Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-1 Business Statistics, 3e by Ken Black Chapter 9 Statistical Inference: Hypothesis Testing for Single Populations

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 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. Use large samples to test hypotheses about a single population mean and about a single population proportion. Test hypotheses about a single population mean using small samples when  is unknown and the population is normally distributed. Test hypotheses about a single population variance.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-3 Method of Indirect Proof X X Y Either X or Y is true but not both X is demonstrated not to be true Y Y Y is true by default

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-4 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.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-5 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.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-6 Null and Alternative Hypotheses: Example A soft drink company is filling 12 oz. cans with cola. The company hopes that the cans are averaging 12 ounces.

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-7 Rejection and Non Rejection Regions  =12 oz Non Rejection Region Rejection Region Critical Value Rejection Region Critical Value

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-8 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 .

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-9 Decision Table for Hypothesis Testing ( () Null TrueNull False Fail to reject null Correct Decision Type II error  ) Reject nullType I error  Correct Decision

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-10 One-tailed Tests One-tailed and Two-tailed Tests Two-tailed Test

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-11 One-tailed Tests  =12 oz Rejection Region Non Rejection Region Critical Value  =12 oz Rejection Region Non Rejection Region Critical Value

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-12 Two-tailed Tests  =12 oz Rejection Region Non Rejection Region Critical Values Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-13 CPA Net Income Example: Two-tailed Test Rejection Region Non Rejection Region  =0 Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-14 MINITAB Computer Printout of Z Test for CPA Net Income Test of mu = vs. mu not = The assumed sigma = VariableNMEANSTDEVSE MEANZP VALUE Net income

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-15 CPA Net Income Example: Critical Value Method (Part 1) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-16 CPA Net Income Example: Critical Value Method (Part 2) Rejection Region Non Rejection Region  =0 Rejection Region 72,22377,605

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-17 Demonstration Problem 9.1 (Part 1) Rejection Region Non Rejection Region 0  =.05

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-18 Demonstration Problem 9.1 (Part 2) Rejection Region Non Rejection Region 0  =

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-19 Rejection Region Non Rejection Region 0  =.05 Demonstration Problem 9.1 (Part 3)

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-20 Demonstration Problem 9.1 (Part 4) Test of mu = vs mu < The assumed sigma = VariableNMEANSTDEVSE MEANZP VALUE Impcusat

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-21 Two-tailed Test: Small Sample,  Unknown,  =.05 (Part 1) Weights in Pounds of a Sample of 20 Plates

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-22 Two-tailed Test: Small Sample,  Unknown,  =.05 (part 2) Critical Values Non Rejection Region Rejection Regions

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-23 Two-tailed Test: Small Sample,  Unknown,  =.05 (part 3) Critical Values Non Rejection Region Rejection Regions

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-24 MINITAB Computer Printout for the Machine Plate Example Test of mu = vs mu not = VariableNMEANSTDEVSE MEANTP VALUE Platewt

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-25 Demonstration Problem 9.2 (Part 1) Size in Acres of 23 Farms

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-26 Demonstration Problem 9.2 (Part 2) Critical Value Non Rejection Region Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-27 Demonstration Problem 9.2 (Part 3) Critical Value Non Rejection Region Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-28 Z Test of Population Proportion

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-29 Testing Hypotheses about a Proportion: Manufacturer Example (Part 1) Critical Values Non Rejection Region Rejection Regions

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-30 Testing Hypotheses about a Proportion: Manufacturer Example (Part 2) Critical Values Non Rejection Region Rejection Regions

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-31 Demonstration Problem 9.3 (Part 1) Critical Value Non Rejection Region Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-32 Demonstration Problem 9.3 (Part 2) Critical Value Non Rejection Region Rejection Region

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-33 Hypothesis Test for  2: Demonstration Problem 9.4 (Part 1) 0 df =

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-34 Hypothesis Test for  2: Demonstration Problem 9.4 (Part 2) 0 df =

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-35 Solving for Type II Errors: The Beverage Example Rejectio n Region Non Rejection Region  =0  =.05

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-36 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

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-37 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

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-38  Values and Power Values for the Soft-Drink Example  Power

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-39 Operating Characteristic Curve for the Soft-Drink Example Probability 

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 9-40 Power Curve for the Soft-Drink Example Probability 