{"id":309,"date":"2021-07-14T15:59:11","date_gmt":"2021-07-14T15:59:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/prediction\/"},"modified":"2023-12-05T09:48:19","modified_gmt":"2023-12-05T09:48:19","slug":"prediction","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/prediction\/","title":{"raw":"Using the Least-Squares Regression Equation to Make Predictions","rendered":"Using the Least-Squares Regression Equation to Make Predictions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul>\r\n \t<li>Make a prediction for a given value of the independent value using a regression equation<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nRecall the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">third exam\/final exam example<\/a>\u00a0(example 2).\r\n\r\nWe examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the best-fit line for the final exam grade as a function of the grade on the third exam. We can now use the least-squares regression line for prediction.\r\n\r\nSuppose you want to estimate, or predict, the mean final exam score of statistics students who received 73 on the third exam. The exam scores (x-values) range from 65 to 75. <strong>Since 73 is between the x-values 65 and 75<\/strong>, substitute <em>x<\/em> = 73 into the equation. Then:\r\n<p style=\"text-align: center;\">[latex]\\hat{y}[\/latex] = -173.51 + 4.83(73) = 179.08<\/p>\r\nWe predict that statistics students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse\u00a0the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">third exam\/final exam example<\/a>\u00a0(example 2).\r\n<ol>\r\n \t<li>What would you predict the final exam score to be for a student who scored a 66 on the third exam?<\/li>\r\n \t<li>What would you predict the final exam score to be for a student who scored a 90 on the third exam?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"110010\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"110010\"]\r\n<ol>\r\n \t<li>145.27<\/li>\r\n \t<li>The <em>x<\/em> values in the data are between 65 and 75. Ninety is outside of the domain of the observed <em>x<\/em> values in the data (independent variable), so you cannot reliably predict the final exam score for this student. (Even though it is possible to enter 90 into the equation for <em>x<\/em> and calculate a corresponding <em>y<\/em> value, the <em>y<\/em> value that you get will not be reliable). To understand really how unreliable the prediction can be outside of the observed <em>x<\/em> values observed in the data, make the substitution <em>x\u00a0<\/em>= 90 into the equation. [latex]\\displaystyle\\hat{{y}}=-{173.51}+{4.83}{({90})}={261.19}[\/latex] The final-exam score is predicted to be 261.19. The largest the final-exam score can be is 200.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h4>Note<\/h4>\r\nThe process of predicting inside of the observed <em>x<\/em> values observed in the data is called <strong>interpolation<\/strong>. The process of predicting outside of the observed <em>x <\/em>values observed in the data is called <strong>extrapolation<\/strong>.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nData are collected on the relationship between the number of hours per week practicing a musical instrument and scores on a math test. The line of best fit is as follows:\r\n\r\n[latex]\\displaystyle\\hat{{y}}={72.5}+{2.8}{x}[\/latex]\r\n\r\nWhat would you predict the score on a math test would be for a student who practices a musical instrument for five hours a week?\r\n[reveal-answer q=\"125929\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"125929\"]\r\n\r\n86.5\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul>\n<li>Make a prediction for a given value of the independent value using a regression equation<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>Recall the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">third exam\/final exam example<\/a>\u00a0(example 2).<\/p>\n<p>We examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the best-fit line for the final exam grade as a function of the grade on the third exam. We can now use the least-squares regression line for prediction.<\/p>\n<p>Suppose you want to estimate, or predict, the mean final exam score of statistics students who received 73 on the third exam. The exam scores (x-values) range from 65 to 75. <strong>Since 73 is between the x-values 65 and 75<\/strong>, substitute <em>x<\/em> = 73 into the equation. Then:<\/p>\n<p style=\"text-align: center;\">[latex]\\hat{y}[\/latex] = -173.51 + 4.83(73) = 179.08<\/p>\n<p>We predict that statistics students who earn a grade of 73 on the third exam will earn a grade of 179.08 on the final exam, on average.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use\u00a0the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">third exam\/final exam example<\/a>\u00a0(example 2).<\/p>\n<ol>\n<li>What would you predict the final exam score to be for a student who scored a 66 on the third exam?<\/li>\n<li>What would you predict the final exam score to be for a student who scored a 90 on the third exam?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q110010\">Show Answer<\/span><\/p>\n<div id=\"q110010\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>145.27<\/li>\n<li>The <em>x<\/em> values in the data are between 65 and 75. Ninety is outside of the domain of the observed <em>x<\/em> values in the data (independent variable), so you cannot reliably predict the final exam score for this student. (Even though it is possible to enter 90 into the equation for <em>x<\/em> and calculate a corresponding <em>y<\/em> value, the <em>y<\/em> value that you get will not be reliable). To understand really how unreliable the prediction can be outside of the observed <em>x<\/em> values observed in the data, make the substitution <em>x\u00a0<\/em>= 90 into the equation. [latex]\\displaystyle\\hat{{y}}=-{173.51}+{4.83}{({90})}={261.19}[\/latex] The final-exam score is predicted to be 261.19. The largest the final-exam score can be is 200.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h4>Note<\/h4>\n<p>The process of predicting inside of the observed <em>x<\/em> values observed in the data is called <strong>interpolation<\/strong>. The process of predicting outside of the observed <em>x <\/em>values observed in the data is called <strong>extrapolation<\/strong>.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Data are collected on the relationship between the number of hours per week practicing a musical instrument and scores on a math test. The line of best fit is as follows:<\/p>\n<p>[latex]\\displaystyle\\hat{{y}}={72.5}+{2.8}{x}[\/latex]<\/p>\n<p>What would you predict the score on a math test would be for a student who practices a musical instrument for five hours a week?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125929\">Show Answer<\/span><\/p>\n<div id=\"q125929\" class=\"hidden-answer\" style=\"display: none\">\n<p>86.5<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-309\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Prediction. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-5-prediction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-5-prediction<\/a>. <strong>Project<\/strong>: Introductory Statistics. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prediction\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/12-5-prediction\",\"project\":\"Introductory Statistics\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-309","chapter","type-chapter","status-publish","hentry"],"part":303,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/309\/revisions"}],"predecessor-version":[{"id":4013,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/309\/revisions\/4013"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/303"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}