1. 1 1 Slide © 2012 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 2 Slide © 2012 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 14, 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 Residual Analysis: Outliers and Influential Observations Observations n Computer Solution

3. 3 3 Slide © 2012 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 Excel’s Regression Tool The Regression tool can be used to perform a The Regression tool can be used to perform a complete regression analysis. complete regression analysis. Excel also has a comprehensive tool in its Data Excel also has a comprehensive tool in its Data Analysis package called Regression. Analysis package called Regression. Up to this point, you have seen how Excel can be Up to this point, you have seen how Excel can be used for various parts of a regression analysis. used for various parts of a regression analysis.

4. 4 4 Slide © 2012 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 Excel Worksheet (showing data) Estimated Regression Equation

5. 5 5 Slide © 2012 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 Excel’s Regression Tool n Performing the Regression Analysis Step 3 Choose Regression from the list of Analysis Tools Analysis Tools Step 2 Choose the Data Analysis option Step 1 Select the Tools menu

6. 6 6 Slide © 2012 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 Excel’s Regression Tool n Excel Regression Dialog Box

7. 7 7 Slide © 2012 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 Excel’s Regression Tool n Excel Value Worksheet ANOVA Output Regression Statistics Output Data Estimated Regression Equation Output

8. 8 8 Slide © 2012 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 Excel’s Regression Tool Note: Columns F-I are not shown. n Excel Value Worksheet (bottom-left portion)

9. 9 9 Slide © 2012 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 Excel’s Regression Tool Note: Columns C-E are hidden. n Excel Value Worksheet (bottom-right portion)

10. 10 Slide © 2012 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 Excel’s Regression Tool n Excel Value Worksheet (middle portion)

11. 11 Slide © 2012 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 Excel’s Regression Tool n Excel Value Worksheet (top portion)

12. 12 Slide © 2012 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.

13. 13 Slide © 2012 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 *

14. 14 Slide © 2012 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 = 10 + 5(3) = 25 cars

15. 15 Slide © 2012 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 * )

16. 16 Slide © 2012 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 * ) 25 + 4.61 25 + 3.1824(1.4491) 20.39 to 29.61 cars

17. 17 Slide © 2012 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 *

18. 18 Slide © 2012 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 * 25 + 8.28 25 + 3.1824(2.6013) 16.72 to 33.28 cars

19. 19 Slide © 2012 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.

20. 20 Slide © 2012 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

21. 21 Slide © 2012 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

22. 22 Slide © 2012 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

23. 23 Slide © 2012 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

24. 24 Slide © 2012 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 Plot Against x

25. 25 Slide © 2012 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 Using Excel to Produce a Residual Plot The output will include two new items: The output will include two new items: A plot of the residuals against the A plot of the residuals against the independent variable, and independent variable, and A list of predicted values of y and the A list of predicted values of y and the corresponding residual values. corresponding residual values. When the Regression dialog box appears, we must When the Regression dialog box appears, we must also select the Residual Plot option. also select the Residual Plot option. The steps outlined earlier to obtain the regression The steps outlined earlier to obtain the regression output are performed with one change. output are performed with one change. Residual Plot Against x

26. 26 Slide © 2012 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

27. 27 Slide © 2012 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 Standardized Residual for Observation i Standardized Residuals where:

28. 28 Slide © 2012 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. Standardized Residual Plot The standardized residual plot can provide insight about the assumption that the error term  has a normal distribution. The standardized residual plot can provide insight about the assumption that the error term  has a normal distribution. n If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution.

29. 29 Slide © 2012 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 Standardized Residuals Standardized Residual Plot

30. 30 Slide © 2012 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 Standardized Residual Plot Standardized Residual Plot

31. 31 Slide © 2012 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. Standardized Residual Plot All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that  has a normal distribution. All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that  has a normal distribution.

32. 32 Slide © 2012 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. Outliers and Influential Observations n Detecting Outliers Minitab classifies an observation as an outlier if its standardized residual value is +2. Minitab classifies an observation as an outlier if its standardized residual value is +2. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. This rule’s shortcoming can be circumvented by using studentized deleted residuals. This rule’s shortcoming can be circumvented by using studentized deleted residuals. The | i th studentized deleted residual| will be larger than the | i th standardized residual|. The | i th studentized deleted residual| will be larger than the | i th standardized residual|. An outlier is an observation that is unusual in comparison with the other data. An outlier is an observation that is unusual in comparison with the other data.

33. 33 Slide © 2012 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 14, Part B