1. 1 1 Slide © 2014 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 Simple Linear Regression n Simple Linear Regression Model n Least Squares Method n Coefficient of Determination n Model Assumptions n Testing for Significance n Using the Estimated Regression Equation for Estimation and Prediction for Estimation and Prediction n Residual Analysis: Validating Model Assumptions n Outliers and Influential Observations

2. 2 2 Slide © 2014 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. Simple Linear Regression Regression analysis can be used to develop an Regression analysis can be used to develop an equation showing how the variables are related. equation showing how the variables are related. Managerial decisions often are based on the Managerial decisions often are based on the relationship between two or more variables. relationship between two or more variables. The variables being used to predict the value of the The variables being used to predict the value of the dependent variable are called the independent dependent variable are called the independent variables and are denoted by x. variables and are denoted by x. The variable being predicted is called the dependent The variable being predicted is called the dependent variable and is denoted by y. variable and is denoted by y.

3. 3 3 Slide © 2014 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. Simple Linear Regression The relationship between the two variables is The relationship between the two variables is approximated by a straight line. approximated by a straight line. Simple linear regression involves one independent Simple linear regression involves one independent variable and one dependent variable. variable and one dependent variable. Regression analysis involving two or more Regression analysis involving two or more independent variables is called multiple regression. independent variables is called multiple regression.

4. 4 4 Slide © 2014 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. Simple Linear Regression Model y =  0 +  1 x +  where:  0 and  1 are called parameters of the model,  is a random variable called the error term.  is a random variable called the error term. The simple linear regression model is: The simple linear regression model is: The equation that describes how y is related to x and The equation that describes how y is related to x and an error term is called the regression model. an error term is called the regression model.

5. 5 5 Slide © 2014 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. Simple Linear Regression Equation n The simple linear regression equation is: E ( y ) is the expected value of y for a given x value. E ( y ) is the expected value of y for a given x value.  1 is the slope of the regression line.  1 is the slope of the regression line.  0 is the y intercept of the regression line.  0 is the y intercept of the regression line. Graph of the regression equation is a straight line. Graph of the regression equation is a straight line. E ( y ) =  0 +  1 x

6. 6 6 Slide © 2014 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. Simple Linear Regression Equation n Positive Linear Relationship E(y)E(y)E(y)E(y) x Slope  1 is positive Regression line Intercept  0

7. 7 7 Slide © 2014 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. Simple Linear Regression Equation n Negative Linear Relationship E(y)E(y)E(y)E(y)x Slope  1 is negative Regression line Intercept  0

8. 8 8 Slide © 2014 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. Simple Linear Regression Equation n No Relationship E(y)E(y)E(y)E(y)x Slope  1 is 0 Regression line Intercept  0

9. 9 9 Slide © 2014 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. Estimated Simple Linear Regression Equation n The estimated simple linear regression equation is the estimated value of y for a given x value. is the estimated value of y for a given x value. b 1 is the slope of the line. b 1 is the slope of the line. b 0 is the y intercept of the line. b 0 is the y intercept of the line. The graph is called the estimated regression line. The graph is called the estimated regression line.

10. 10 Slide © 2014 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. Estimation Process Regression Model y =  0 +  1 x +  Regression Equation E ( y ) =  0 +  1 x Unknown Parameters  0,  1 Regression Model y =  0 +  1 x +  Regression Equation E ( y ) =  0 +  1 x Unknown Parameters  0,  1 Sample Data: x y x 1 y 1...... x n y n Sample Data: x y x 1 y 1...... x n y n b 0 and b 1 provide estimates of  0 and  1 b 0 and b 1 provide estimates of  0 and  1 Estimated Regression Equation Sample Statistics b 0, b 1 Estimated Regression Equation Sample Statistics b 0, b 1

11. 11 Slide © 2014 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. Least Squares Method n Least Squares Criterion where: y i = observed value of the dependent variable for the i th observation for the i th observation^ y i = estimated value of the dependent variable for the i th observation for the i th observation

12. 12 Slide © 2014 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 Slope for the Estimated Regression Equation Least Squares Method where: x i = value of independent variable for i th observation observation_ y = mean value for dependent variable _ x = mean value for independent variable y i = value of dependent variable for i th observation observation

13. 13 Slide © 2014 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 y -Intercept for the Estimated Regression Equation Least Squares Method

14. 14 Slide © 2014 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. Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide. Simple Linear Regression n Example: Reed Auto Sales

15. 15 Slide © 2014 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. Simple Linear Regression n Example: Reed Auto Sales Number of TV Ads ( x ) TV Ads ( x ) Number of Cars Sold ( y ) 1 3 2 1 3 14 24 18 17 27  x = 10  y = 100

16. 16 Slide © 2014 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. Estimated Regression Equation n Slope for the Estimated Regression Equation n y -Intercept for the Estimated Regression Equation n Estimated Regression Equation

17. 17 Slide © 2014 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. Coefficient of Determination n Relationship Among SST, SSR, SSE where: SST = total sum of squares SST = total sum of squares SSR = sum of squares due to regression SSR = sum of squares due to regression SSE = sum of squares due to error SSE = sum of squares due to error SST = SSR + SSE

18. 18 Slide © 2014 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 The coefficient of determination is: Coefficient of Determination where: SSR = sum of squares due to regression SST = total sum of squares r 2 = SSR/SST

19. 19 Slide © 2014 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. Coefficient of Determination r 2 = SSR/SST = 100/114 =.8772 The regression relationship is very strong; 87.72% The regression relationship is very strong; 87.72% of the variability in the number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

20. 20 Slide © 2014 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. Sample Correlation Coefficient where: b 1 = the slope of the estimated regression b 1 = the slope of the estimated regression equation equation

21. 21 Slide © 2014 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 sign of b 1 in the equation is “+”. Sample Correlation Coefficient r xy = +.9366

22. 22 Slide © 2014 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. Assumptions About the Error Term  1. The error  is a random variable with mean of zero. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 2. The variance of , denoted by  2, is the same for all values of the independent variable. all values of the independent variable. 3. The values of  are independent. 4. The error  is a normally distributed random variable. variable. 4. The error  is a normally distributed random variable. variable.

23. 23 Slide © 2014 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. Testing for Significance To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. To test for a significant regression relationship, we To test for a significant regression relationship, we must conduct a hypothesis test to determine whether must conduct a hypothesis test to determine whether the value of  1 is zero. the value of  1 is zero. Two tests are commonly used: Two tests are commonly used: t Test and F Test Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model. Both the t test and F test require an estimate of  2, Both the t test and F test require an estimate of  2, the variance of  in the regression model. the variance of  in the regression model.

24. 24 Slide © 2014 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. An Estimate of  2 An Estimate of  2 Testing for Significance where: s 2 = MSE = SSE/( n  2) The mean square error (MSE) provides the estimate of  2, and the notation s 2 is also used.

25. 25 Slide © 2014 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. Testing for Significance An Estimate of  An Estimate of  To estimate  we take the square root of  2. To estimate  we take the square root of  2. The resulting s is called the standard error of The resulting s is called the standard error of the estimate. the estimate.

26. 26 Slide © 2014 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 Hypotheses n Test Statistic Testing for Significance: t Test where

27. 27 Slide © 2014 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 Rejection Rule Testing for Significance: t Test where: t  is based on a t distribution with n - 2 degrees of freedom Reject H 0 if p -value <  or t t 

28. 28 Slide © 2014 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. 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  =.05 4. State the rejection rule. Reject H 0 if p -value <.05 or | t| > 3.182 (with 3 degrees of freedom) Testing for Significance: t Test

29. 29 Slide © 2014 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. Testing for Significance: t Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. t = 4.541 provides an area of.01 in the upper tail. Hence, the p -value is less than.02. (Also, t = 4.63 > 3.182.) We can reject H 0.

30. 30 Slide © 2014 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. Confidence Interval for  1 H 0 is rejected if the hypothesized value of  1 is not H 0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1. included in the confidence interval for  1. We can use a 95% confidence interval for  1 to test We can use a 95% confidence interval for  1 to test the hypotheses just used in the t test. the hypotheses just used in the t test.

31. 31 Slide © 2014 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 form of a confidence interval for  1 is: The form of a confidence interval for  1 is: Confidence Interval for  1 where is the t value providing an area of  /2 in the upper tail of a t distribution with n - 2 degrees of freedom b 1 is the pointestimator pointestimator is the margin of error is the margin of error

32. 32 Slide © 2014 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. Confidence Interval for  1 Reject H 0 if 0 is not included in the confidence interval for  1. 0 is not included in the confidence interval. Reject H 0 = 5 +/- 3.182(1.08) = 5 +/- 3.44 or 1.56 to 8.44 n Rejection Rule 95% Confidence Interval for  1 95% Confidence Interval for  1 n Conclusion

33. 33 Slide © 2014 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 Hypotheses n Test Statistic Testing for Significance: F Test F = MSR/MSE

34. 34 Slide © 2014 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 Rejection Rule Testing for Significance: F Test where: F  is based on an F distribution with 1 degree of freedom in the numerator and n - 2 degrees of freedom in the denominator Reject H 0 if p -value <  p -value <  or F > F 

35. 35 Slide © 2014 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. 1. Determine the hypotheses. 2. Specify the level of significance. 3. Select the test statistic.  =.05 4. State the rejection rule. Reject H 0 if p -value <.05 or F > 10.13 (with 1 d.f. in numerator and 3 d.f. in denominator) 3 d.f. in denominator) Testing for Significance: F Test F = MSR/MSE

36. 36 Slide © 2014 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. Testing for Significance: F Test 5. Compute the value of the test statistic. 6. Determine whether to reject H 0. F = 17.44 provides an area of.025 in the upper tail. Thus, the p -value corresponding to F = 21.43 is less than.025. Hence, we reject H 0. F = 17.44 provides an area of.025 in the upper tail. Thus, the p -value corresponding to F = 21.43 is less than.025. Hence, we reject H 0. F = MSR/MSE = 100/4.667 = 21.43 The statistical evidence is sufficient to conclude The statistical evidence is sufficient to conclude that we have a significant relationship between the number of TV ads aired and the number of cars sold.

37. 37 Slide © 2014 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. Some Cautions about the Interpretation of Significance Tests Just because we are able to reject H 0 :  1 = 0 and Just because we are able to reject H 0 :  1 = 0 and demonstrate statistical significance does not enable demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y. Rejecting H 0 :  1 = 0 and concluding that the Rejecting H 0 :  1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.

38. 38 Slide © 2014 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.

39. 39 Slide © 2014 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 *

40. 40 Slide © 2014 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

41. 41 Slide © 2014 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 * )

42. 42 Slide © 2014 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

43. 43 Slide © 2014 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 *

44. 44 Slide © 2014 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

45. 45 Slide © 2014 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.

46. 46 Slide © 2014 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

47. 47 Slide © 2014 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

48. 48 Slide © 2014 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

49. 49 Slide © 2014 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

50. 50 Slide © 2014 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

51. 51 Slide © 2014 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

52. 52 Slide © 2014 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:

53. 53 Slide © 2014 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.

54. 54 Slide © 2014 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

55. 55 Slide © 2014 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

56. 56 Slide © 2014 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.

57. 57 Slide © 2014 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.

58. 58 Slide © 2014 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