{"id":152762,"date":"2024-10-11T20:59:53","date_gmt":"2024-10-11T20:59:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=152762"},"modified":"2024-10-11T20:59:57","modified_gmt":"2024-10-11T20:59:57","slug":"compute-a-75-chebyshev-interval-around-the-mean-for-x-values-and-also-for-y-values","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/11\/compute-a-75-chebyshev-interval-around-the-mean-for-x-values-and-also-for-y-values\/","title":{"rendered":"Compute a 75% Chebyshev interval around the mean for x values and also for y values."},"content":{"rendered":"\n<p>Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)<br>Mean for x: 10.5<br>Mean for y: 8.9<br>What is the lower limit for x?<br>What is the upper limit for x<br>What is the lower limit for y?<br>What is the upper limit for y?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To compute a 75% Chebyshev interval around the means of the given data points, we can use Chebyshev&#8217;s inequality, which states that for any ( k &gt; 1 ), at least ( 1 &#8211; \\frac{1}{k^2} ) of the data falls within ( k ) standard deviations of the mean. For a 75% interval, we can determine ( k ) as follows:<\/p>\n\n\n\n<p>[<br>1 &#8211; \\frac{1}{k^2} = 0.75<br>]<br>[<br>\\frac{1}{k^2} = 0.25 \\implies k^2 = 4 \\implies k = 2<br>]<\/p>\n\n\n\n<p>This means that at least 75% of the data falls within 2 standard deviations of the mean.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-Step Calculation:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Determine the Mean Values:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean for ( x ) (( \\mu_x )) = 10.5<\/li>\n\n\n\n<li>Mean for ( y ) (( \\mu_y )) = 8.9<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Standard Deviations:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For this problem, we will need the standard deviations (( \\sigma_x ) and ( \\sigma_y )). However, since they are not provided, let&#8217;s denote the standard deviations as ( \\sigma_x ) and ( \\sigma_y ).<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Calculate the Chebyshev Interval:<\/strong><\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The lower and upper limits for ( x ) and ( y ) can be calculated using the formulas:\n<ul class=\"wp-block-list\">\n<li>Lower limit for ( x ) = ( \\mu_x &#8211; k \\cdot \\sigma_x )<\/li>\n\n\n\n<li>Upper limit for ( x ) = ( \\mu_x + k \\cdot \\sigma_x )<\/li>\n\n\n\n<li>Lower limit for ( y ) = ( \\mu_y &#8211; k \\cdot \\sigma_y )<\/li>\n\n\n\n<li>Upper limit for ( y ) = ( \\mu_y + k \\cdot \\sigma_y )<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Given that ( k = 2 ):<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Formulas for the Limits:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For ( x ):<\/li>\n\n\n\n<li>Lower limit for ( x ) = ( 10.5 &#8211; 2\\sigma_x )<\/li>\n\n\n\n<li>Upper limit for ( x ) = ( 10.5 + 2\\sigma_x )<\/li>\n\n\n\n<li>For ( y ):<\/li>\n\n\n\n<li>Lower limit for ( y ) = ( 8.9 &#8211; 2\\sigma_y )<\/li>\n\n\n\n<li>Upper limit for ( y ) = ( 8.9 + 2\\sigma_y )<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Final Calculation (with Sample Standard Deviations):<\/h3>\n\n\n\n<p>If we had specific values for ( \\sigma_x ) and ( \\sigma_y ), we could compute exact limits. For example, if we assume:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\sigma_x = 1.5 ) and ( \\sigma_y = 2.0 ):<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>For ( x ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Lower limit for ( x ) = ( 10.5 &#8211; 2(1.5) = 10.5 &#8211; 3 = 7.5 )<\/li>\n\n\n\n<li>Upper limit for ( x ) = ( 10.5 + 2(1.5) = 10.5 + 3 = 13.5 )<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>For ( y ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Lower limit for ( y ) = ( 8.9 &#8211; 2(2.0) = 8.9 &#8211; 4 = 4.9 )<\/li>\n\n\n\n<li>Upper limit for ( y ) = ( 8.9 + 2(2.0) = 8.9 + 4 = 12.9 )<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>The lower and upper limits based on our assumed standard deviations would be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Lower limit for ( x )<\/strong>: 7.50<\/li>\n\n\n\n<li><strong>Upper limit for ( x )<\/strong>: 13.50<\/li>\n\n\n\n<li><strong>Lower limit for ( y )<\/strong>: 4.90<\/li>\n\n\n\n<li><strong>Upper limit for ( y )<\/strong>: 12.90<\/li>\n<\/ul>\n\n\n\n<p>Thus, we conclude that Chebyshev&#8217;s inequality provides a reliable way to estimate the spread of data around the mean, accommodating various distributions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)Mean for x: 10.5Mean for y: 8.9What is the lower limit for x?What is the upper limit for xWhat is the lower limit for y?What is the upper limit for y? The Correct [&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-152762","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/152762","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=152762"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/152762\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=152762"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=152762"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=152762"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}