{"id":274,"date":"2023-02-20T17:14:20","date_gmt":"2023-02-20T17:14:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/assessing-the-fit-of-a-line-learn-it-1\/"},"modified":"2025-05-11T23:22:04","modified_gmt":"2025-05-11T23:22:04","slug":"assessing-the-fit-of-a-line-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/assessing-the-fit-of-a-line-learn-it-1\/","title":{"raw":"Assessing the Fit of a Line: Learn It 1","rendered":"Assessing the Fit of a Line: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe the connection between the residual and the position of a data point relative to the line of best fit.<\/li>\r\n\t<li>Create and use a residual plot to identify influential points and determine the most appropriate regression model.<\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals&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;}\">Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Analyzing Residuals<\/h2>\r\n<p>We have used scatterplots of data and constructed lines of best fit to describe the relationship in bivariate data. You have learned about the correlation coefficient [latex]r[\/latex]\u00a0and the coefficient of determination [latex]R^{2}[\/latex], which are tools we have for determining whether the line of best fit is a useful model and how well the line fits the data.<\/p>\r\n<p>Another tool we have is the <strong>analysis of residuals<\/strong>. When we fit a line to the data, one thing we are interested in is how similar the linear model\u2019s prediction is to the observed data. In other words, we want to know how closely the model matches the data.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>residuals<\/h3>\r\n<p>The <strong>residual<\/strong> for a data point is the difference between the observed value of the response variable and the linear model\u2019s prediction.<\/p>\r\n<p style=\"text-align: center;\">Residual = observed value \u2013 predicted value<\/p>\r\n<p style=\"text-align: center;\">Residual = [latex]y-\\hat y[\/latex]<\/p>\r\n<p><strong>Vocabulary:<\/strong> The word \u201cresidual\u201d means \u201cleft over\u201d or \u201cremaining.\u201d One way to relate the term \u201cresidual\u201d to the concept above is to think of the residual as the quantity left over that can\u2019t be explained by the linear relationship between the response variable and the explanatory variable.<\/p>\r\n<\/section>\r\n<section class=\"textbox recall\"><strong><strong>Predicted values: <\/strong><\/strong>To calculate the predicted value, input a value of the explanatory variable, [latex]x[\/latex], to get a predicted value of the response variable, [latex]\\hat y[\/latex]. For example, suppose you have the following equation:\r\n\r\n<p style=\"text-align: center;\">[latex]\\hat y=5+3.4x[\/latex].<\/p>\r\n<p>You can calculate the predicted value of the response variable for a value of the explanatory variable [latex]x=6[\/latex] in the following way:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\hat y=5+3.4(6)=5+20.4=25.4[\/latex]<\/p>\r\n<p>Thus, when [latex]x=6[\/latex], the predicted value of [latex]\\hat y[\/latex] will be [latex]25.4[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1335[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1337[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1341[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe the connection between the residual and the position of a data point relative to the line of best fit.<\/li>\n<li>Create and use a residual plot to identify influential points and determine the most appropriate regression model.<\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals&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;}\">Determine the reliability of predictions from the line of best fit using the residuals and standard error of the residuals.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Analyzing Residuals<\/h2>\n<p>We have used scatterplots of data and constructed lines of best fit to describe the relationship in bivariate data. You have learned about the correlation coefficient [latex]r[\/latex]\u00a0and the coefficient of determination [latex]R^{2}[\/latex], which are tools we have for determining whether the line of best fit is a useful model and how well the line fits the data.<\/p>\n<p>Another tool we have is the <strong>analysis of residuals<\/strong>. When we fit a line to the data, one thing we are interested in is how similar the linear model\u2019s prediction is to the observed data. In other words, we want to know how closely the model matches the data.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>residuals<\/h3>\n<p>The <strong>residual<\/strong> for a data point is the difference between the observed value of the response variable and the linear model\u2019s prediction.<\/p>\n<p style=\"text-align: center;\">Residual = observed value \u2013 predicted value<\/p>\n<p style=\"text-align: center;\">Residual = [latex]y-\\hat y[\/latex]<\/p>\n<p><strong>Vocabulary:<\/strong> The word \u201cresidual\u201d means \u201cleft over\u201d or \u201cremaining.\u201d One way to relate the term \u201cresidual\u201d to the concept above is to think of the residual as the quantity left over that can\u2019t be explained by the linear relationship between the response variable and the explanatory variable.<\/p>\n<\/section>\n<section class=\"textbox recall\"><strong><strong>Predicted values: <\/strong><\/strong>To calculate the predicted value, input a value of the explanatory variable, [latex]x[\/latex], to get a predicted value of the response variable, [latex]\\hat y[\/latex]. For example, suppose you have the following equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\hat y=5+3.4x[\/latex].<\/p>\n<p>You can calculate the predicted value of the response variable for a value of the explanatory variable [latex]x=6[\/latex] in the following way:<\/p>\n<p style=\"text-align: center;\">[latex]\\hat y=5+3.4(6)=5+20.4=25.4[\/latex]<\/p>\n<p>Thus, when [latex]x=6[\/latex], the predicted value of [latex]\\hat y[\/latex] will be [latex]25.4[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1335\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1335&theme=lumen&iframe_resize_id=ohm1335&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1337\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1337&theme=lumen&iframe_resize_id=ohm1337&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1341\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1341&theme=lumen&iframe_resize_id=ohm1341&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":28,"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":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/274"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/274\/revisions"}],"predecessor-version":[{"id":6662,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/274\/revisions\/6662"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/225"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/274\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=274"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=274"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=274"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}