{"id":242427,"date":"2025-07-04T07:31:03","date_gmt":"2025-07-04T07:31:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=242427"},"modified":"2025-07-04T07:31:05","modified_gmt":"2025-07-04T07:31:05","slug":"in-the-fall-harvest-are-normally-distributed-with-mean-weight-of-200","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/04\/in-the-fall-harvest-are-normally-distributed-with-mean-weight-of-200\/","title":{"rendered":"In the fall harvest are normally distributed, with mean weight of 200"},"content":{"rendered":"\n<p>&#8216;in the fall harvest are normally distributed, with mean weight of 200 The weights of apples grams and standard deviation of 8 grams What is the percentage of apples between 184 and 216 grams? 12 (a) 50% (0) 682 (c) 47.5% (d) 34% (e) 95%&#8217;<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To solve this, we can use the properties of the normal distribution. Given that the weights of apples follow a normal distribution with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean (\u03bc) = 200 grams<\/li>\n\n\n\n<li>Standard deviation (\u03c3) = 8 grams<\/li>\n<\/ul>\n\n\n\n<p>We need to find the percentage of apples that weigh between 184 grams and 216 grams.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Standardize the values using the Z-score formula<\/h3>\n\n\n\n<p>The Z-score formula is: Z=X\u2212\u03bc\u03c3Z = \\frac{X &#8211; \\mu}{\\sigma}Z=\u03c3X\u2212\u03bc\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>XXX is the data point,<\/li>\n\n\n\n<li>\u03bc\\mu\u03bc is the mean,<\/li>\n\n\n\n<li>\u03c3\\sigma\u03c3 is the standard deviation.<\/li>\n<\/ul>\n\n\n\n<p>Now, calculate the Z-scores for 184 grams and 216 grams.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">For 184 grams:<\/h4>\n\n\n\n<p>Z=184\u22122008=\u2212168=\u22122Z = \\frac{184 &#8211; 200}{8} = \\frac{-16}{8} = -2Z=8184\u2212200\u200b=8\u221216\u200b=\u22122<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">For 216 grams:<\/h4>\n\n\n\n<p>Z=216\u22122008=168=2Z = \\frac{216 &#8211; 200}{8} = \\frac{16}{8} = 2Z=8216\u2212200\u200b=816\u200b=2<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use the Z-table (standard normal distribution table) to find the probabilities<\/h3>\n\n\n\n<p>For Z = -2 and Z = 2, we look up the corresponding probabilities from the Z-table:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The probability for Z = -2 is approximately <strong>0.0228<\/strong>.<\/li>\n\n\n\n<li>The probability for Z = 2 is approximately <strong>0.9772<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Calculate the percentage between 184 and 216 grams<\/h3>\n\n\n\n<p>To find the percentage of apples that fall between 184 grams and 216 grams, we subtract the probability at Z = -2 from the probability at Z = 2: P(184\u2264X\u2264216)=P(Z=2)\u2212P(Z=\u22122)=0.9772\u22120.0228=0.9544P(184 \\leq X \\leq 216) = P(Z = 2) &#8211; P(Z = -2) = 0.9772 &#8211; 0.0228 = 0.9544P(184\u2264X\u2264216)=P(Z=2)\u2212P(Z=\u22122)=0.9772\u22120.0228=0.9544<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Convert to percentage<\/h3>\n\n\n\n<p>To convert this to a percentage: 0.9544\u00d7100=95.44%0.9544 \\times 100 = 95.44\\%0.9544\u00d7100=95.44%<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>The percentage of apples weighing between 184 grams and 216 grams is approximately <strong>95%<\/strong>. Therefore, the correct answer is:<\/p>\n\n\n\n<p><strong>(e) 95%<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-241.jpeg\" alt=\"\" class=\"wp-image-242432\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>&#8216;in the fall harvest are normally distributed, with mean weight of 200 The weights of apples grams and standard deviation of 8 grams What is the percentage of apples between 184 and 216 grams? 12 (a) 50% (0) 682 (c) 47.5% (d) 34% (e) 95%&#8217; The Correct Answer and Explanation is: To solve this, we [&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-242427","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242427","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=242427"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/242427\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=242427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=242427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=242427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}