{"id":3862,"date":"2022-03-15T23:20:52","date_gmt":"2022-03-15T23:20:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3862"},"modified":"2022-06-03T20:51:39","modified_gmt":"2022-06-03T20:51:39","slug":"forming-connections-in-6-c","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/forming-connections-in-6-c\/","title":{"raw":"Forming Connections in 6.C: Understanding the Coefficient of Determination","rendered":"Forming Connections in 6.C: Understanding the Coefficient of Determination"},"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\">Interpret [latex]R^2[\/latex] values and determine their utility in different tasks (gauging prediction strength vs. determining a causal relationship).<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the\u00a0<em>What to Know\u00a0<\/em>assignment leading to this activity, you developed an intuition about how [latex]R^{2}[\/latex] is related to the shape of a scatterplot and practiced using technology to calculate and interpret\u00a0[latex]R^{2}[\/latex] for a dataset. In this activity, you'll utilize these skills to examine a real world situation in which linear analysis plays a role in educational policy decision-making. You'll also gain experience attempting to determine a causal relationship between variables during this activity. As you complete the questions below, you'll gain understanding of\u00a0[latex]R^{2}[\/latex] as a measure of prediction strength in a linear relationship by seeing how\u00a0[latex]R^{2}[\/latex] can be interpreted as the percentage of variation in the response explained by the linear relationship. You'll also see that a high\u00a0[latex]R^{2}[\/latex] value does not indicate a causal relationship.\r\n<h2>Thinking About Education<\/h2>\r\nYou will approach this in-class activity from the perspective of the secretary of education in your state. You notice that many public school students in your state are not showing good results on their high school math exams. You\u2019d like to introduce a policy change that will lead to better results. Your first step is to collect data about high school students to see what factors best predict their math performance.\r\n\r\n<img class=\"alignnone wp-image-1230\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193005\/Picture146-300x213.jpg\" alt=\"Children with backpacks on smiling and running out of a building\" width=\"1151\" height=\"817\" \/>\r\n<div class=\"textbox tryit\">\r\n<h3>Guidance<\/h3>\r\n<span style=\"background-color: #e6daf7;\">[Intro: Note that the dataset used in this activity is simulated due to privacy concerns with sharing real student data. It does, however, represent results that real school districts have found when studying these variables. The policies discussed in this activity and their results are also representative of policies that real school districts have implemented.]<\/span>\r\n\r\n<\/div>\r\nAs an introduction to the scenario presented in this activity, read and answer Question 1 independently, then discuss your answers with a partner. Once you feel comfortable with your answer, move on to Question 2 to begin the activity.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nThink about the kinds of data that schools collect about their students and teachers. Which variables do you think will be the best predictors of math exam performance? List at least three variables and explain why you chose each one.\r\n\r\n[reveal-answer q=\"318390\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"318390\"]List an explanation with each variable you choose.[\/hidden-answer]\r\n\r\n<\/div>\r\nThe scenario for this activity is given in Question 2. Read it carefully to ensure that you understand each scatterplot is driven by a different explanatory variable. In each set of variables, the response is math scores on a state algebra exam. Each scatterplot and regression line shown pairs the test scores as the proposed outcome for a different input variable. Use the anticipated [latex]R^{2}[\/latex] as a tool to decide which of the variable pairs would be most appropriate to choose to analyze possible causes for low math test scores in your district.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nYour department\u2019s head statistician gathers data from a random sample of 100 students in the state. In separate scatterplots, she visualizes the relationship between their algebra state exam scores and three explanatory variables: their math teachers\u2019 experience levels (years teaching), their attendance (percentages of school days attended), and their schools\u2019 math department discretionary budgets per teacher. These plots are visualized below, along with their linear regression models:\r\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Teacher Experience<\/strong><\/p>\r\n<img class=\"alignnone wp-image-1231\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193010\/Picture147-300x115.png\" alt=\"A scatterplot labeled &quot;Teacher Experience (Years)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is seen going from (1, 64) to (22, 90). One of the points is located at approximately (17, 63).\" width=\"1095\" height=\"420\" \/>\r\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Attendance<\/strong><\/p>\r\n<img class=\"alignnone wp-image-1233\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193018\/Picture149-300x116.png\" alt=\"A scatterplot labeled &quot;Number of School Days Attended&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is shown reaches from approximately (50, 53) to (100, 95).\" width=\"1210\" height=\"469\" \/>\r\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Math Dept. Budget<\/strong><\/p>\r\n<img class=\"alignnone wp-image-1235\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193026\/Picture151-300x118.png\" alt=\"A scatterplot labeled &quot;Math Department Budget ($ per teacher)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. The line of best fit is shown extending from approximately (223, 66) to approximately (1900, 82).\" width=\"1096\" height=\"431\" \/>\r\n\r\nPart A: Rank in order (from highest to lowest) the [latex]R^2[\/latex] values you\u2019d expect from each of these linear models.\r\n\r\n[reveal-answer q=\"126932\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"126932\"]Use what you know about how [latex]R^{2}[\/latex] is related to the shape and spread of a scatterplot.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: If you had to use one of these explanatory variables to predict the exam score of a newly sampled student, which would you use? Explain.\r\n\r\n[reveal-answer q=\"6891\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6891\"]Rely solely on what you know about the strength of a linear relationship apparent visually in a scatterplot to answer.[\/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: What did you choose to use as the explanatory variable for your analysis? Did you rely solely on evidence in the data to make your choice or did you permit opinion to bias your decision? In statistics, it is important to understand that we rely on mathematical tools to make decisions in order to minimize the opportunity for opinion to drive our conclusions.]<\/span>\r\n\r\n<\/div>\r\nIn Question 3, you are given the precise\u00a0[latex]R^{2}[\/latex] values for each of the scatterplots you examined in Question 2. Use them to make a final determination of which explanatory variable seems most reasonable to use as a predictor of low test scores.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nHere are the [latex]R^2[\/latex] values from each of the previous models:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Teacher experience model: 53.8%<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Attendance model: 84.2%<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Math budget model: 13.5%<\/li>\r\n<\/ul>\r\nInterpret the [latex]R^2[\/latex] value from the attendance model. Make sure you interpret in context.\r\n\r\n[reveal-answer q=\"919105\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"919105\"]Use what you know about how\u00a0[latex]R^{2}[\/latex] can be interpreted in a statistical model. [\/hidden-answer]\r\n\r\n<\/div>\r\nYou probably have a fairly good idea of which model would be most appropriate to use to predict low test scores. But there are other aspects of the models to consider. For example, see the Teacher Experience scatterplot again to answer Question 4.\r\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Teacher Experience<\/strong><\/p>\r\n<img class=\"alignnone wp-image-1231\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193010\/Picture147-300x115.png\" alt=\"A scatterplot labeled &quot;Teacher Experience (Years)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is seen going from (1, 64) to (22, 90). One of the points is located at approximately (17, 63).\" width=\"1095\" height=\"420\" \/>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nOn the Teacher Experience scatterplot, note that there\u2019s one data point representing a student whose teacher has more than 15 years of teaching experience but who scored fairly low on the assessment.\r\n\r\n&nbsp;\r\n\r\nPart A: Locate the data point on the Teacher Experience scatterplot.\r\n\r\n[reveal-answer q=\"279768\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"279768\"]Inspect the plot visually to locate a low test score intersecting with more than 15 years of experience.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: If we removed this data point from the graphic, would the [latex]R^2[\/latex] value increase, decrease, or stay the same? Explain.\r\n\r\n[reveal-answer q=\"584799\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"584799\"]Recall how to interpret [latex]R^2[\/latex] with regard to variability in the response variable due to the linear relationship.[\/hidden-answer]\r\n\r\n<\/div>\r\nNow that you have thoroughly examined all three potential explanatory variables, it's time to propose a policy change based on the data. Answer Question 5 independently before discussing it in your group or with a partner. Be sure to consider both pros and cons as you develop your response.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 5<\/h3>\r\nYour staff proposes three new education policies: one aims to recruit more experienced teachers, another aims to improve student attendance rates, and the last aims to increase math department budgets. Assume these policies would have the same costs and popularity. Which would you choose to implement? Explain.\r\n\r\n[reveal-answer q=\"908304\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"908304\"]Make sure to include a clear explanation with your answer.[\/hidden-answer]\r\n\r\n<\/div>\r\nNo matter which policy you chose to implement, in real school districts in the United States, superintendents have implemented initiatives designed to improve attendance as a measure to support improving test scores. The surprising result of these initiatives is that while attendance increased over time, test scores did not.\r\n\r\nWork together in pairs or in groups to consider this result as you answer Question 6. Include specific reasoning in your answer as to why attendance may not be causally related to exam scores. For example, could there be alternative causal mechanisms that result in a correlation between attendance and test scores?\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nLet\u2019s move from this hypothetical example to real examples. Seeing similar patterns to what you previously analyzed, superintendents in school districts across the United States have piloted large-scale (and sometimes quite expensive) initiatives to improve student attendance. These included:\r\n<ul>\r\n \t<li>Call programs for chronically-absent students<\/li>\r\n \t<li>Hiring attendance case managers and coordinators<\/li>\r\n \t<li>Using Uber\/Lyft for students with transportation issues<\/li>\r\n<\/ul>\r\nThe results have often looked like this:\r\n\r\n<img class=\"alignnone wp-image-1237\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193033\/Picture153-300x155.png\" alt=\"A graph with Time on its x-axis and Attendance on its y-axis. There is a line on the graph with a positive slope.\" width=\"328\" height=\"169\" \/>.\u00a0<img class=\"alignnone wp-image-1238\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193038\/Picture154-300x162.png\" alt=\"A graph with Time on its x-axis and Test Scores on its y-axis. There is a horizontal line on the graph.\" width=\"330\" height=\"179\" \/>\r\n\r\nHow is it possible that, in school districts with strong correlations between test scores and attendance rates, improving attendance didn\u2019t lead to an overall improvement in student test scores?\r\n\r\n[reveal-answer q=\"141115\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"141115\"]Could alternative variables be acting as confounders in this situation? Be specific and clear in your answer.[\/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: What sort of alternative causes for the correlation between attendance and low test scores did you come up with? It is likely that confounding variables could explain at least some of the apparent causal relationship. Consider this graph:<\/span>\r\n\r\n<span style=\"background-color: #e6daf7;\"><img class=\"aligncenter size-full wp-image-4849\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/03155107\/Attend_TestScores_Confounders.jpg\" alt=\"A graph shows two causal relationships. The first is labeled &quot;Assumption.&quot; It shows the words Low Attendance with an arrow pointing directly to Low Scores. That is &quot;Low Attendance causes Low Scores.&quot; The second is labeled &quot;Reality.&quot; It shows the word Poverty pointing to Low Attendance and to Hunger, Worse Schools, Less Study Time (job, taking care of family). An arrow labeled Cause is drawn from the larger list to Low Scores.\" width=\"1813\" height=\"753\" \/>Discuss the difference between the assumption made by the data analysis, that low attendance causes low test scores and the reality that groups of students who tend to experience low attendance also tend to experience several other factors that can also lead to low scores. When performing analysis, it is crucial to keep an open mind for alternative factors that may be driving a response.]<\/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\">Interpret [latex]R^2[\/latex] values and determine their utility in different tasks (gauging prediction strength vs. determining a causal relationship).<\/li>\n<\/ul>\n<\/div>\n<p>In the\u00a0<em>What to Know\u00a0<\/em>assignment leading to this activity, you developed an intuition about how [latex]R^{2}[\/latex] is related to the shape of a scatterplot and practiced using technology to calculate and interpret\u00a0[latex]R^{2}[\/latex] for a dataset. In this activity, you&#8217;ll utilize these skills to examine a real world situation in which linear analysis plays a role in educational policy decision-making. You&#8217;ll also gain experience attempting to determine a causal relationship between variables during this activity. As you complete the questions below, you&#8217;ll gain understanding of\u00a0[latex]R^{2}[\/latex] as a measure of prediction strength in a linear relationship by seeing how\u00a0[latex]R^{2}[\/latex] can be interpreted as the percentage of variation in the response explained by the linear relationship. You&#8217;ll also see that a high\u00a0[latex]R^{2}[\/latex] value does not indicate a causal relationship.<\/p>\n<h2>Thinking About Education<\/h2>\n<p>You will approach this in-class activity from the perspective of the secretary of education in your state. You notice that many public school students in your state are not showing good results on their high school math exams. You\u2019d like to introduce a policy change that will lead to better results. Your first step is to collect data about high school students to see what factors best predict their math performance.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1230\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193005\/Picture146-300x213.jpg\" alt=\"Children with backpacks on smiling and running out of a building\" width=\"1151\" height=\"817\" \/><\/p>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Intro: Note that the dataset used in this activity is simulated due to privacy concerns with sharing real student data. It does, however, represent results that real school districts have found when studying these variables. The policies discussed in this activity and their results are also representative of policies that real school districts have implemented.]<\/span><\/p>\n<\/div>\n<p>As an introduction to the scenario presented in this activity, read and answer Question 1 independently, then discuss your answers with a partner. Once you feel comfortable with your answer, move on to Question 2 to begin the activity.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>Think about the kinds of data that schools collect about their students and teachers. Which variables do you think will be the best predictors of math exam performance? List at least three variables and explain why you chose each one.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q318390\">Hint<\/span><\/p>\n<div id=\"q318390\" class=\"hidden-answer\" style=\"display: none\">List an explanation with each variable you choose.<\/div>\n<\/div>\n<\/div>\n<p>The scenario for this activity is given in Question 2. Read it carefully to ensure that you understand each scatterplot is driven by a different explanatory variable. In each set of variables, the response is math scores on a state algebra exam. Each scatterplot and regression line shown pairs the test scores as the proposed outcome for a different input variable. Use the anticipated [latex]R^{2}[\/latex] as a tool to decide which of the variable pairs would be most appropriate to choose to analyze possible causes for low math test scores in your district.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>Your department\u2019s head statistician gathers data from a random sample of 100 students in the state. In separate scatterplots, she visualizes the relationship between their algebra state exam scores and three explanatory variables: their math teachers\u2019 experience levels (years teaching), their attendance (percentages of school days attended), and their schools\u2019 math department discretionary budgets per teacher. These plots are visualized below, along with their linear regression models:<\/p>\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Teacher Experience<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1231\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193010\/Picture147-300x115.png\" alt=\"A scatterplot labeled &quot;Teacher Experience (Years)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is seen going from (1, 64) to (22, 90). One of the points is located at approximately (17, 63).\" width=\"1095\" height=\"420\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Attendance<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1233\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193018\/Picture149-300x116.png\" alt=\"A scatterplot labeled &quot;Number of School Days Attended&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is shown reaches from approximately (50, 53) to (100, 95).\" width=\"1210\" height=\"469\" \/><\/p>\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Math Dept. Budget<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1235\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193026\/Picture151-300x118.png\" alt=\"A scatterplot labeled &quot;Math Department Budget ($ per teacher)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. The line of best fit is shown extending from approximately (223, 66) to approximately (1900, 82).\" width=\"1096\" height=\"431\" \/><\/p>\n<p>Part A: Rank in order (from highest to lowest) the [latex]R^2[\/latex] values you\u2019d expect from each of these linear models.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q126932\">Hint<\/span><\/p>\n<div id=\"q126932\" class=\"hidden-answer\" style=\"display: none\">Use what you know about how [latex]R^{2}[\/latex] is related to the shape and spread of a scatterplot.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: If you had to use one of these explanatory variables to predict the exam score of a newly sampled student, which would you use? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6891\">Hint<\/span><\/p>\n<div id=\"q6891\" class=\"hidden-answer\" style=\"display: none\">Rely solely on what you know about the strength of a linear relationship apparent visually in a scatterplot to answer.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Summary: What did you choose to use as the explanatory variable for your analysis? Did you rely solely on evidence in the data to make your choice or did you permit opinion to bias your decision? In statistics, it is important to understand that we rely on mathematical tools to make decisions in order to minimize the opportunity for opinion to drive our conclusions.]<\/span><\/p>\n<\/div>\n<p>In Question 3, you are given the precise\u00a0[latex]R^{2}[\/latex] values for each of the scatterplots you examined in Question 2. Use them to make a final determination of which explanatory variable seems most reasonable to use as a predictor of low test scores.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>Here are the [latex]R^2[\/latex] values from each of the previous models:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Teacher experience model: 53.8%<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Attendance model: 84.2%<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Math budget model: 13.5%<\/li>\n<\/ul>\n<p>Interpret the [latex]R^2[\/latex] value from the attendance model. Make sure you interpret in context.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q919105\">Hint<\/span><\/p>\n<div id=\"q919105\" class=\"hidden-answer\" style=\"display: none\">Use what you know about how\u00a0[latex]R^{2}[\/latex] can be interpreted in a statistical model. <\/div>\n<\/div>\n<\/div>\n<p>You probably have a fairly good idea of which model would be most appropriate to use to predict low test scores. But there are other aspects of the models to consider. For example, see the Teacher Experience scatterplot again to answer Question 4.<\/p>\n<p style=\"text-align: center;\"><strong>Explanatory Variable: Teacher Experience<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1231\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193010\/Picture147-300x115.png\" alt=\"A scatterplot labeled &quot;Teacher Experience (Years)&quot; on the horizontal axis and &quot;Algebra Exam Scores (%)&quot; on the vertical axis. There is a line of best fit that is seen going from (1, 64) to (22, 90). One of the points is located at approximately (17, 63).\" width=\"1095\" height=\"420\" \/><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>On the Teacher Experience scatterplot, note that there\u2019s one data point representing a student whose teacher has more than 15 years of teaching experience but who scored fairly low on the assessment.<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: Locate the data point on the Teacher Experience scatterplot.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279768\">Hint<\/span><\/p>\n<div id=\"q279768\" class=\"hidden-answer\" style=\"display: none\">Inspect the plot visually to locate a low test score intersecting with more than 15 years of experience.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: If we removed this data point from the graphic, would the [latex]R^2[\/latex] value increase, decrease, or stay the same? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584799\">Hint<\/span><\/p>\n<div id=\"q584799\" class=\"hidden-answer\" style=\"display: none\">Recall how to interpret [latex]R^2[\/latex] with regard to variability in the response variable due to the linear relationship.<\/div>\n<\/div>\n<\/div>\n<p>Now that you have thoroughly examined all three potential explanatory variables, it&#8217;s time to propose a policy change based on the data. Answer Question 5 independently before discussing it in your group or with a partner. Be sure to consider both pros and cons as you develop your response.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 5<\/h3>\n<p>Your staff proposes three new education policies: one aims to recruit more experienced teachers, another aims to improve student attendance rates, and the last aims to increase math department budgets. Assume these policies would have the same costs and popularity. Which would you choose to implement? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q908304\">Hint<\/span><\/p>\n<div id=\"q908304\" class=\"hidden-answer\" style=\"display: none\">Make sure to include a clear explanation with your answer.<\/div>\n<\/div>\n<\/div>\n<p>No matter which policy you chose to implement, in real school districts in the United States, superintendents have implemented initiatives designed to improve attendance as a measure to support improving test scores. The surprising result of these initiatives is that while attendance increased over time, test scores did not.<\/p>\n<p>Work together in pairs or in groups to consider this result as you answer Question 6. Include specific reasoning in your answer as to why attendance may not be causally related to exam scores. For example, could there be alternative causal mechanisms that result in a correlation between attendance and test scores?<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>Let\u2019s move from this hypothetical example to real examples. Seeing similar patterns to what you previously analyzed, superintendents in school districts across the United States have piloted large-scale (and sometimes quite expensive) initiatives to improve student attendance. These included:<\/p>\n<ul>\n<li>Call programs for chronically-absent students<\/li>\n<li>Hiring attendance case managers and coordinators<\/li>\n<li>Using Uber\/Lyft for students with transportation issues<\/li>\n<\/ul>\n<p>The results have often looked like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1237\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193033\/Picture153-300x155.png\" alt=\"A graph with Time on its x-axis and Attendance on its y-axis. There is a line on the graph with a positive slope.\" width=\"328\" height=\"169\" \/>.\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1238\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12193038\/Picture154-300x162.png\" alt=\"A graph with Time on its x-axis and Test Scores on its y-axis. There is a horizontal line on the graph.\" width=\"330\" height=\"179\" \/><\/p>\n<p>How is it possible that, in school districts with strong correlations between test scores and attendance rates, improving attendance didn\u2019t lead to an overall improvement in student test scores?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q141115\">Hint<\/span><\/p>\n<div id=\"q141115\" class=\"hidden-answer\" style=\"display: none\">Could alternative variables be acting as confounders in this situation? Be specific and clear in your answer.<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Guidance<\/h3>\n<p><span style=\"background-color: #e6daf7;\">[Wrap-up: What sort of alternative causes for the correlation between attendance and low test scores did you come up with? It is likely that confounding variables could explain at least some of the apparent causal relationship. Consider this graph:<\/span><\/p>\n<p><span style=\"background-color: #e6daf7;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4849\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/03\/03155107\/Attend_TestScores_Confounders.jpg\" alt=\"A graph shows two causal relationships. The first is labeled &quot;Assumption.&quot; It shows the words Low Attendance with an arrow pointing directly to Low Scores. That is &quot;Low Attendance causes Low Scores.&quot; The second is labeled &quot;Reality.&quot; It shows the word Poverty pointing to Low Attendance and to Hunger, Worse Schools, Less Study Time (job, taking care of family). An arrow labeled Cause is drawn from the larger list to Low Scores.\" width=\"1813\" height=\"753\" \/>Discuss the difference between the assumption made by the data analysis, that low attendance causes low test scores and the reality that groups of students who tend to experience low attendance also tend to experience several other factors that can also lead to low scores. When performing analysis, it is crucial to keep an open mind for alternative factors that may be driving a response.]<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":428269,"menu_order":14,"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-3862","chapter","type-chapter","status-publish","hentry"],"part":4241,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3862","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":12,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3862\/revisions"}],"predecessor-version":[{"id":4857,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3862\/revisions\/4857"}],"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\/3862\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3862"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3862"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3862"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}