1. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-1 Business Statistics, 4e by Ken Black Chapter 9 Statistical Inference: Hypothesis Testing for Single Populations

2. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-2 Learning Objectives Understand the logic of hypothesis testing, and know how to establish null and alternate hypotheses. Understand Type I and Type II errors, and know how to solve for Type II errors. Know how to implement the HTAB system to test hypotheses. Test hypotheses about a single population mean when  is known. Test hypotheses about a single population mean when  is unknown. Test hypotheses about a single population proportion. Test hypotheses about a single population variance.

3. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-3 Types of Hypotheses Research Hypothesis –a statement of what the researcher believes will be the outcome of an experiment or a study. Statistical Hypotheses –a more formal structure derived from the research hypothesis. Substantive Hypotheses –a statistically significant difference does not imply or mean a material, substantive difference.

4. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-4 Example Research Hypotheses Older workers are more loyal to a company Companies with more than $1 billion of assets spend a higher percentage of their annual budget on advertising than do companies with less than $1 billion of assets. The price of scrap metal is a good indicator of the industrial production index six months later.

5. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-5 Statistical Hypotheses Two Parts –a null hypothesis –an alternative hypothesis Null Hypothesis – nothing new is happening Alternative Hypothesis – something new is happening Notation –null: H 0 –alternative: H a

6. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-6 Null and Alternative Hypotheses The Null and Alternative Hypotheses are mutually exclusive. Only one of them can be true. The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.) The Null Hypothesis is assumed to be true. The burden of proof falls on the Alternative Hypothesis.

7. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-7 Null and Alternative Hypotheses: Example A manufacturer is filling 40 oz. packages with flour. The company wants the package contents to average 40 ounces.

8. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-8 One-tailed Tests One-tailed and Two-tailed Tests Two-tailed Test

9. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-9 HTAB System to Test Hypotheses Task 1: HYPOTHESIZE Task 2: TEST Task 3: TAKE STATISTICAL ACTION Task 4: DETERMINING THE BUSINESS IMPLICATIONS

10. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-10 Steps in Testing Hypotheses 1. Establish hypotheses: state the null and alternative hypotheses. 2. Determine the appropriate statistical test and sampling distribution. 3. Specify the Type I error rate (  4. State the decision rule. 5. Gather sample data. 6. Calculate the value of the test statistic. 7. State the statistical conclusion. 8. Make a managerial decision.

11. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-11 HTAB Paradigm – Task 1 Task 1: Hypotheses Step 1. Establish hypotheses: state the null and alternative hypotheses.

12. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-12 HTAB Paradigm – Task 2 Task 2: Test Step 2. Determine the appropriate statistical test and sampling distribution. Step 3. Specify the Type I error rate (  Step 4. State the decision rule. Step 5. Gather sample data. Step 6. Calculate the value of the test statistic.

13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-13 HTAB Paradigm – Task 3 Task 3: Take Statistical Action Step 7. State the statistical conclusion.

14. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-14 HTAB Paradigm – Task 4 Task 4: Determine the business implications Step 8. Make a managerial decision.

15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-15 Rejection and Non Rejection Regions  =40 oz Non Rejection Region Rejection Region Critical Value Rejection Region Critical Value

16. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-16 Type I and Type II Errors Type I Error –Rejecting a true null hypothesis –The probability of committing a Type I error is called , the level of significance. Type II Error –Failing to reject a false null hypothesis –The probability of committing a Type II error is called .

17. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-17 Decision Table for Hypothesis Testing ( () Null TrueNull False Fail to reject null Correct Decision Type II error  ) Reject nullType I error  Correct Decision

18. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-18 One-tailed Tests  =40 oz Rejection Region Non Rejection Region Critical Value  =40 oz Rejection Region Non Rejection Region Critical Value

19. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-19 Two-tailed Tests  =12 oz Rejection Region Non Rejection Region Critical Values Rejection Region

20. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-20 CPA Net Income Example: Two-tailed Test (Part 1) Rejection Region Non Rejection Region  =0 Rejection Region

21. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-21 CPA Net Income Example: Two-tailed Test (Part 2)

22. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-22 CPA Net Income Example: Critical Value Method (Part 1) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

23. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-23 CPA Net Income Example: Critical Value Method (Part 2) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

24. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-24 Demonstration Problem 9.1: z Test (Part 1) Rejection Region Non Rejection Region 0  =.05

25. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-25 Demonstration Problem 9.1: z Test (Part 2) Rejection Region Non Rejection Region 0  =.05

26. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-26 Demonstration Problem 9.1: Critical Value (Part 1) Rejection Region Non Rejection Region 0  =.05 4.30

27. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-27 Demonstration Problem 9.1: Critical Value (Part 2) Rejection Region Non Rejection Region 0  =.05 4.30

28. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-28 Rejection Region Non Rejection Region 0  =.05 Demonstration Problem 9.1: Using the p-Value

29. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-29 Demonstration Problem 9.1: MINITAB Test of mu = 4.300 vs mu < 4.300 The assumed sigma = 0.574 VariableNMEANSTDEVSE MEANZP VALUE Ratings324.1560.5740.101-1.420.078

30. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-30 Demonstration Problem 9.1: Excel (Part 1)

31. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-31 Demonstration Problem 9.1: Excel (Part 2) H 0 :  = 4.3 H a :  < 4.3 34554554 44444445 44434443 54454445 n ==COUNT(A4:H7)  = 0.05 Mean ==AVERAGE(A4:H7) S ==STDEV(A4:H7) Std Error ==B12/SQRT(B9) Z ==(B11-B1)/B13 p-Value=NORMSDIST(B14)

32. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-32 Two-tailed Test:  Unknown,  =.05 (Part 1) Weights in Pounds of a Sample of 20 Plates 22.622.223.227.424.5 27.026.628.126.924.9 26.225.323.124.226.1 25.830.428.623.523.6

33. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-33 Two-tailed Test:  Unknown,  =.05 (part 2) Critical Values Non Rejection Region Rejection Regions

34. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-34 Two-tailed Test:  Unknown,  =.05 (part 3) Critical Values Non Rejection Region Rejection Regions

35. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-35 MINITAB Computer Printout for the Machine Plate Example Test of mu = 25.000 vs mu not = 25.000 VariableNMEANSTDEVSE MEANTP VALUE Platewt2025.5102.1930.4901.040.31

36. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-36 Machine Plate Example: Excel (Part 1)

37. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-37 Machine Plate Example: Excel (Part 2) AB CDE 1 H 0 :  = 25 2 H a :   25 3 4 22.622.223.227.424.5 5 2726.628.126.924.9 6 26.225.323.124.226.1 7 25.830.428.623.523.6 8 9 n ==COUNT(A4:E7) 10  = 0.05 11 Mean ==AVERAGE(A4:E7) 12 S ==STDEV(A4:E7) 13 Std Error ==B12/SQRT(B9) 14 t ==(B11-B1)/B13 15 p-Value=TDIST(B14,B9-1,2)

38. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-38 Demonstration Problem 9.2 (Part 1) Size in Acres of 23 Farms 445489474505553477545 463466557502449438500 466477557433545511590 561560

39. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-39 Demonstration Problem 9.2 (Part 2) Critical Value Non Rejection Region Rejection Region

40. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-40 Demonstration Problem 9.2 (Part 3) Critical Value Non Rejection Region Rejection Region

41. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-41 z Test of Population Proportion

42. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-42 Testing Hypotheses about a Proportion: Manufacturer Example (Part 1) Critical Values Non Rejection Region Rejection Regions

43. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-43 Testing Hypotheses about a Proportion: Manufacturer Example (Part 2) Critical Values Non Rejection Region Rejection Regions

44. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-44 Demonstration Problem 9.3 (Part 1) Critical Value Non Rejection Region Rejection Region

45. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-45 Demonstration Problem 9.3 (Part 2) Critical Value Non Rejection Region Rejection Region

46. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-46 Hypothesis Test for  2: Demonstration Problem 9.4 (Part 1) 0 df = 15.05.95 7.2609424.9958

47. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-47 Hypothesis Test for  2: Demonstration Problem 9.4 (Part 2) 0 df = 15.05.95 7.2609424.9958

48. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-48 Solving for Type II Errors: The Beverage Example Rejectio n Region Non Rejection Region  =0  =.05

49. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-49 Type II Error for Beverage Example with  =11.99 oz  =.05 Reject H o Do Not Reject H o   H o is True H o is False 95%  =.8023 Correct Decision Type I Error Type II Error Correct Decision 19.77%    Z0Z0 Z1Z1

50. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-50 Type II Error for Demonstration Problem 9.5, with  =11.96 oz  =.05  H o is True H o is False 95%  Reject H o Do Not Reject H o  =.0708 Correct Decision Type I Error Type II Error Correct Decision 92.92%   Z0Z0 Z1Z1

51. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-51  Values and Power Values for the Soft-Drink Example  Power 11.999.94.06 11.995.89.11 11.990.80.20 11.980.53.47 11.970.24.76 11.960.07.93 11.950.01.99

52. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-52 Operating Characteristic Curve for the Soft-Drink Example 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11.9511.9611.9711.9811.9912 Probability 

53. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-53 Power Curve for the Soft-Drink Example 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11.9511.9611.9711.9811.9912 Probability 