{"id":267063,"date":"2025-07-23T09:05:40","date_gmt":"2025-07-23T09:05:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=267063"},"modified":"2025-07-23T09:05:43","modified_gmt":"2025-07-23T09:05:43","slug":"consider-the-following-data-for-a-dependent-variable-y-and-two-independent-variables","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/consider-the-following-data-for-a-dependent-variable-y-and-two-independent-variables\/","title":{"rendered":"Consider the following data for a dependent variable y and two independent variables"},"content":{"rendered":"\n<p>Consider the following data for a dependent variable y and two independent variables, x1 and x2. x1: 30, 47, 24, 51, 40, 51, 75, 36, 59, 77; x2: 13, 11, 18, 17, 5, 19, 8, 12, 14, 17; y: 95, 109, 112, 179, 94, 175, 170, 118, 143, 211 Round your all answers to two decimal places. Enter negative values as negative numbers, if necessary. Develop an estimated regression equation relating y to x2.<\/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 develop the estimated regression equation relating yyy to x2x_2x2\u200b, we can perform a simple linear regression using x2x_2x2\u200b as the independent variable and yyy as the dependent variable. The regression equation has the general form:y=\u03b20+\u03b21x2y = \\beta_0 + \\beta_1 x_2y=\u03b20\u200b+\u03b21\u200bx2\u200b<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>yyy is the dependent variable,<\/li>\n\n\n\n<li>x2x_2x2\u200b is the independent variable,<\/li>\n\n\n\n<li>\u03b20\\beta_0\u03b20\u200b is the intercept,<\/li>\n\n\n\n<li>\u03b21\\beta_1\u03b21\u200b is the coefficient (slope) of x2x_2x2\u200b.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Organize the Data<\/h3>\n\n\n\n<p>Given values for x2x_2x2\u200b and yyy are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x2=[13,11,18,17,5,19,8,12,14,17]x_2 = [13, 11, 18, 17, 5, 19, 8, 12, 14, 17]x2\u200b=[13,11,18,17,5,19,8,12,14,17]<\/li>\n\n\n\n<li>y=[95,109,112,179,94,175,170,118,143,211]y = [95, 109, 112, 179, 94, 175, 170, 118, 143, 211]y=[95,109,112,179,94,175,170,118,143,211]<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Calculate the Required Sums<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Sum of x2x_2x2\u200b:<\/li>\n<\/ol>\n\n\n\n<p>\u2211x2=13+11+18+17+5+19+8+12+14+17=124\\sum x_2 = 13 + 11 + 18 + 17 + 5 + 19 + 8 + 12 + 14 + 17 = 124\u2211x2\u200b=13+11+18+17+5+19+8+12+14+17=124<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>Sum of yyy:<\/li>\n<\/ol>\n\n\n\n<p>\u2211y=95+109+112+179+94+175+170+118+143+211=1406\\sum y = 95 + 109 + 112 + 179 + 94 + 175 + 170 + 118 + 143 + 211 = 1406\u2211y=95+109+112+179+94+175+170+118+143+211=1406<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li>Sum of x22x_2^2&#215;22\u200b (the square of each value in x2x_2x2\u200b):<\/li>\n<\/ol>\n\n\n\n<p>\u2211x22=132+112+182+172+52+192+82+122+142+172=169+121+324+289+25+361+64+144+196+289=1962\\sum x_2^2 = 13^2 + 11^2 + 18^2 + 17^2 + 5^2 + 19^2 + 8^2 + 12^2 + 14^2 + 17^2 = 169 + 121 + 324 + 289 + 25 + 361 + 64 + 144 + 196 + 289 = 1962\u2211x22\u200b=132+112+182+172+52+192+82+122+142+172=169+121+324+289+25+361+64+144+196+289=1962<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li>Sum of x2\u22c5yx_2 \\cdot yx2\u200b\u22c5y (the product of each corresponding pair of x2x_2x2\u200b and yyy):<\/li>\n<\/ol>\n\n\n\n<p>\u2211x2\u22c5y=(13\u00d795)+(11\u00d7109)+(18\u00d7112)+(17\u00d7179)+(5\u00d794)+(19\u00d7175)+(8\u00d7170)+(12\u00d7118)+(14\u00d7143)+(17\u00d7211)=1235+1199+2016+3043+470+3325+1360+1416+2002+3587=17053\\sum x_2 \\cdot y = (13 \\times 95) + (11 \\times 109) + (18 \\times 112) + (17 \\times 179) + (5 \\times 94) + (19 \\times 175) + (8 \\times 170) + (12 \\times 118) + (14 \\times 143) + (17 \\times 211) = 1235 + 1199 + 2016 + 3043 + 470 + 3325 + 1360 + 1416 + 2002 + 3587 = 17053\u2211x2\u200b\u22c5y=(13\u00d795)+(11\u00d7109)+(18\u00d7112)+(17\u00d7179)+(5\u00d794)+(19\u00d7175)+(8\u00d7170)+(12\u00d7118)+(14\u00d7143)+(17\u00d7211)=1235+1199+2016+3043+470+3325+1360+1416+2002+3587=17053<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Compute the Slope (\u03b21\\beta_1\u03b21\u200b) and Intercept (\u03b20\\beta_0\u03b20\u200b)<\/h3>\n\n\n\n<p>Using the formulas for the slope and intercept in simple linear regression:\u03b21=n\u2211x2y\u2212\u2211x2\u2211yn\u2211x22\u2212(\u2211x2)2\\beta_1 = \\frac{n \\sum x_2 y &#8211; \\sum x_2 \\sum y}{n \\sum x_2^2 &#8211; (\\sum x_2)^2}\u03b21\u200b=n\u2211x22\u200b\u2212(\u2211x2\u200b)2n\u2211x2\u200by\u2212\u2211x2\u200b\u2211y\u200b\u03b20=\u2211y\u2212\u03b21\u2211x2n\\beta_0 = \\frac{\\sum y &#8211; \\beta_1 \\sum x_2}{n}\u03b20\u200b=n\u2211y\u2212\u03b21\u200b\u2211x2\u200b\u200b<\/p>\n\n\n\n<p>Where nnn is the number of data points (in this case, n=10n = 10n=10).<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Calculate \u03b21\\beta_1\u03b21\u200b:<\/li>\n<\/ol>\n\n\n\n<p>\u03b21=10(17053)\u2212(124)(1406)10(1962)\u2212(124)2=170530\u221217412419620\u221215376=\u221235944244\u2248\u22120.85\\beta_1 = \\frac{10(17053) &#8211; (124)(1406)}{10(1962) &#8211; (124)^2} = \\frac{170530 &#8211; 174124}{19620 &#8211; 15376} = \\frac{-3594}{4244} \\approx -0.85\u03b21\u200b=10(1962)\u2212(124)210(17053)\u2212(124)(1406)\u200b=19620\u221215376170530\u2212174124\u200b=4244\u22123594\u200b\u2248\u22120.85<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>Calculate \u03b20\\beta_0\u03b20\u200b:<\/li>\n<\/ol>\n\n\n\n<p>\u03b20=1406\u2212(\u22120.85)(124)10=1406+105.410=1511.410\u2248151.14\\beta_0 = \\frac{1406 &#8211; (-0.85)(124)}{10} = \\frac{1406 + 105.4}{10} = \\frac{1511.4}{10} \\approx 151.14\u03b20\u200b=101406\u2212(\u22120.85)(124)\u200b=101406+105.4\u200b=101511.4\u200b\u2248151.14<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Form the Regression Equation<\/h3>\n\n\n\n<p>Now that we have the values for \u03b20\\beta_0\u03b20\u200b and \u03b21\\beta_1\u03b21\u200b, we can write the estimated regression equation:y=151.14\u22120.85x2y = 151.14 &#8211; 0.85 x_2y=151.14\u22120.85&#215;2\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>The estimated regression equation that relates yyy to x2x_2x2\u200b is:y=151.14\u22120.85x2y = 151.14 &#8211; 0.85 x_2y=151.14\u22120.85&#215;2\u200b<\/p>\n\n\n\n<p>This equation can now be used to predict yyy for any given value of x2x_2x2\u200b.<\/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-banner6-1653.jpeg\" alt=\"\" class=\"wp-image-267064\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the following data for a dependent variable y and two independent variables, x1 and x2. x1: 30, 47, 24, 51, 40, 51, 75, 36, 59, 77; x2: 13, 11, 18, 17, 5, 19, 8, 12, 14, 17; y: 95, 109, 112, 179, 94, 175, 170, 118, 143, 211 Round your all answers to two [&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-267063","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/267063","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=267063"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/267063\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=267063"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=267063"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=267063"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}