1. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-1 Business Statistics, 3e by Ken Black Chapter 11 Analysis of Variance & Design of Experiments

2. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-2 Learning Objectives Understand the differences between various experimental designs and when to use them. Compute and interpret the results of a one-way ANOVA. Compute and interpret the results of a random block design. Compute and interpret the results of a two-way ANOVA. Understand and interpret interaction. Know when and how to use multiple comparison techniques.

3. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-3 Introduction to Design of Experiments, #1 Experimental Design - a plan and a structure to test hypotheses in which the researcher controls or manipulates one or more variables.

4. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-4 Introduction to Design of Experiments, #2 Independent Variable Treatment variable is one that the experimenter controls or modifies in the experiment. Classification variable is a characteristic of the experimental subjects that was present prior to the experiment, and is not a result of the experimenter’s manipulations or control. Levels or Classifications are the subcategories of the independent variable used by the researcher in the experimental design.

5. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-5 Introduction to Design of Experiments, #3 Dependent Variable - the response to the different levels of the independent variables.

6. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-6 Three Types of Experimental Designs Completely Randomized Design Randomized Block Design Factorial Experiments

7. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-7 Completely Randomized Design Machine Operator 123.................. Valve Opening Measurements

8. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-8 Example: Number of Foreign Freighters Docking in each Port per Day Long Beach 57425742 Houston 2354623546 New York 846798846798 New Orleans 3534235342

9. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-9 Analysis of Variance: Assumptions Observations are drawn from normally distributed populations. Observations represent random samples from the populations. Variances of the populations are equal.

10. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-10 One-Way ANOVA: Procedural Overview

11. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-11 One-Way ANOVA: Sums of Squares Definitions

12. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-12 Partitioning Total Sum of Squares of Variation SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares)

13. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-13 One-Way ANOVA: Computational Formulas

14. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-14 One-Way ANOVA: Preliminary Calculations Long Beach 5 7 4 2 T 1 = 18 n 1 = 4 Houston 2 3 5 4 6 T 2 = 20 n 2 = 5 New York 8 4 6 7 9 8 T 3 = 42 n 3 = 6 New Orleans 3 5 3 4 2 T 4 = 17 n 4 = 5 T = 97 N = 20

15. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-15 One-Way ANOVA: Sum of Squares Calculations

16. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-16 One-Way ANOVA: Sum of Squares Calculations

17. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-17 One-Way ANOVA: Mean Square and F Calculations

18. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-18 Analysis of Variance for the Freighter Example Source of VariancedfSSMSF Factor342.3514.125.12 Error1644.202.76 Total1986.55

19. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-19 A Portion of the F Table for  = 0.05 123456789 1161.45199.50215.71224.58230.16233.99236.77238.88240.54... 154.543.683.293.062.902.792.712.642.59 164.493.633.243.012.852.742.662.592.54 174.453.593.202.962.812.702.612.552.49 Denominator Degrees of Freedom Numerator Degrees of Freedom

20. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-20 One-Way ANOVA: Procedural Summary Rejection Region  Critical Value Non rejection Region

21. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-21 MINITAB Output for the Freighter Example ANALYSIS OF VARIANCE SourcedfSSMSFp Factor342.3514.125.110.011 Error1644.202.76 Total1986.55 LEVELNMeanStDev Long B44.5002.082 Houston54.0001.581 New York67.0001.789 NewOrlns53.4001.140 Pooled StDev = 1.662

22. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-22 Excel Output for the Freighter Example Anova: Single Factor SUMMARY GroupsCountSumAverageVariance Long Beach4184.54.3333 Houston52042.5 New York64273.2 New Orleans5173.41.3 ANOVA Source of VariationSSdfMSFP-valueF crit Between Groups42.35314.1175.11010.01143.2389 Within Groups44.2162.7625 Total86.5519

23. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-23 Multiple Comparison Tests An analysis of variance (ANOVA) test is an overall test of differences among groups. Multiple Comparison techniques are used to identify which pairs of means are significantly different given that the ANOVA test reveals overall significance. Tukey’s honestly significant difference (HSD) test requires equal sample sizes Tukey-Kramer Procedure is used when sample sizes are unequal.

24. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-24 Tukey’s Honestly Significant Difference (HSD) Test

25. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-25 Data for Demonstration Problem 11.1 PLANT (Employee Age) 1 2 3 293225 273324 303124 273425 283026 Group Means28.232.024.8 n j 555 C = 3 df E = N - C = 12MSE = 1.63

26. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-26 q Values for  =.01 Degrees of Freedom 1 2 3 4. 11 12 2345 90135164186 141922.324.7 8.2610.612.213.3 6.518.129.179.96 4.395.145.625.97 4.325.045.505.84.... Number of Populations

27. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-27 Tukey’s HSD Test for the Employee Age Data

28. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-28 Tukey-Kramer Procedure: The Case of Unequal Sample Sizes

29. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-29 Freighter Example: Means and Sample Sizes for the Four Ports PortSample SizeMean Long Beach44.50 Houston54.00 New York67.00 New Orleans53.40

30. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-30 Tukey- Kramer Results for the Freighter Example Pair Critical Difference |Actual Differences| 1 and 23.190.50 (Long Beach and Houston) 1 and 33.072.50 (Long Beach and New York) 1 and 43.190.35 (Long Beach and New Orleans) 2 and 32.883.00* (Houston and New York) 2 and 43.010.60 (Houston and New Orleans) 3 and 42.883.60* (New York and New Orleans) *denotes significant at .05

31. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-31 Partitioning the Total Sum of Squares in the Randomized Block Design SST (Total Sum of Squares) SSC (Treatment Sum of Squares) SSE (Error Sum of Squares) SSR (Sum of Squares Blocks) SSE’ (Sum of Squares Error)

32. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-32 A Randomized Block Design Individual observations........................ Single Independent Variable Blocking Variable.....

33. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-33 Randomized Block Design Treatment Effects: Procedural Overview

34. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-34 Randomized Block Design: Computational Formulas

35. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-35 Randomized Block Design: Tread-Wear Example Supplier 1 2 3 4 SlowMediumFast Block Means ( ) 3.74.53.13.77 3.43.92.83.37 3.54.13.03.53 3.23.52.63.10 5 Treatment Means( ) 3.94.83.44.03 3.544.162.983.56 Speed C = 3 n = 5 N = 15

36. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-36 Randomized Block Design: Sum of Squares Calculations (Part 1)

37. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-37 Randomized Block Design: Sum of Squares Calculations (Part 2)

38. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-38 Randomized Block Design: Mean Square Calculations

39. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-39 Analysis of Variance for the Tread-Wear Example Source of VarianceSSdfMSF Treatment3.48421.74296.78 Block1.54940.38721.50 Error0.14380.018 Total5.17614

40. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-40 Randomized Block Design Treatment Effects: Procedural Summary

41. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-41 Randomized Block Design Blocking Effects: Procedural Overview

42. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-42 Excel Output for Tread-Wear Example: Randomized Block Design Anova: Two-Factor Without Replication SUMMARYCountSumAverageVariance Suplier 1311.33.76666670.4933333 Suplier 2310.13.36666670.3033333 Suplier 3310.63.53333330.3033333 Suplier 439.33.10.21 Suplier 5312.14.03333330.5033333 Slow517.73.540.073 Medium520.84.160.258 Fast514.92.980.092 ANOVA Source of VariationSSdfMSFP-valueF crit Rows1.549333340.387333321.7196260.00023577.0060651 Columns3.48421.74297.6822432.395E-068.6490672 Error0.142666780.0178333 Total5.17614

43. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-43 Two-Way Factorial Design Cells........................ Column Treatment Row Treatment.....

44. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-44 Two-Way ANOVA: Hypotheses

45. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-45 Formulas for Computing a Two-Way ANOVA

46. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-46 A 2  3 Factorial Design with Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

47. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-47 A 2  3 Factorial Design with Some Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

48. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-48 A 2  3 Factorial Design with No Interaction Cell Means C1C1 C2C2 C3C3 Row effects R1R1 R2R2 Column

49. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-49 A 2  3 Factorial Design: Data and Measurements for CEO Dividend Example N = 24 n = 4 X=2.7083 1.752.753.625 Location Where Company Stock is Traded How Stockholders are Informed of Dividends NYSEAMEXOTC Annual/Quarterly Reports 21212121 23322332 43434343 2.5 Presentations to Analysts 23122312 33243324 44344434 2.9167 XjXj XiXi X 11 =1.5 X 23 =3.75X 22 =3.0X 21 =2.0 X 13 =3.5X 12 =2.5

50. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-50 A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 1)

51. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-51 A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 2)

52. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-52 A 2  3 Factorial Design: Calculations for the CEO Dividend Example (Part 3)

53. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-53 Analysis of Variance for the CEO Dividend Problem Source of VarianceSSdfMSF Row1.041811.04182.42 Column14.083327.041716.35 * Interaction0.083320.04170.10 Error7.7500180.4306 Total22.958323 * Denotes significance at  =.01.

54. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-54 Excel Output for the CEO Dividend Example (Part 1) Anova: Two-Factor With Replication SUMMARYNYSEASEOTCTotal AQReport Count44412 Sum6101430 Average1.52.53.52.5 Variance0.3333 1 Presentation Count44412 Sum8121535 Average233.752.9167 Variance0.6667 0.250.9924 Total Count888 Sum142229 Average1.752.753.625 Variance0.5 0.2679

55. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 11-55 Excel Output for the CEO Dividend Example (Part 2) ANOVA Source of VariationSSdfMSFP-valueF crit Sample1.04171 2.41940.13734.4139 Columns14.08327.041716.3559E-053.5546 Interaction0.083320.04170.09680.90823.5546 Within7.75180.4306 Total22.95823