{"id":178864,"date":"2024-12-28T14:43:02","date_gmt":"2024-12-28T14:43:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=178864"},"modified":"2024-12-28T14:43:04","modified_gmt":"2024-12-28T14:43:04","slug":"the-height-and-age-of-each-child-in-a-random-sample-of-children-was-recorded","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/12\/28\/the-height-and-age-of-each-child-in-a-random-sample-of-children-was-recorded\/","title":{"rendered":"The height and age of each child in a random sample of children was recorded"},"content":{"rendered":"\n<p>The height and age of each child in a random sample of children was recorded. The value of the correlation coefficient between height and age for the children in the sample was . Based on the least-squares regression line created from the data to predict the height of a child based on age, which of the following is a correct statement?<\/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 provide a comprehensive explanation, I need the specific value of the correlation coefficient between height and age in your sample. The correlation coefficient, denoted as ( r ), quantifies the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>( r = 1 )<\/strong>: Perfect positive linear relationship. As one variable increases, the other increases proportionally.<\/li>\n\n\n\n<li><strong>( r = -1 )<\/strong>: Perfect negative linear relationship. As one variable increases, the other decreases proportionally.<\/li>\n\n\n\n<li><strong>( r = 0 )<\/strong>: No linear relationship between the variables.<\/li>\n<\/ul>\n\n\n\n<p>In the context of predicting a child&#8217;s height based on age using the least-squares regression line, the correlation coefficient influences the slope of the regression line. The slope ( m ) of the least-squares regression line is calculated using the formula:<\/p>\n\n\n\n<p>[ m = r \\times \\frac{s_y}{s_x} ]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( s_y ) is the standard deviation of the dependent variable (height),<\/li>\n\n\n\n<li>( s_x ) is the standard deviation of the independent variable (age),<\/li>\n\n\n\n<li>( r ) is the correlation coefficient.<\/li>\n<\/ul>\n\n\n\n<p>This formula indicates that the slope depends on both the strength and direction of the linear relationship between age and height. A positive ( r ) results in a positive slope, meaning that as age increases, height is expected to increase. Conversely, a negative ( r ) leads to a negative slope, indicating that as age increases, height decreases.<\/p>\n\n\n\n<p>The correlation coefficient also affects the coefficient of determination, ( r^2 ), which represents the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher ( r^2 ) value indicates a better fit of the regression model to the data.<\/p>\n\n\n\n<p>Without the specific value of ( r ), I cannot provide a precise statement regarding the relationship between height and age in your sample. If you can provide the value of the correlation coefficient, I can offer a more detailed interpretation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The height and age of each child in a random sample of children was recorded. The value of the correlation coefficient between height and age for the children in the sample was . Based on the least-squares regression line created from the data to predict the height of a child based on age, which of [&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-178864","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/178864","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=178864"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/178864\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=178864"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=178864"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=178864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}