{"id":219724,"date":"2025-05-26T17:41:09","date_gmt":"2025-05-26T17:41:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=219724"},"modified":"2025-05-26T17:41:12","modified_gmt":"2025-05-26T17:41:12","slug":"where-msr-is-the-mean-square-due-to-regression-and-mse-is-the-mse-ssr-mean-square-error","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/26\/where-msr-is-the-mean-square-due-to-regression-and-mse-is-the-mse-ssr-mean-square-error\/","title":{"rendered":"Where MSR is the mean square due to regression and MSE is the MSE SSR mean square error."},"content":{"rendered":"\n<p>Step 2 MSR The F test statistic is calculated as F =. where MSR is the mean square due to regression and MSE is the MSE SSR mean square error. The MSR is calculated as MSR = where SSR is the sum of squares due to regression p and p is the number of independent variables in the estimated regression equation. The MSE is calculated as SSE MSE = where SSE is the sum of square error, n is the total number of observations used to create n- p &#8211; 1 the estimated regression equation, and p is again the number of independent variables. The values for SST and SSE are given, so SSR must be found. Recall the relationship between the sum of squares. SST = SSR + SSE We are given SST = 1,525 and SSE = 580. Therefore, we can find SSR as follows. SSR = SST &#8211; SSE = 1,525 &#8211; 580 The estimated regression equation given below was based on 27 observations. \u00fd = 25.2 + 5.5x The estimated regression equation has Since there were 27 observations, we have n = independent variable(s), so p =<\/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 problem step by step and calculate the <strong>F-test statistic<\/strong> for the regression, we need to follow these procedures:<\/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 1: Find SSR (Sum of Squares due to Regression)<\/strong><\/h3>\n\n\n\n<p>We are given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Total Sum of Squares (SST) = 1525<\/li>\n\n\n\n<li>Sum of Squares Error (SSE) = 580<\/li>\n<\/ul>\n\n\n\n<p>Using the identity: SSR=SST\u2212SSE=1525\u2212580=945SSR = SST &#8211; SSE = 1525 &#8211; 580 = 945<\/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: Determine p and n<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The estimated regression equation is:<\/li>\n<\/ul>\n\n\n\n<p>y^=25.2+5.5x\\hat{y} = 25.2 + 5.5x<\/p>\n\n\n\n<p>This has <strong>one independent variable<\/strong>, so p=1p = 1.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The number of observations is given as 27, so n=27n = 27<\/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 3: Calculate MSR and MSE<\/strong><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>MSR (Mean Square Regression)<\/strong>:<\/h4>\n\n\n\n<p>MSR=SSRp=9451=945MSR = \\frac{SSR}{p} = \\frac{945}{1} = 945<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>MSE (Mean Square Error)<\/strong>:<\/h4>\n\n\n\n<p>MSE=SSEn\u2212p\u22121=58027\u22121\u22121=58025=23.2MSE = \\frac{SSE}{n &#8211; p &#8211; 1} = \\frac{580}{27 &#8211; 1 &#8211; 1} = \\frac{580}{25} = 23.2<\/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 4: Calculate the F-Statistic<\/strong><\/h3>\n\n\n\n<p>F=MSRMSE=94523.2\u224840.73F = \\frac{MSR}{MSE} = \\frac{945}{23.2} \\approx 40.73<\/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>Final Answer:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>F = 40.73<\/strong><\/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\">\ud83d\udcd8 <strong>Explanation <\/strong><\/h3>\n\n\n\n<p>In linear regression, the <strong>F-test statistic<\/strong> is used to determine whether the regression model provides a better fit to the data than a model with no predictors. It compares the model\u2019s explained variance to the unexplained variance. This comparison is made through the ratio of two quantities: the Mean Square due to Regression (MSR) and the Mean Square Error (MSE).<\/p>\n\n\n\n<p>MSR measures the average variation explained by the model per independent variable, and MSE measures the average variation in the residuals (unexplained part) per degree of freedom left after fitting the model.<\/p>\n\n\n\n<p>To compute the F-statistic:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>We first find the <strong>SSR<\/strong> by subtracting the SSE (unexplained variation) from SST (total variation). This gives us the variation explained by the regression model.<\/li>\n\n\n\n<li>We then divide SSR by the number of predictors (p) to get MSR.<\/li>\n\n\n\n<li>Similarly, we divide SSE by the degrees of freedom left after fitting the model (n \u2212 p \u2212 1) to get MSE.<\/li>\n\n\n\n<li>Finally, dividing MSR by MSE gives the <strong>F-statistic<\/strong>, which tells us whether the explained variance is significantly greater than the unexplained variance.<\/li>\n<\/ol>\n\n\n\n<p>In our case, a large F-value of <strong>40.73<\/strong> suggests that the regression model explains a significant portion of the variance in the response variable, indicating that the independent variable xx contributes meaningfully to the model. This is a strong indication that the predictor has a statistically significant relationship with the response variable.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner7-29.jpeg\" alt=\"\" class=\"wp-image-219725\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Step 2 MSR The F test statistic is calculated as F =. where MSR is the mean square due to regression and MSE is the MSE SSR mean square error. The MSR is calculated as MSR = where SSR is the sum of squares due to regression p and p is the number of independent [&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-219724","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219724","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=219724"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219724\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219724"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219724"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219724"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}