{"id":3853,"date":"2022-03-15T23:18:44","date_gmt":"2022-03-15T23:18:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3853"},"modified":"2022-06-03T21:04:03","modified_gmt":"2022-06-03T21:04:03","slug":"forming-connections-in-6-b","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/forming-connections-in-6-b\/","title":{"raw":"Forming Connections in 6.B: Interpreting Estimated Slopes and Y-Intercepts","rendered":"Forming Connections in 6.B: Interpreting Estimated Slopes and Y-Intercepts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Objectives for this activity<\/h3>\r\nDuring this activity you will:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Identify the estimated slope and estimated y-intercept given the equation of the line of best fit.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Interpret the estimated slope and interpret the estimated y-intercept in the specific context of a problem.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Assess the reasonableness of estimations.<\/li>\r\n<\/ul>\r\n<\/div>\r\nHopefully, you are beginning to feel proficient using the data analysis tool to generate equations for lines of best fit to identify the estimated slope and y-intercept. In this activity, you'll focus on the interpretation of these measures in the context of the given scenario. You'll also assess the reasonableness (or unreasonableness) of using a linear model to extrapolate from the data. Along the way, you'll develop a deeper understanding that the equation of the the line of best fit is based on sample data and will change from dataset to dataset.\r\n<div class=\"textbox tryit\">\r\n<h3>Guidance<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Intro: As you work through this activity, look for these key concepts:<\/span>\r\n<ul>\r\n \t<li><span style=\"background-color: #e6daf7;\">It is important to look for trends in quantitative data early and via both visual and technical methods.<\/span><\/li>\r\n \t<li><span style=\"background-color: #e6daf7;\">It is crucial for you to be able to identify the explanatory and response variables on your own. Try defending your answers individually to Question 1 Part B.<\/span><\/li>\r\n \t<li><span style=\"background-color: #e6daf7;\">A foundational fact about lines of best fit is that each dataset will generate its own different linear equation. Each equation provides estimates of the slope and y-intercept unique to that dataset.\u00a0]<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nLet's get started with an interesting set of data that relates cricket chirps with the outside temperature.\r\n<h2>Learning from a Cricket and a Scientist Who\u00a0Died a Long Time Ago<\/h2>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Chirps per second<\/strong><\/td>\r\n<td><strong>Temperature (in [latex]^\\circ \\text{F}[\/latex] )<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20<\/td>\r\n<td>88.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>16<\/td>\r\n<td>71.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>19.8<\/td>\r\n<td>93.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>18.4<\/td>\r\n<td>84.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>17.1<\/td>\r\n<td>80.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>15.5<\/td>\r\n<td>75.2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>14.7<\/td>\r\n<td>69.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>17.1<\/td>\r\n<td>82<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>15.4<\/td>\r\n<td>69.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>16.2<\/td>\r\n<td>83.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>15<\/td>\r\n<td>79.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>17.2<\/td>\r\n<td>82.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>16<\/td>\r\n<td>80.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>17<\/td>\r\n<td>83.5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>14.4<\/td>\r\n<td>76.3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nIn a previous assignment, you were introduced to a famous study conducted in the 1940s on the relationship between the number of chirps made by a striped ground cricket (measured in number of wing vibrations per second) and the surrounding ground temperature (measured in degrees Fahrenheit).\r\n\r\nDr. Pierce wanted to see if it seemed reasonable to predict the temperature based on the number of chirps.\r\n\r\n&nbsp;\r\n\r\nPart A: What did you think when you first read about this study? Can you think of situations where we might learn from the behavior of animals using quantitative variables?\r\n\r\n[reveal-answer q=\"659806\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"659806\"]What do <em>you\u00a0<\/em>think? Consider other weather related animal behaviors such as the change in polar bear activity over time. Does a plot of carbon emissions over the same time indicate a positive trend in temperature change?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: Identify the explanatory and response variables in Dr. Pierce\u2019s experiment.\r\n\r\n[reveal-answer q=\"500644\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"500644\"]Answer this question individually to challenge your own understanding of how to identify these variables in a given context.[\/hidden-answer]\r\n\r\n<\/div>\r\nWork in small groups or pairs to answer Questions 2 and 3. Don't just answer Parts A through C in Question 2, but support your claims using the outputs from the data analysis tool.\r\n<h3>Estimated Slope and Y-Intercept in Context<\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nThe following is the output from the DCMP Linear Regression tool for the cricket data:\r\n\r\n<img class=\"alignnone wp-image-1209\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184323\/Picture130-300x124.png\" alt=\"A scatterplot labeled &quot;Chirps per second&quot; on the x-axis and &quot;Temperature in degrees Fahrenheit.&quot; Above the graph, it says &quot;Regression Line: y = 25.2 + 3.29x&quot; The scatterplot has points at approximately (14.5, 76.5), (14.7, 69.5), (15, 79.5), (15.35, 69), (15.5, 75), (16, 72), (16, 81), (16.2, 83), (17, 83), (17.1, 81), (17.1, 82.5), (17.2, 83), (18.4, 84), (19.75, 87.5), (20, 88.5).\" width=\"992\" height=\"410\" \/>\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Linear Regression Equation:<\/strong><\/span>\r\n<table style=\"border-collapse: collapse; width: 100%; height: 36px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\"><strong>Parameter<\/strong><\/td>\r\n<td style=\"width: 50%; height: 12px;\"><strong>Estimate<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">Intercept<\/td>\r\n<td style=\"width: 50%; height: 12px;\">25.23<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">Slope (Chirps per Second)<\/td>\r\n<td style=\"width: 50%; height: 12px;\">3.291<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<span style=\"text-decoration: underline;\"><strong>Model Summary:<\/strong><\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>Statistic<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>Value<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Correlation Coefficient\u00a0<em>r<\/em><\/td>\r\n<td style=\"width: 50%;\">0.835<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Coefficient of Determination\u00a0<em>r^2<\/em><\/td>\r\n<td style=\"width: 50%;\">69.7%<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Residual Standard Deviation<\/td>\r\n<td style=\"width: 50%;\">3.829<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPart A:\u00a0Does it seem reasonable to use a linear model to describe the relationship between Chirps per second and Temperature?\u00a0Clearly answer \u201cyes\u201d or \u201cno.\u201d Cite at least two pieces of evidence to support your claim.\r\n\r\n[reveal-answer q=\"415743\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"415743\"]Support your answer with output from the data analysis tool.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B:\u00a0Is the association positive or negative? Strong or weak?\u00a0Cite evidence to support your claim.\r\n\r\n[reveal-answer q=\"798545\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"798545\"]Support your answer with output from the data analysis tool.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C:\u00a0Describe what the variables in the line of best fit represent.\r\n\r\n[reveal-answer q=\"559633\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"559633\"]Support your answer with output from the data analysis tool.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart D: Interpret the slope of the line of best fit in context.\r\n\r\n[reveal-answer q=\"687878\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"687878\"]In this case, the change is estimated \"on average,\" given all values of chirps per second. This will connect data points representing the mean (average) temperature values for each value of chirps per second.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart E:\u00a0The line of best fit was generated by the 15 data points produced by a cricket that lived in the 1940s.\u00a0If you repeated the same study today, do you think the results would be exactly the same, similar, or very different? Explain.\r\n\r\n[reveal-answer q=\"729520\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"729520\"]Would the observed values naturally be different each time data is collected? Why is that? [\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart F: Is it reasonable to interpret the estimated y-intercept in this scenario? Support your answer.\r\n\r\n[reveal-answer q=\"772567\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"772567\"]Where is the y-intercept located on the graph? Is it outside or within the range of values of the explanatory variable used to create this model? [\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nWhich of the following statements are correct? Choose all that apply.\r\n\r\nThis study of cricket chirps per second and temperature demonstrates that you can use linear modeling in the following way:\r\n<ol>\r\n \t<li>a) Number of chirps per second can be used to predict the temperature.<\/li>\r\n \t<li>b) Number of chirps per second can be used to calculate the estimated temperature.<\/li>\r\n \t<li>c) Number of chirps per second can be used to determine the actual temperature.<\/li>\r\n \t<li>d) An increase in the number of chirps means it is likely that there will be an increase in the temperature.<\/li>\r\n \t<li>e) An increase in the number of chirps per second will cause an increase in the temperature.<\/li>\r\n \t<li>f) All of the above<\/li>\r\n \t<li>g) None of the above<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"865293\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"865293\"]Use what you understand about the model in this scenario to choose correct statements. Carefully consider the wording of each statement.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Guidance<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Summary: In Question 2, Part F, you were asked to extrapolate. That is, you were asked to discuss a value of the explanatory variable that did not exist in the observed data. There are dangers inherent in the act of extrapolation from the known variable values in a dataset. Let's discuss this briefly.<\/span>\r\n<ul>\r\n \t<li><span style=\"background-color: #e6daf7;\">Extrapolation is the prediction of a response value using an explanatory variable value that is outside the range of the original data. <\/span><\/li>\r\n \t<li><span style=\"background-color: #e6daf7;\">We must always be cautious regarding the reasonableness of extrapolation. If the known input does not include or even come close to zero, we may obtain a fully unreasonable interpretation of the y-intercept in the given dataset. ]<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nContinue to practice writing interpretations and answering questions using statistical terminology and support from the data and analysis tool as you answer Question 4.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nRecall the Beer Alcohol example from the preview assignment. The model uses alcohol % (explanatory variable) to predict the calories in a bottle of beer (response variable). The equation for the line of best fit is:\r\n\r\n[latex]\\hat{y} =8.27+28.2x[\/latex]\r\n\r\nThe equation can be rewritten using context and proper notation as follows:\r\n\r\n(Estimated calories = 8.27 + 28.2(alcohol%)\r\n\r\n&nbsp;\r\n\r\nPart A: Visualize the data on 227 bottles of beer by going to the <em>DCMP Linear Regression<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/\">https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/<\/a>. Write a sentence describing the trend in the data.\r\n\r\n[reveal-answer q=\"735929\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"735929\"]Use what you know about the shape and spread of the data.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: Does the trend you identified in Part A make sense?\r\n\r\n[reveal-answer q=\"135711\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"135711\"]Does it seem reasonable in the context of this scenario?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C: Is it reasonable to interpret the estimated y-intercept in this scenario? Specifically, does it make sense that a bottle of beer with zero alcohol in it could have a calorie count of 8.27 calories?\r\n\r\n[reveal-answer q=\"731282\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"731282\"]Does it seem reasonable in the context of this scenario?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart D: Interpret the estimated y-intercept, [latex]a=8.27[\/latex].\r\n\r\n[reveal-answer q=\"647205\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"647205\"]Use both the input and estimate output values with their units in a sentence.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Guidance<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Wrap-Up: Take a look back at the objectives for this activity and identify where they were located in the questions your answered in this activity.\u00a0 ]<\/span>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Objectives for this activity<\/h3>\n<p>During this activity you will:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Identify the estimated slope and estimated y-intercept given the equation of the line of best fit.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Interpret the estimated slope and interpret the estimated y-intercept in the specific context of a problem.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Assess the reasonableness of estimations.<\/li>\n<\/ul>\n<\/div>\n<p>Hopefully, you are beginning to feel proficient using the data analysis tool to generate equations for lines of best fit to identify the estimated slope and y-intercept. In this activity, you&#8217;ll focus on the interpretation of these measures in the context of the given scenario. You&#8217;ll also assess the reasonableness (or unreasonableness) of using a linear model to extrapolate from the data. Along the way, you&#8217;ll develop a deeper understanding that the equation of the the line of best fit is based on sample data and will change from dataset to dataset.<\/p>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Intro: As you work through this activity, look for these key concepts:<\/span><\/p>\n<ul>\n<li><span style=\"background-color: #e6daf7;\">It is important to look for trends in quantitative data early and via both visual and technical methods.<\/span><\/li>\n<li><span style=\"background-color: #e6daf7;\">It is crucial for you to be able to identify the explanatory and response variables on your own. Try defending your answers individually to Question 1 Part B.<\/span><\/li>\n<li><span style=\"background-color: #e6daf7;\">A foundational fact about lines of best fit is that each dataset will generate its own different linear equation. Each equation provides estimates of the slope and y-intercept unique to that dataset.\u00a0]<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Let&#8217;s get started with an interesting set of data that relates cricket chirps with the outside temperature.<\/p>\n<h2>Learning from a Cricket and a Scientist Who\u00a0Died a Long Time Ago<\/h2>\n<table>\n<tbody>\n<tr>\n<td><strong>Chirps per second<\/strong><\/td>\n<td><strong>Temperature (in [latex]^\\circ \\text{F}[\/latex] )<\/strong><\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>88.6<\/td>\n<\/tr>\n<tr>\n<td>16<\/td>\n<td>71.6<\/td>\n<\/tr>\n<tr>\n<td>19.8<\/td>\n<td>93.3<\/td>\n<\/tr>\n<tr>\n<td>18.4<\/td>\n<td>84.3<\/td>\n<\/tr>\n<tr>\n<td>17.1<\/td>\n<td>80.6<\/td>\n<\/tr>\n<tr>\n<td>15.5<\/td>\n<td>75.2<\/td>\n<\/tr>\n<tr>\n<td>14.7<\/td>\n<td>69.7<\/td>\n<\/tr>\n<tr>\n<td>17.1<\/td>\n<td>82<\/td>\n<\/tr>\n<tr>\n<td>15.4<\/td>\n<td>69.4<\/td>\n<\/tr>\n<tr>\n<td>16.2<\/td>\n<td>83.3<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>79.6<\/td>\n<\/tr>\n<tr>\n<td>17.2<\/td>\n<td>82.6<\/td>\n<\/tr>\n<tr>\n<td>16<\/td>\n<td>80.6<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>83.5<\/td>\n<\/tr>\n<tr>\n<td>14.4<\/td>\n<td>76.3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>In a previous assignment, you were introduced to a famous study conducted in the 1940s on the relationship between the number of chirps made by a striped ground cricket (measured in number of wing vibrations per second) and the surrounding ground temperature (measured in degrees Fahrenheit).<\/p>\n<p>Dr. Pierce wanted to see if it seemed reasonable to predict the temperature based on the number of chirps.<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: What did you think when you first read about this study? Can you think of situations where we might learn from the behavior of animals using quantitative variables?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q659806\">Hint<\/span><\/p>\n<div id=\"q659806\" class=\"hidden-answer\" style=\"display: none\">What do <em>you\u00a0<\/em>think? Consider other weather related animal behaviors such as the change in polar bear activity over time. Does a plot of carbon emissions over the same time indicate a positive trend in temperature change?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: Identify the explanatory and response variables in Dr. Pierce\u2019s experiment.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q500644\">Hint<\/span><\/p>\n<div id=\"q500644\" class=\"hidden-answer\" style=\"display: none\">Answer this question individually to challenge your own understanding of how to identify these variables in a given context.<\/div>\n<\/div>\n<\/div>\n<p>Work in small groups or pairs to answer Questions 2 and 3. Don&#8217;t just answer Parts A through C in Question 2, but support your claims using the outputs from the data analysis tool.<\/p>\n<h3>Estimated Slope and Y-Intercept in Context<\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>The following is the output from the DCMP Linear Regression tool for the cricket data:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1209\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184323\/Picture130-300x124.png\" alt=\"A scatterplot labeled &quot;Chirps per second&quot; on the x-axis and &quot;Temperature in degrees Fahrenheit.&quot; Above the graph, it says &quot;Regression Line: y = 25.2 + 3.29x&quot; The scatterplot has points at approximately (14.5, 76.5), (14.7, 69.5), (15, 79.5), (15.35, 69), (15.5, 75), (16, 72), (16, 81), (16.2, 83), (17, 83), (17.1, 81), (17.1, 82.5), (17.2, 83), (18.4, 84), (19.75, 87.5), (20, 88.5).\" width=\"992\" height=\"410\" \/><\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Linear Regression Equation:<\/strong><\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 36px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\"><strong>Parameter<\/strong><\/td>\n<td style=\"width: 50%; height: 12px;\"><strong>Estimate<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">Intercept<\/td>\n<td style=\"width: 50%; height: 12px;\">25.23<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">Slope (Chirps per Second)<\/td>\n<td style=\"width: 50%; height: 12px;\">3.291<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Model Summary:<\/strong><\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>Statistic<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>Value<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Correlation Coefficient\u00a0<em>r<\/em><\/td>\n<td style=\"width: 50%;\">0.835<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Coefficient of Determination\u00a0<em>r^2<\/em><\/td>\n<td style=\"width: 50%;\">69.7%<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Residual Standard Deviation<\/td>\n<td style=\"width: 50%;\">3.829<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Part A:\u00a0Does it seem reasonable to use a linear model to describe the relationship between Chirps per second and Temperature?\u00a0Clearly answer \u201cyes\u201d or \u201cno.\u201d Cite at least two pieces of evidence to support your claim.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q415743\">Hint<\/span><\/p>\n<div id=\"q415743\" class=\"hidden-answer\" style=\"display: none\">Support your answer with output from the data analysis tool.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B:\u00a0Is the association positive or negative? Strong or weak?\u00a0Cite evidence to support your claim.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798545\">Hint<\/span><\/p>\n<div id=\"q798545\" class=\"hidden-answer\" style=\"display: none\">Support your answer with output from the data analysis tool.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C:\u00a0Describe what the variables in the line of best fit represent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q559633\">Hint<\/span><\/p>\n<div id=\"q559633\" class=\"hidden-answer\" style=\"display: none\">Support your answer with output from the data analysis tool.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part D: Interpret the slope of the line of best fit in context.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q687878\">Hint<\/span><\/p>\n<div id=\"q687878\" class=\"hidden-answer\" style=\"display: none\">In this case, the change is estimated &#8220;on average,&#8221; given all values of chirps per second. This will connect data points representing the mean (average) temperature values for each value of chirps per second.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part E:\u00a0The line of best fit was generated by the 15 data points produced by a cricket that lived in the 1940s.\u00a0If you repeated the same study today, do you think the results would be exactly the same, similar, or very different? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729520\">Hint<\/span><\/p>\n<div id=\"q729520\" class=\"hidden-answer\" style=\"display: none\">Would the observed values naturally be different each time data is collected? Why is that? <\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part F: Is it reasonable to interpret the estimated y-intercept in this scenario? Support your answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q772567\">Hint<\/span><\/p>\n<div id=\"q772567\" class=\"hidden-answer\" style=\"display: none\">Where is the y-intercept located on the graph? Is it outside or within the range of values of the explanatory variable used to create this model? <\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>Which of the following statements are correct? Choose all that apply.<\/p>\n<p>This study of cricket chirps per second and temperature demonstrates that you can use linear modeling in the following way:<\/p>\n<ol>\n<li>a) Number of chirps per second can be used to predict the temperature.<\/li>\n<li>b) Number of chirps per second can be used to calculate the estimated temperature.<\/li>\n<li>c) Number of chirps per second can be used to determine the actual temperature.<\/li>\n<li>d) An increase in the number of chirps means it is likely that there will be an increase in the temperature.<\/li>\n<li>e) An increase in the number of chirps per second will cause an increase in the temperature.<\/li>\n<li>f) All of the above<\/li>\n<li>g) None of the above<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q865293\">Hint<\/span><\/p>\n<div id=\"q865293\" class=\"hidden-answer\" style=\"display: none\">Use what you understand about the model in this scenario to choose correct statements. Carefully consider the wording of each statement.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Summary: In Question 2, Part F, you were asked to extrapolate. That is, you were asked to discuss a value of the explanatory variable that did not exist in the observed data. There are dangers inherent in the act of extrapolation from the known variable values in a dataset. Let&#8217;s discuss this briefly.<\/span><\/p>\n<ul>\n<li><span style=\"background-color: #e6daf7;\">Extrapolation is the prediction of a response value using an explanatory variable value that is outside the range of the original data. <\/span><\/li>\n<li><span style=\"background-color: #e6daf7;\">We must always be cautious regarding the reasonableness of extrapolation. If the known input does not include or even come close to zero, we may obtain a fully unreasonable interpretation of the y-intercept in the given dataset. ]<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Continue to practice writing interpretations and answering questions using statistical terminology and support from the data and analysis tool as you answer Question 4.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>Recall the Beer Alcohol example from the preview assignment. The model uses alcohol % (explanatory variable) to predict the calories in a bottle of beer (response variable). The equation for the line of best fit is:<\/p>\n<p>[latex]\\hat{y} =8.27+28.2x[\/latex]<\/p>\n<p>The equation can be rewritten using context and proper notation as follows:<\/p>\n<p>(Estimated calories = 8.27 + 28.2(alcohol%)<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: Visualize the data on 227 bottles of beer by going to the <em>DCMP Linear Regression<\/em> tool at <a href=\"https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/\">https:\/\/dcmathpathways.shinyapps.io\/LinearRegression\/<\/a>. Write a sentence describing the trend in the data.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q735929\">Hint<\/span><\/p>\n<div id=\"q735929\" class=\"hidden-answer\" style=\"display: none\">Use what you know about the shape and spread of the data.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: Does the trend you identified in Part A make sense?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q135711\">Hint<\/span><\/p>\n<div id=\"q135711\" class=\"hidden-answer\" style=\"display: none\">Does it seem reasonable in the context of this scenario?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C: Is it reasonable to interpret the estimated y-intercept in this scenario? Specifically, does it make sense that a bottle of beer with zero alcohol in it could have a calorie count of 8.27 calories?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731282\">Hint<\/span><\/p>\n<div id=\"q731282\" class=\"hidden-answer\" style=\"display: none\">Does it seem reasonable in the context of this scenario?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part D: Interpret the estimated y-intercept, [latex]a=8.27[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647205\">Hint<\/span><\/p>\n<div id=\"q647205\" class=\"hidden-answer\" style=\"display: none\">Use both the input and estimate output values with their units in a sentence.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Wrap-Up: Take a look back at the objectives for this activity and identify where they were located in the questions your answered in this activity.\u00a0 ]<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":428269,"menu_order":9,"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-3853","chapter","type-chapter","status-publish","hentry"],"part":4241,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3853","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":13,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3853\/revisions"}],"predecessor-version":[{"id":5569,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3853\/revisions\/5569"}],"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\/3853\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3853"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3853"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3853"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3853"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}