1. 1 1 Slide Simple Linear Regression Estimation and Residuals Chapter 14 BA 303 – Spring 2011

2. 2 2 Slide Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: ^ y = 10 + 5(3) = 25 cars

3. 3 3 Slide 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 p ) The CI is an interval estimate of the mean value of y for a given value of x. Confidence Interval of E(y p )

4. 4 4 Slide n Estimate of the Standard Deviation of Confidence Interval for E ( y p )

5. 5 5 Slide The 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: Confidence Interval for E ( y p ) 25 - 4.61 25 + 3.182(1.4491) 20.39 to 29.61 cars 25 + 4.61

6. 6 6 Slide where: confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 degrees of freedom n Prediction Interval Estimate of y p The PI is an interval estimate of an individual value of y for a given value of x. The margin of error is larger than for a CI. Prediction Interval

7. 7 7 Slide n Estimate of the Standard Deviation of an Individual Value of y p Prediction Interval for y p

8. 8 8 Slide 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 p 25 - 8.28 25 + 3.1824(2.6013) 16.72 to 33.28 cars 25 + 8.28

9. 9 9 Slide Comparison 16.72 to 33.28 carsPrediction Interval: Confidence Interval:20.39 to 29.61 cars Point Estimate:25

10. 10 Slide PRACTICE PREDICTION INTERVALS AND CONFIDENCE INTERVALS

11. 11 Slide Data 12.8 38.0 513.2 t table 3.182  =0.05,  /2=0.025 d.f. = n – 2 = 3 s2.033 3 10

12. 12 Slide Confidence Interval LowerUpper 12.8 38.0 513.2

13. 13 Slide Prediction Interval LowerUpper 12.8 38.0 513.2

14. 14 Slide RESIDUAL ANALYSIS

15. 15 Slide Residual Analysis Much of the residual analysis is based on an examination of graphical plots. Residual for Observation i The residuals provide the best information about . If the assumptions about the error term  appear questionable, the hypothesis tests about the significance of the regression relationship and the interval estimation results may not be valid.

16. 16 Slide 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 The residual plot should give an overall impression of a horizontal band of points

17. 17 Slide x 0 Good Pattern Residual Residual Plot Against x

18. 18 Slide Residual Plot Against x x 0 Residual Nonconstant Variance

19. 19 Slide Residual Plot Against x x 0 Residual Model Form Not Adequate

20. 20 Slide Residuals 11415 32425 21820-2 117152 327252

21. 21 Slide Residual Plot Against x

22. 22 Slide n Standardized Residual for Observation i Standardized Residuals : where:

23. 23 Slide Standardized Residuals 110.25000.45001.6020 310.25000.45001.6020 200.00000.20001.9321 110.25000.45001.6020 310.25000.45001.6020 4 s=2.1602x=2

24. 24 Slide Standardized Residuals 114151.6020-0.6242 324251.6020-0.6242 218201.9321-1.0351 117151.60201.2484 327251.60201.2484

25. 25 Slide Standardized Residual Plot The standardized residual plot can provide insight about the assumption that the error term  has a normal distribution. n n If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution.

26. 26 Slide Standardized Residual Plot

27. 27 Slide 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.

28. 28 Slide Outliers and Influential Observations Detecting Outliers 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 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|. An outlier is an observation that is unusual in comparison with the other data.

29. 29 Slide PRACTICE STANDARDIZED RESIDUALS

30. 30 Slide Standardized Residuals 1 2 3 4 5 10 3 2.0330 s

31. 31 Slide Standardized Residuals 132.80002.0330 275.40001.9287 358.00001.8184 41110.60001.7009 51413.20001.5748

32. 32 Slide COMPUTER SOLUTIONS

33. 33 Slide Computer Solution Performing the regression analysis computations without the help of a computer can be quite time consuming.

34. 34 Slide Our Solution – Calculations

35. 35 Slide Our Solution – Calculations

36. 36 Slide Basic MiniTab Output

37. 37 Slide MiniTab Residuals, Prediction Intervals, and Confidence Intervals

38. 38 Slide Excel Output

39. 39 Slide