{"id":167935,"date":"2024-11-16T12:15:13","date_gmt":"2024-11-16T12:15:13","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=167935"},"modified":"2024-11-16T12:15:15","modified_gmt":"2024-11-16T12:15:15","slug":"bicycling-the-worlds-leading-cycling-magazine-reviews-hundreds-of-bicycles-throughout-the-year","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/16\/bicycling-the-worlds-leading-cycling-magazine-reviews-hundreds-of-bicycles-throughout-the-year\/","title":{"rendered":"Bicycling, the world&#8217;s leading cycling magazine, reviews hundreds of bicycles throughout the year."},"content":{"rendered":"\n<p>Bicycling, the world&#8217;s leading cycling magazine, reviews hundreds of bicycles throughout the year. Their &#8220;&#8221;Road-Race&#8221;&#8221; category contains reviews of bikes used by riders primarily interested in racing. one of the most important factors in selecting a bike for racing is the weight of the bike. the following data show the weight (pounds) and price ($) for 10 racing bikes reviewed by the magazine. Brand Weight (pound) Price ($)<br>A 17.8 2100<br>B 16.1 6250<br>C 14.9 8370<br>D 15.9 6200<br>E 17.2 4000<br>F 13.1 8600<br>G 16.2 6000<br>H 17.1 2580<br>I 17.6 3400<br>J 14.1 8000<br>Use the data to develop an estimated regression equation that could be used to estimate the price for the bike given the weight. (hint: y on x)<\/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 develop the estimated regression equation for predicting the price of a racing bike based on its weight, we need to use the formula for the least squares regression line, which is:y=b0+b1xy = b_0 + b_1 xy=b0\u200b+b1\u200bx<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>yyy is the dependent variable (price in this case),<\/li>\n\n\n\n<li>xxx is the independent variable (weight in this case),<\/li>\n\n\n\n<li>b1b_1b1\u200b is the slope of the line (which represents the change in price for each unit increase in weight),<\/li>\n\n\n\n<li>b0b_0b0\u200b is the y-intercept of the line.<\/li>\n<\/ul>\n\n\n\n<p>We can find the slope (b1b_1b1\u200b) and y-intercept (b0b_0b0\u200b) using the following formulas:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>b1=n\u2211(xy)\u2212\u2211x\u2211yn\u2211x2\u2212(\u2211x)2b_1 = \\frac{n \\sum{(xy)} &#8211; \\sum{x} \\sum{y}}{n \\sum{x^2} &#8211; (\\sum{x})^2}b1\u200b=n\u2211x2\u2212(\u2211x)2n\u2211(xy)\u2212\u2211x\u2211y\u200b<\/li>\n\n\n\n<li>b0=\u2211y\u2212b1\u2211xnb_0 = \\frac{\\sum{y} &#8211; b_1 \\sum{x}}{n}b0\u200b=n\u2211y\u2212b1\u200b\u2211x\u200b<\/li>\n<\/ul>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>nnn is the number of data points,<\/li>\n\n\n\n<li>xxx represents the weight of the bikes,<\/li>\n\n\n\n<li>yyy represents the price of the bikes,<\/li>\n\n\n\n<li>\u2211\\sum\u2211 represents the sum of the corresponding values.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-Step Calculation:<\/h3>\n\n\n\n<p>We are given the following data for the 10 bikes:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Brand<\/th><th>Weight (pounds) xxx<\/th><th>Price ($) yyy<\/th><\/tr><\/thead><tbody><tr><td>A<\/td><td>17.8<\/td><td>2100<\/td><\/tr><tr><td>B<\/td><td>16.1<\/td><td>6250<\/td><\/tr><tr><td>C<\/td><td>14.9<\/td><td>8370<\/td><\/tr><tr><td>D<\/td><td>15.9<\/td><td>6200<\/td><\/tr><tr><td>E<\/td><td>17.2<\/td><td>4000<\/td><\/tr><tr><td>F<\/td><td>13.1<\/td><td>8600<\/td><\/tr><tr><td>G<\/td><td>16.2<\/td><td>6000<\/td><\/tr><tr><td>H<\/td><td>17.1<\/td><td>2580<\/td><\/tr><tr><td>I<\/td><td>17.6<\/td><td>3400<\/td><\/tr><tr><td>J<\/td><td>14.1<\/td><td>8000<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>We need to compute the necessary sums for the formula:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2211x\\sum{x}\u2211x, \u2211y\\sum{y}\u2211y, \u2211xy\\sum{xy}\u2211xy, \u2211x2\\sum{x^2}\u2211x2<\/li>\n<\/ul>\n\n\n\n<p>First, let&#8217;s compute these sums:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Sum of weights (\u2211x\\sum{x}\u2211x)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>17.8+16.1+14.9+15.9+17.2+13.1+16.2+17.1+17.6+14.1=150.917.8 + 16.1 + 14.9 + 15.9 + 17.2 + 13.1 + 16.2 + 17.1 + 17.6 + 14.1 = 150.917.8+16.1+14.9+15.9+17.2+13.1+16.2+17.1+17.6+14.1=150.9<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li><strong>Sum of prices (\u2211y\\sum{y}\u2211y)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>2100+6250+8370+6200+4000+8600+6000+2580+3400+8000=561002100 + 6250 + 8370 + 6200 + 4000 + 8600 + 6000 + 2580 + 3400 + 8000 = 561002100+6250+8370+6200+4000+8600+6000+2580+3400+8000=56100<\/p>\n\n\n\n<ol start=\"3\" class=\"wp-block-list\">\n<li><strong>Sum of products of weight and price (\u2211xy\\sum{xy}\u2211xy)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>(17.8\u00d72100)+(16.1\u00d76250)+(14.9\u00d78370)+(15.9\u00d76200)+(17.2\u00d74000)+(13.1\u00d78600)+(16.2\u00d76000)+(17.1\u00d72580)+(17.6\u00d73400)+(14.1\u00d78000)(17.8 \\times 2100) + (16.1 \\times 6250) + (14.9 \\times 8370) + (15.9 \\times 6200) + (17.2 \\times 4000) + (13.1 \\times 8600) + (16.2 \\times 6000) + (17.1 \\times 2580) + (17.6 \\times 3400) + (14.1 \\times 8000)(17.8\u00d72100)+(16.1\u00d76250)+(14.9\u00d78370)+(15.9\u00d76200)+(17.2\u00d74000)+(13.1\u00d78600)+(16.2\u00d76000)+(17.1\u00d72580)+(17.6\u00d73400)+(14.1\u00d78000) =37380+100750+124413+98780+68800+112660+97200+44118+59840+112800=703741= 37380 + 100750 + 124413 + 98780 + 68800 + 112660 + 97200 + 44118 + 59840 + 112800 = 703741=37380+100750+124413+98780+68800+112660+97200+44118+59840+112800=703741<\/p>\n\n\n\n<ol start=\"4\" class=\"wp-block-list\">\n<li><strong>Sum of squares of weights (\u2211x2\\sum{x^2}\u2211x2)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<p>(17.82)+(16.12)+(14.92)+(15.92)+(17.22)+(13.12)+(16.22)+(17.12)+(17.62)+(14.12)(17.8^2) + (16.1^2) + (14.9^2) + (15.9^2) + (17.2^2) + (13.1^2) + (16.2^2) + (17.1^2) + (17.6^2) + (14.1^2)(17.82)+(16.12)+(14.92)+(15.92)+(17.22)+(13.12)+(16.22)+(17.12)+(17.62)+(14.12) =317.64+259.21+222.01+252.81+295.84+171.61+262.44+292.41+309.76+198.81=2639.54= 317.64 + 259.21 + 222.01 + 252.81 + 295.84 + 171.61 + 262.44 + 292.41 + 309.76 + 198.81 = 2639.54=317.64+259.21+222.01+252.81+295.84+171.61+262.44+292.41+309.76+198.81=2639.54<\/p>\n\n\n\n<p>Now, let&#8217;s substitute these sums into the formulas:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>b1=10\u00d7703741\u2212150.9\u00d75610010\u00d72639.54\u2212(150.9)2=7037410\u2212846249026395.4\u221222770.81=\u221214250803624.59\u2248\u2212393.22b_1 = \\frac{10 \\times 703741 &#8211; 150.9 \\times 56100}{10 \\times 2639.54 &#8211; (150.9)^2} = \\frac{7037410 &#8211; 8462490}{26395.4 &#8211; 22770.81} = \\frac{-1425080}{3624.59} \\approx -393.22b1\u200b=10\u00d72639.54\u2212(150.9)210\u00d7703741\u2212150.9\u00d756100\u200b=26395.4\u221222770.817037410\u22128462490\u200b=3624.59\u22121425080\u200b\u2248\u2212393.22<\/li>\n\n\n\n<li>b0=56100\u2212(\u2212393.22\u00d7150.9)10=56100+59369.9910=115469.9910\u224811546.99b_0 = \\frac{56100 &#8211; (-393.22 \\times 150.9)}{10} = \\frac{56100 + 59369.99}{10} = \\frac{115469.99}{10} \\approx 11546.99b0\u200b=1056100\u2212(\u2212393.22\u00d7150.9)\u200b=1056100+59369.99\u200b=10115469.99\u200b\u224811546.99<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Estimated Regression Equation:<\/h3>\n\n\n\n<p>Thus, the regression equation for estimating the price based on weight is:y=11546.99\u2212393.22xy = 11546.99 &#8211; 393.22xy=11546.99\u2212393.22x<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This equation suggests that for each additional pound in weight, the price of the bike decreases by approximately $393.22. The y-intercept of 11546.99 indicates the estimated price when the weight of the bike is zero, though this is not practically meaningful for this scenario, as no bike weighs zero pounds.<\/p>\n\n\n\n<p>This regression model can now be used to predict the price of a racing bike given its weight. For example, if a bike weighs 16 pounds, the estimated price would be:y=11546.99\u2212393.22(16)=11546.99\u22126285.52=5261.47y = 11546.99 &#8211; 393.22(16) = 11546.99 &#8211; 6285.52 = 5261.47y=11546.99\u2212393.22(16)=11546.99\u22126285.52=5261.47<\/p>\n\n\n\n<p>Therefore, the estimated price for a 16-pound bike is approximately $5261.47.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bicycling, the world&#8217;s leading cycling magazine, reviews hundreds of bicycles throughout the year. Their &#8220;&#8221;Road-Race&#8221;&#8221; category contains reviews of bikes used by riders primarily interested in racing. one of the most important factors in selecting a bike for racing is the weight of the bike. the following data show the weight (pounds) and price ($) [&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-167935","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167935","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=167935"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167935\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=167935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=167935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=167935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}