{"id":3876,"date":"2022-03-15T23:24:49","date_gmt":"2022-03-15T23:24:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3876"},"modified":"2022-06-03T05:25:43","modified_gmt":"2022-06-03T05:25:43","slug":"what-to-know-about-6-e","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/what-to-know-about-6-e\/","title":{"raw":"What to Know About 6.E: Calculating Predicted Values of the Response Variable","rendered":"What to Know About 6.E: Calculating Predicted Values of the Response Variable"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Goals<\/h3>\r\nAt the end of this page, you should feel comfortable performing these skills:\r\n<ul>\r\n \t<li>Use a scatterplot to describe bivariate relationships.<\/li>\r\n \t<li>Approximate predicted values from a scatterplot.<\/li>\r\n \t<li>Calculate predictions using the line of best fit.<\/li>\r\n \t<li>Assess reliability of a prediction calculated using the line of best fit.<\/li>\r\n<\/ul>\r\n<\/div>\r\nFor the upcoming\u00a0<em>Forming Connections<\/em>\u00a0activity, you will need to use technology to make a scatterplot, calculate the line of best fit, and use the line to calculate predicted values of the response variable. Let's walk through a scenario to practice those skills before you dive into that activity. The questions on this page will provide a refresher for the ideas you've learned previously in this module, so it should feel fairly comfortable. As such, there is no video demonstration for these questions. Use this page to assess your understanding before you finish this module.\r\n<h2>Linear Analysis<\/h2>\r\nWhat is the relationship between the size and price of a house in Florida? To answer this question, you will analyze data for 100 houses in Gainesville, Florida that sold in 2003. The data are from the \u201cHouse Prices in FL\u201d dataset available in the <em>DCMP Linear Regression<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/\">https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/<\/a>.\r\n\r\nThe data contain the sizes of the homes (in square feet) and prices of the homes (in thousands of U.S. dollars).\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nThe goal of this analysis is to predict the price of a house based on its size. Identify the explanatory and response variables.\r\n\r\n[reveal-answer q=\"255171\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"255171\"]Which variable will measure the outcome and which will be the variable that drives the outcome?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Visualizing the Relationship<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nUse the tool to make a scatterplot to visualize the relationship between the two variables. Describe the relationship between the two variables. Select all that apply.\r\n<ol>\r\n \t<li>a) Positive<\/li>\r\n \t<li>b) Negative<\/li>\r\n \t<li>c) Quadratic<\/li>\r\n \t<li>d) Linear<\/li>\r\n \t<li>e) No relationship<\/li>\r\n \t<li>f) Weak<\/li>\r\n \t<li>g) Strong<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"605410\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"605410\"]Recall what you know about the shape and spread of a plot of bivariate data.[\/hidden-answer]\r\n\r\n<\/div>\r\nUse the scatterplot from Question 2 to answer Questions 3 and 4. Don't calculate and fit a regression line yet. You'll do that in Question 5.\r\n<h3>Approximating From a Scatterplot<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nSuppose there is a house for sale that has a size of 1,500 square feet. Based on the scatterplot, which of the following is the most likely value for its price?\r\n<ol>\r\n \t<li>a) $100,000<\/li>\r\n \t<li>b) $110,000<\/li>\r\n \t<li>c) $125,000<\/li>\r\n \t<li>d) $150,000<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"715888\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"715888\"]What is the approximate mean price for houses that have a size of 1,500 square feet?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nSuppose you fit a line of best fit for these data. The line would predict the price of a house that has a size of 1,000 square feet to be closest to what value?\r\n<ol>\r\n \t<li>a) $90,000<\/li>\r\n \t<li>b) $100,000<\/li>\r\n \t<li>c) $110,000<\/li>\r\n \t<li>d) $120,000<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"306923\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"306923\"]What is the approximate mean price for houses that have a size of 1,000 square feet?[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>\u00a0The Regression Line Equation<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nUse technology to calculate the line of best fit and write the equation for the line.\r\n\r\n&nbsp;\r\n\r\nPart A: Write the equation using appropriate notation and customize the names of the variables.\r\n\r\n[reveal-answer q=\"564913\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"564913\"]Rather than using [latex]\\hat{y}[\/latex] and [latex]x[\/latex], customize the variables using words or meaningful letters.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: Interpret the slope, in context. Select the best answer.\r\n<p style=\"padding-left: 30px;\">a) For every one square foot increase in the size, there will be an increase of $77.01 in the price of a house in Florida.<\/p>\r\n<p style=\"padding-left: 30px;\">b) For every one square foot increase in the size, we predict an average increase of $77.01 in the price of a house in Florida.<\/p>\r\n<p style=\"padding-left: 30px;\">c) For every $1 increase in the price, there will be an increase of 77.01 square feet in the size of a house in Florida.<\/p>\r\n<p style=\"padding-left: 30px;\">d) For every $1 increase in the price, we predict an average increase of 77.01 square feet in the size of a house in Florida.<\/p>\r\n[reveal-answer q=\"396809\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"396809\"]Recall the template for interpreting slope.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C: Does the intercept have a meaningful interpretation? Select the best answer.\r\n<p style=\"padding-left: 30px;\">a) Yes. At a size of zero square feet, the predicted price of a house in Florida is $9,161.<\/p>\r\n<p style=\"padding-left: 30px;\">b) Yes. At a price of $0, the predicted size of a house in Florida is 9,161 square feet.<\/p>\r\n<p style=\"padding-left: 30px;\">c) No. It does not make sense to have a house that is zero square feet.<\/p>\r\n<p style=\"padding-left: 30px;\">d) No. It does not make sense to have a house that costs $0.<\/p>\r\n[reveal-answer q=\"78978\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"78978\"]What do <em>you\u00a0<\/em>think?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart D: Use the line to calculate the predicted price for a house that is 1,852 square feet. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"76000\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"76000\"]Use the equation of the line to make this calculation.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart E: Use the line to calculate the predicted price for a house that is 2,860 square feet. Round your answer to 2 decimal places.\r\n\r\n[reveal-answer q=\"495551\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"495551\"]Use the equation of the line to make this calculation.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart F: Do you think the model prediction is more reliable for houses that are 1,852 square feet or for those that are 2,860 square feet? Explain.\r\n\r\n[reveal-answer q=\"678859\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"678859\"]Consider the number of observations at a given value of the explanatory variable.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nIn this <em>What to Know\u00a0<\/em>page,\u00a0you practiced the basic skills necessary to perform a linear regression analysis. Let\u2019s summarize these skills.\r\n<ul>\r\n \t<li>In Question 1, you identified the explanatory and response variables for a given scenario.<\/li>\r\n \t<li>In Question 2, you used technology to make a scatterplot.<\/li>\r\n \t<li>In Question 2, you also used a scatterplot to describe bivariate relationships.<\/li>\r\n \t<li>In Questions 3 and 4, you approximated predicted values from a scatterplot.<\/li>\r\n \t<li>In Question 5, Part A, you used technology to calculate a line of best fit.<\/li>\r\n \t<li>In Question 5, Parts B and C, you interpreted the slope and intercept of the line of best fit.<\/li>\r\n \t<li>In Question 5, Parts D through F, you calculated predictions using the line of best fit and assessed reliability of the predictions.<\/li>\r\n<\/ul>\r\nThis page provided a summary and practice of the ideas you've learned in this section (and in [Section 5A] previously). Feel free to return to it any time to practice. Hopefully, it felt comfortable and familiar. Let's move on to <em>Forming Connections<\/em> to put it all together.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Goals<\/h3>\n<p>At the end of this page, you should feel comfortable performing these skills:<\/p>\n<ul>\n<li>Use a scatterplot to describe bivariate relationships.<\/li>\n<li>Approximate predicted values from a scatterplot.<\/li>\n<li>Calculate predictions using the line of best fit.<\/li>\n<li>Assess reliability of a prediction calculated using the line of best fit.<\/li>\n<\/ul>\n<\/div>\n<p>For the upcoming\u00a0<em>Forming Connections<\/em>\u00a0activity, you will need to use technology to make a scatterplot, calculate the line of best fit, and use the line to calculate predicted values of the response variable. Let&#8217;s walk through a scenario to practice those skills before you dive into that activity. The questions on this page will provide a refresher for the ideas you&#8217;ve learned previously in this module, so it should feel fairly comfortable. As such, there is no video demonstration for these questions. Use this page to assess your understanding before you finish this module.<\/p>\n<h2>Linear Analysis<\/h2>\n<p>What is the relationship between the size and price of a house in Florida? To answer this question, you will analyze data for 100 houses in Gainesville, Florida that sold in 2003. The data are from the \u201cHouse Prices in FL\u201d dataset available in the <em>DCMP Linear Regression<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/\">https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/<\/a>.<\/p>\n<p>The data contain the sizes of the homes (in square feet) and prices of the homes (in thousands of U.S. dollars).<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>The goal of this analysis is to predict the price of a house based on its size. Identify the explanatory and response variables.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255171\">Hint<\/span><\/p>\n<div id=\"q255171\" class=\"hidden-answer\" style=\"display: none\">Which variable will measure the outcome and which will be the variable that drives the outcome?<\/div>\n<\/div>\n<\/div>\n<h3>Visualizing the Relationship<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>Use the tool to make a scatterplot to visualize the relationship between the two variables. Describe the relationship between the two variables. Select all that apply.<\/p>\n<ol>\n<li>a) Positive<\/li>\n<li>b) Negative<\/li>\n<li>c) Quadratic<\/li>\n<li>d) Linear<\/li>\n<li>e) No relationship<\/li>\n<li>f) Weak<\/li>\n<li>g) Strong<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q605410\">Hint<\/span><\/p>\n<div id=\"q605410\" class=\"hidden-answer\" style=\"display: none\">Recall what you know about the shape and spread of a plot of bivariate data.<\/div>\n<\/div>\n<\/div>\n<p>Use the scatterplot from Question 2 to answer Questions 3 and 4. Don&#8217;t calculate and fit a regression line yet. You&#8217;ll do that in Question 5.<\/p>\n<h3>Approximating From a Scatterplot<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>Suppose there is a house for sale that has a size of 1,500 square feet. Based on the scatterplot, which of the following is the most likely value for its price?<\/p>\n<ol>\n<li>a) $100,000<\/li>\n<li>b) $110,000<\/li>\n<li>c) $125,000<\/li>\n<li>d) $150,000<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q715888\">Hint<\/span><\/p>\n<div id=\"q715888\" class=\"hidden-answer\" style=\"display: none\">What is the approximate mean price for houses that have a size of 1,500 square feet?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>Suppose you fit a line of best fit for these data. The line would predict the price of a house that has a size of 1,000 square feet to be closest to what value?<\/p>\n<ol>\n<li>a) $90,000<\/li>\n<li>b) $100,000<\/li>\n<li>c) $110,000<\/li>\n<li>d) $120,000<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q306923\">Hint<\/span><\/p>\n<div id=\"q306923\" class=\"hidden-answer\" style=\"display: none\">What is the approximate mean price for houses that have a size of 1,000 square feet?<\/div>\n<\/div>\n<\/div>\n<h3>\u00a0The Regression Line Equation<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>Use technology to calculate the line of best fit and write the equation for the line.<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: Write the equation using appropriate notation and customize the names of the variables.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q564913\">Hint<\/span><\/p>\n<div id=\"q564913\" class=\"hidden-answer\" style=\"display: none\">Rather than using [latex]\\hat{y}[\/latex] and [latex]x[\/latex], customize the variables using words or meaningful letters.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: Interpret the slope, in context. Select the best answer.<\/p>\n<p style=\"padding-left: 30px;\">a) For every one square foot increase in the size, there will be an increase of $77.01 in the price of a house in Florida.<\/p>\n<p style=\"padding-left: 30px;\">b) For every one square foot increase in the size, we predict an average increase of $77.01 in the price of a house in Florida.<\/p>\n<p style=\"padding-left: 30px;\">c) For every $1 increase in the price, there will be an increase of 77.01 square feet in the size of a house in Florida.<\/p>\n<p style=\"padding-left: 30px;\">d) For every $1 increase in the price, we predict an average increase of 77.01 square feet in the size of a house in Florida.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396809\">Hint<\/span><\/p>\n<div id=\"q396809\" class=\"hidden-answer\" style=\"display: none\">Recall the template for interpreting slope.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C: Does the intercept have a meaningful interpretation? Select the best answer.<\/p>\n<p style=\"padding-left: 30px;\">a) Yes. At a size of zero square feet, the predicted price of a house in Florida is $9,161.<\/p>\n<p style=\"padding-left: 30px;\">b) Yes. At a price of $0, the predicted size of a house in Florida is 9,161 square feet.<\/p>\n<p style=\"padding-left: 30px;\">c) No. It does not make sense to have a house that is zero square feet.<\/p>\n<p style=\"padding-left: 30px;\">d) No. It does not make sense to have a house that costs $0.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q78978\">Hint<\/span><\/p>\n<div id=\"q78978\" class=\"hidden-answer\" style=\"display: none\">What do <em>you\u00a0<\/em>think?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part D: Use the line to calculate the predicted price for a house that is 1,852 square feet. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q76000\">Hint<\/span><\/p>\n<div id=\"q76000\" class=\"hidden-answer\" style=\"display: none\">Use the equation of the line to make this calculation.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part E: Use the line to calculate the predicted price for a house that is 2,860 square feet. Round your answer to 2 decimal places.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495551\">Hint<\/span><\/p>\n<div id=\"q495551\" class=\"hidden-answer\" style=\"display: none\">Use the equation of the line to make this calculation.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part F: Do you think the model prediction is more reliable for houses that are 1,852 square feet or for those that are 2,860 square feet? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q678859\">Hint<\/span><\/p>\n<div id=\"q678859\" class=\"hidden-answer\" style=\"display: none\">Consider the number of observations at a given value of the explanatory variable.<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>In this <em>What to Know\u00a0<\/em>page,\u00a0you practiced the basic skills necessary to perform a linear regression analysis. Let\u2019s summarize these skills.<\/p>\n<ul>\n<li>In Question 1, you identified the explanatory and response variables for a given scenario.<\/li>\n<li>In Question 2, you used technology to make a scatterplot.<\/li>\n<li>In Question 2, you also used a scatterplot to describe bivariate relationships.<\/li>\n<li>In Questions 3 and 4, you approximated predicted values from a scatterplot.<\/li>\n<li>In Question 5, Part A, you used technology to calculate a line of best fit.<\/li>\n<li>In Question 5, Parts B and C, you interpreted the slope and intercept of the line of best fit.<\/li>\n<li>In Question 5, Parts D through F, you calculated predictions using the line of best fit and assessed reliability of the predictions.<\/li>\n<\/ul>\n<p>This page provided a summary and practice of the ideas you&#8217;ve learned in this section (and in [Section 5A] previously). Feel free to return to it any time to practice. Hopefully, it felt comfortable and familiar. Let&#8217;s move on to <em>Forming Connections<\/em> to put it all together.<\/p>\n","protected":false},"author":428269,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3876","chapter","type-chapter","status-publish","hentry"],"part":4241,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3876","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/428269"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3876\/revisions"}],"predecessor-version":[{"id":4845,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3876\/revisions\/4845"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4241"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3876\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3876"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3876"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3876"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3876"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}