1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 12.2 Linear Regression

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Model a given set of data using a regression line.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Least-Squares Regression Line The least-squares regression line is the line for which the average variation from the data is the smallest, also called the line of best fit, given by

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Slope of the Least-Squares Regression Line The slope of the least-squares regression line for paired data from a sample is given by where n is the number of data pairs in the sample, x i is the i th value of the explanatory variable, and y i is the i th value of the response variable.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. y-Intercept of the Least-Squares Regression Line The y-intercept of the least-squares regression line for paired data from a sample is given by where n is the number of data pairs in the sample, x i is the i th value of the explanatory variable, y i is the i th value of the response variable, and b 1 is the slope of the least-squares regression line.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator The local school board wants to evaluate the relationship between class size and performance on the state achievement test. It decides to collect data from various schools in the district, and the data from a sample of eight classes are shown in the following table. Each pair of data values represents the class size and corresponding average score on the achievement test for one class.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) Class Sizes and Average Test Scores Class Size1517182021242629 Average Test Score85.386.28582.781.978.875.372.1 Determine if there is a significant linear relationship between class size and average test score at the 0.05 level of significance. If the relationship is significant, find the least-squares regression line for these data.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) Solution First, we must decide which variable should be the x-variable and which variable should be the y-variable. Consider whether there is a possibility that one of these variables influences the values of the other. In this case, we are interested in the possibility that class size influences the average test score. Thus, class size is the explanatory variable, x, and average test score is the response variable, y.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) Press. Select option 1:Edit. Enter the values for class size (x) in List 1 ( L1 ) and the values for average test score (y) in List 2 ( L2 ). Press. Select CALC. Choose option 4:LinReg(ax+b). Press twice.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) The output will include the correlation coefficient, r, and the coefficient of determination, r 2, that we discussed in the previous section. In addition, we see the slope, a, and the y-intercept, b, of the least-squares regression line.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) Before we write out the equation for the regression line, we must be sure that there is sufficient evidence to claim that a linear relationship exists between these two variables. Based on the critical value for a 0.05 level of significance, the absolute value of the correlation coefficient must be greater than or equal to 0.707. The absolute value of the correlation coefficient, |r| ≈ 0.978, is certainly greater than 0.707, so there is enough evidence at the 0.05 level of significance to conclude that the correlation between the two variables is statistically significant.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.) Next we need to consider the shape of the data in the scatter plot. Looking at the following graph, we can verify that the data points fall in a somewhat linear fashion. It is now appropriate to consider the linear regression model. The slope of the regression line is a = b 1 ≈ −1.043 and the y-intercept of the regression line is b = b 0 ≈ 103.085. Thus, the equation of the regression line, in the form is as follows.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.10: Finding a Least-Squares Regression Line Using a TI-83/84 Plus Calculator (cont.)

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A prediction should not be made with a regression model if… 1.The data do not fall in a linear pattern when graphed on a scatter plot. 2.The correlation coefficient is not statistically significant. 3.You wish to make a prediction about a value outside the range of the sample data. 4.The population is different than that from which the sample data were drawn.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.11: Making Predictions Using a Least-Squares Regression Line Use the equation of the regression line from the previous example, to predict what the average achievement test score will be for the following class sizes. a.16 b.19 c.25 d.45

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.11: Making Predictions Using a Least-Squares Regression Line (cont.) Solution d.It is not meaningful to predict the value of y for this class size because the value x = 45 is outside the range of the original data. The original data only considered class sizes between 15 and 29, so we should only predict the average achievement test scores for class sizes within this range.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set The following table lists data collected on the selling prices of used Land Rover Freelanders and their ages in years. Find a linear regression model for predicting the price of a used Land Rover Freelander based on its age in years, if appropriate at the 0.05 level of significance. Selling Prices and Ages of Used Land Rover Freelanders Age (in Years) 3412234341 Price (in Dollars) 15,50014,99530,79528,99523,99520,90020,50019,99519,88829,995

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) Solution To begin, we must determine which variable is the explanatory variable (x) and which variable is the response variable (y). We want to use the age of a used Freelander to predict its selling price, thus age (x) is the explanatory variable and price (y) is the response variable.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) Press. Select option 1:Edit. Enter the values for age (x) in L1 and the values for price (y) in L2. Press. Select CALC. Choose option 4:LinReg(ax+b). Press twice.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) The output will include the correlation coefficient, r, the coefficient of determination, r 2, the slope, a, and the y-intercept, b, of the regression line.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) Before we write out the equation for the regression line, we must be sure that there is sufficient evidence to claim that a linear relationship exists between these two variables. First, notice that the scatter plot depicts a possible linear relationship.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.)

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) Next, based on the critical value for a 0.05 level of significance, the absolute value of the correlation coefficient must be greater than or equal to 0.632. The absolute value of the correlation coefficient, |r| ≈ 0.863, is greater than 0.632, so there is enough evidence at the 0.05 level of significance to conclude that a possible linear relationship exists between these two variables.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.12: Finding the Least-Squares Regression Line for a Given Data Set (cont.) Thus, it is appropriate to consider the linear regression model. The slope of the regression line is a = b 1 ≈  4257.818 and the y-intercept of the regression line is b = b 0 ≈ 34,051.909. Therefore the regression line, in the form is as follows.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.13: Making Predictions Using a Linear Regression Model Use the linear regression model from the previous example, to predict the following. a.The selling price of a Land Rover Freelander that is 2.5 years old b.The selling price of a Land Rover Freelander that is 10 years old c.The selling price of a Land Rover Range Rover that is 3 years old

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.13: Making Predictions Using a Linear Regression Model (cont.) Solution a.Substitute the value x = 2.5 into the regression line and solve for Thus, we would predict that a 2.5-year-old Freelander would sell for approximately $23,407.36.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.13: Making Predictions Using a Linear Regression Model (cont.) b.The original sample only contains Land Rover Freelanders that are 1 to 4 years old; therefore, it is inappropriate to use this model to predict the price of a Freelander that is 10 years old. c.The population is Freelanders, not Range Rovers. Thus, it is inappropriate to use this model to predict the price of a Range Rover, no matter how old it is.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish The following table gives the average monthly temperatures and corresponding monthly precipitation totals for one year in Key West, Florida. Average Temperatures and Precipitation Totals in Key West, Florida Average Temperature (in °F) Precipitation (in Inches) 75 2.22 76 1.51 79 1.86 82 2.06 85 3.48 88 4.57 Average Temperature (in °F) Precipitation (in Inches) 89 3.27 90 5.4 88 5.45 85 4.34 81 2.64 77 2.14

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) a.Create a scatter plot for the data. Does there appear to be a linear relationship between x and y? b.Calculate the correlation coefficient, r. c.Verify that the correlation coefficient is statistically significant at the 0.05 level of significance. d.Determine the equation of the line of best fit. e.Calculate and interpret the coefficient of determination, r 2.

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) f.If appropriate, predict the monthly precipitation total in Key West for a month in which the average temperature is 80 degrees. g.If appropriate, predict the monthly precipitation in Destin, Florida, for a month in which the average temperature is 83 degrees.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) Solution a.In order to create a scatter plot, we must first decide which variable will be x and which will be y. Since it is not clear whether one variable influences another, we can choose either variable for the x ‑ axis. Let’s choose the average temperatures for the x-values and the inches of precipitation for the y ‑ values. A scatter plot of the data is shown in the following figure. Notice that there does appear to be a possible positive linear relationship between precipitation and temperature.

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) If r is statistically significant for the variables, a linear regression model would be appropriate.

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) b.Using a TI-83/84 Plus calculator as we have in the previous examples, we calculate r ≈ 0.859.

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) c.Using Table I in Appendix A, we find that the critical value at the 0.05 level of significance is 0.576. Since |0.859| > 0.576, we can conclude that r is indeed statistically significant. d.From the calculator, we see that the equation of the line of best fit is as follows.

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) e.The coefficient of determination is approximately 0.738. This tells us that approximately 73.8% of the variation in precipitation can be attributed to the linear relationship between temperature and precipitation. The remaining 26.2% of the variation is from unknown sources. f.Because r is statistically significant, we can use the regression line to make predictions regarding the variables.

36. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) In addition, 80 is within the range of the x-values from the sample data, so it is appropriate to predict the monthly precipitation total when the average temperature is 80 degrees. Because we designated the average temperatures as the x-values, substitute x = 80 into the regression equation to obtain an estimate for the precipitation total for a month when the average temperature is 80 degrees.

37. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) Thus, a reasonable estimate for the precipitation for a month in which the average temperature is 80 degrees is approximately 2.58 inches.

38. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12.14: A Linear Regression Model, Start to Finish (cont.) g.The data were collected in Key West—not Destin, Florida. Therefore, it is not appropriate to use the linear regression equation to make predictions regarding the precipitation in Destin.