{"id":251,"date":"2023-02-20T17:14:02","date_gmt":"2023-02-20T17:14:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/line-of-best-fit-apply-it-3\/"},"modified":"2024-02-29T19:01:03","modified_gmt":"2024-02-29T19:01:03","slug":"line-of-best-fit-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/line-of-best-fit-apply-it-3\/","title":{"raw":"Line of Best Fit: Apply It 3","rendered":"Line of Best Fit: Apply It 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize when a linear regression model will fit a given data set.<\/li>\r\n\t<li>Use technology to create scatterplots, find the line of best fit, and find the correlation coefficient.<\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the estimated slope and y-intercept for a linear regression model&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find the estimated slope and [latex]y[\/latex]-intercept for a linear regression model.<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the line of best fit to predict values&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use the line of best fit to predict values.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<section>\r\n<section>\r\n<section>\r\n<section class=\"textbox recall\">We use a <strong>Least Squares Regression analysis<\/strong> to determine the equation of a <strong>line of best fit<\/strong> in order to make predictions based on an existing data set.\r\n\r\n<ul>\r\n\t<li>The line of best is a line that best describes a scatterplot of the data by minimizing the total vertical distances (errors) from all the data points to the line.<\/li>\r\n\t<li>The vertical error associated with each data point (the distance from the point to the line of best fit) is called the <strong>residual<\/strong> of that data point. It lets us know how far off the prediction made by the line of best fit is from the actual observation.<\/li>\r\n\t<li>The <strong>correlation coefficient [latex]r[\/latex]<\/strong> describes the strength and direction of the linear relationship between the two quantitative variables in the data set.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox example\">[reveal-answer q=\"178993\"]See the example question[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"178993\"][ohm2_question hide_question_numbers=1]1176[\/ohm2_question][\/hidden-answer]<br \/>\r\n[videopicker divId=\"tnh-video-picker\" title=\"Linear (Least Square) Regression Analysis\" label=\"Select Instructor\"]<br \/>\r\n[videooption displayName=\"Dr. Pamela E. Harris\" value=\"https:\/\/www.youtube.com\/watch?v=kQecG9VTcjw\"][videooption displayName=\"Dr. Aris Winger\" value=\"https:\/\/www.youtube.com\/watch?v=MBU_2G_A7eA\"] [videooption displayName=\"Dr. Lane Fisher\" value=\"https:\/\/www.youtube.com\/watch?v=FndaoZ4yAXc\"]<br \/>\r\n[\/videopicker]<\/section>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox recall\">The slope-intercept form of a linear equation is commonly expressed in statistics using <strong>[latex]\\hat{y}= a + bx[\/latex]<\/strong>, where [latex]b[\/latex] represents the constant rate of change and [latex]a[\/latex] represents the y-intercept.<\/section>\r\n<\/section>\r\n<section class=\"textbox example\">[reveal-answer q=\"9022778\"]See the example question[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"9022778\"][ohm2_question hide_question_numbers=1]1177[\/ohm2_question][\/hidden-answer]<br \/>\r\n[videopicker divId=\"tnh-video-picker\" title=\"Equation of the Line of Best Fit\" label=\"Select Instructor\"]<br \/>\r\n[videooption displayName=\"Dr. Pamela E. Harris\" value=\"https:\/\/www.youtube.com\/watch?v=ltUvWbHZGIg\"][videooption displayName=\"Dr. Aris Winger\" value=\"https:\/\/www.youtube.com\/watch?v=q6Z0YUzXtOw\"] [videooption displayName=\"Dr. Lane Fisher\" value=\"https:\/\/www.youtube.com\/watch?v=pW2P-glTizE\"]<br \/>\r\n[\/videopicker]<\/section>\r\n<iframe src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"850\"><\/iframe>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\r\n<section>\r\n<section class=\"textbox youChoose\"><img class=\"size-full wp-image-7059 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/12\/14011523\/ChooseDatasetImage.png\" alt=\"\" width=\"876\" height=\"156\" \/><br \/>\r\n[choosedataset divId=\"tnh-choose-dataset\" title=\"Choose Your Own Dataset\" label=\"For this problem, you'll find and interpret the equation for the line of best fit for a data set of your choosing.\u200b\" default=\"Choose a Dataset\"]<br \/>\r\n[datasetoption]<br \/>\r\n[displayname]Pink Tax[\/displayname]<br \/>\r\n[ohmid]2759[\/ohmid]<br \/>\r\n[\/datasetoption]<br \/>\r\n[datasetoption]<br \/>\r\n[displayname]YouTube Gaming[\/displayname]<br \/>\r\n[ohmid]2760[\/ohmid]<br \/>\r\n[\/datasetoption]<br \/>\r\n[datasetoption]<br \/>\r\n[displayname]Homelessness[\/displayname]<br \/>\r\n[ohmid]2761[\/ohmid]<br \/>\r\n[\/datasetoption][\/choosedataset]<span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><\/section>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize when a linear regression model will fit a given data set.<\/li>\n<li>Use technology to create scatterplots, find the line of best fit, and find the correlation coefficient.<\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the estimated slope and y-intercept for a linear regression model&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Find the estimated slope and [latex]y[\/latex]-intercept for a linear regression model.<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the line of best fit to predict values&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Use the line of best fit to predict values.<\/span><\/li>\n<\/ul>\n<\/section>\n<section>\n<section>\n<section>\n<section class=\"textbox recall\">We use a <strong>Least Squares Regression analysis<\/strong> to determine the equation of a <strong>line of best fit<\/strong> in order to make predictions based on an existing data set.<\/p>\n<ul>\n<li>The line of best is a line that best describes a scatterplot of the data by minimizing the total vertical distances (errors) from all the data points to the line.<\/li>\n<li>The vertical error associated with each data point (the distance from the point to the line of best fit) is called the <strong>residual<\/strong> of that data point. It lets us know how far off the prediction made by the line of best fit is from the actual observation.<\/li>\n<li>The <strong>correlation coefficient [latex]r[\/latex]<\/strong> describes the strength and direction of the linear relationship between the two quantitative variables in the data set.<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<section class=\"textbox example\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q178993\">See the example question<\/button><\/p>\n<div id=\"q178993\" class=\"hidden-answer\" style=\"display: none\"><iframe loading=\"lazy\" id=\"ohm1176\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1176&theme=lumen&iframe_resize_id=ohm1176&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/div>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Linear (Least Square) Regression Analysis<\/h3>\n<form><label>Select Instructor:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/kQecG9VTcjw\">Dr. Pamela E. Harris<\/option><option value=\"https:\/\/www.youtube.com\/embed\/MBU_2G_A7eA\">Dr. Aris Winger<\/option><option value=\"https:\/\/www.youtube.com\/embed\/FndaoZ4yAXc\">Dr. Lane Fisher<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/kQecG9VTcjw\" allowfullscreen><\/iframe><\/div>\n<\/section>\n<\/section>\n<section>\n<section class=\"textbox recall\">The slope-intercept form of a linear equation is commonly expressed in statistics using <strong>[latex]\\hat{y}= a + bx[\/latex]<\/strong>, where [latex]b[\/latex] represents the constant rate of change and [latex]a[\/latex] represents the y-intercept.<\/section>\n<\/section>\n<section class=\"textbox example\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9022778\">See the example question<\/button><\/p>\n<div id=\"q9022778\" class=\"hidden-answer\" style=\"display: none\"><iframe loading=\"lazy\" id=\"ohm1177\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1177&theme=lumen&iframe_resize_id=ohm1177&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/div>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Equation of the Line of Best Fit<\/h3>\n<form><label>Select Instructor:<\/label><select name=\"video\"><option value=\"https:\/\/www.youtube.com\/embed\/ltUvWbHZGIg\">Dr. Pamela E. Harris<\/option><option value=\"https:\/\/www.youtube.com\/embed\/q6Z0YUzXtOw\">Dr. Aris Winger<\/option><option value=\"https:\/\/www.youtube.com\/embed\/pW2P-glTizE\">Dr. Lane Fisher<\/option><\/select><\/form>\n<div class=\"videoContainer\"><iframe src=\"https:\/\/www.youtube.com\/embed\/ltUvWbHZGIg\" allowfullscreen><\/iframe><\/div>\n<\/section>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" width=\"100%\" height=\"850\"><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/linear_regression\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n<section>\n<section class=\"textbox youChoose\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-7059 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/12\/14011523\/ChooseDatasetImage.png\" alt=\"\" width=\"876\" height=\"156\" \/><\/p>\n<div id=\"tnh-choose-dataset\" class=\"chooseDataset\">\n<h3>Choose Your Own Dataset<\/h3>\n<h4>For this problem, you'll find and interpret the equation for the line of best fit for a data set of your choosing. <\/h4>\n<form><select name=\"dataset\"><option value=\"\">Choose a Dataset<\/option><option value=\"2759\">Pink Tax<\/option><option value=\"2760\">YouTube Gaming<\/option><option value=\"2761\">Homelessness<\/option><\/select><\/form>\n<div class=\"ohmContainer\"><\/div>\n<\/p><\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><\/section>\n<\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":225,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":[{"divId":"tnh-video-picker","title":"Linear (Least Square) Regression Analysis","label":"Select Instructor","video_collection":[{"displayName":"Dr. Pamela E. 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