given sample of passenger cars with a high probability.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A ferry will safely accommodate 68 tons of passenger cars. Assume that the mean weight of a passenger car is 1.8 tons with standard deviation 0.5 tons. If a random sample of 35 cars are loaded onto the ferry, what is the probability that the maximum safe weight will be exceeded? The Correct answer and [&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-149835","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149835","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=149835"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149835\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149835"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149835"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149835"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}