1 1 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES. BY John Loucks St. Edward’s University

2 2 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 12, Part B Simple Linear Regression n Using the Estimated Regression Equation for Estimation and Prediction for Estimation and Prediction n Residual Analysis: Validating Model Assumptions n Computer Solution

3 3 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Using the Estimated Regression Equation for Estimation and Prediction The margin of error is larger for a prediction interval. The margin of error is larger for a prediction interval. A prediction interval is used whenever we want to A prediction interval is used whenever we want to predict an individual value of y for a new observation predict an individual value of y for a new observation corresponding to a given value of x. corresponding to a given value of x. A confidence interval is an interval estimate of the A confidence interval is an interval estimate of the mean value of y for a given value of x. mean value of y for a given value of x.

4 4 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Using the Estimated Regression Equation for Estimation and Prediction where: confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 degrees of freedom n Confidence Interval Estimate of E ( y * ) n Prediction Interval Estimate of y *

5 5 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: Point Estimation ^ y = (3) = 25 cars

6 6 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Estimate of the Standard Deviation of Confidence Interval for E ( y * )

7 7 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: Confidence Interval for E ( y * ) (1.4491) to cars

8 8 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Estimate of the Standard Deviation of an Individual Value of y * of an Individual Value of y * Prediction Interval for y *

9 9 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: Prediction Interval for y * (2.6013) to cars

10 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Computer Solution Recall that the independent variable was named Ads Recall that the independent variable was named Ads and the dependent variable was named Cars in the and the dependent variable was named Cars in the example. example. On the next slide we show Minitab output for the On the next slide we show Minitab output for the Reed Auto Sales example. Reed Auto Sales example. Performing the regression analysis computations Performing the regression analysis computations without the help of a computer can be quite time without the help of a computer can be quite time consuming. consuming.

11 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The regression equation is Cars = Ads PredictorCoefSE CoefTp Constant Ads S = R-sq = 87.7%R-sq(adj) = 83.6% Analysis of Variance SOURCE DFSSMSFp Regression Residual Err Total4114 Predicted Values for New Observations New Obs FitSE Fit 95% C.I. 95% P.I (20.39, 29.61)(16.72, 33.28) Computer Solution n Minitab Output Output EstimatedRegressionEquationEstimatedRegressionEquation ANOVATableANOVATable IntervalEstimatesIntervalEstimates

12 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Minitab Output Minitab prints the standard error of the estimate, s, Minitab prints the standard error of the estimate, s, as well as information about the goodness of fit. as well as information about the goodness of fit. For each of the coefficients b 0 and b 1, the output shows For each of the coefficients b 0 and b 1, the output shows its value, standard deviation, t value, and p -value. its value, standard deviation, t value, and p -value. Minitab prints the estimated regression equation as Minitab prints the estimated regression equation as Cars = Ads. Cars = Ads. The standard ANOVA table is printed. The standard ANOVA table is printed. Also provided are the 95% confidence interval Also provided are the 95% confidence interval estimate of the expected number of cars sold and the estimate of the expected number of cars sold and the 95% prediction interval estimate of the number of 95% prediction interval estimate of the number of cars sold for an individual weekend with 3 ads. cars sold for an individual weekend with 3 ads.

13 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Analysis Much of the residual analysis is based on an Much of the residual analysis is based on an examination of graphical plots. examination of graphical plots. Residual for Observation i Residual for Observation i The residuals provide the best information about . The residuals provide the best information about . If the assumptions about the error term  appear If the assumptions about the error term  appear questionable, the hypothesis tests about the questionable, the hypothesis tests about the significance of the regression relationship and the significance of the regression relationship and the interval estimation results may not be valid. interval estimation results may not be valid.

14 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x If the assumption that the variance of  is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then … If the assumption that the variance of  is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables, then … The residual plot should give an overall The residual plot should give an overall impression of a horizontal band of points impression of a horizontal band of points

15 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. x 0 Good Pattern Residual Residual Plot Against x

16 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x x 0 Residual Nonconstant Variance

17 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x x 0 Residual Model Form Not Adequate

18 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Residuals Residual Analysis Predicted Cars SoldResiduals TV Ads Cars Sold

19 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against x Residual Plot Against TV Ads TV Ads Residuals

20 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Residual Plot Against y ^ Residual Plot Against Predicted y Predicted Cars Sold Residuals Note: The pattern of this residual plot is the same as that of the residual plot against x. as that of the residual plot against x.

21 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 12, Part B