© 2002 Thomson / South-Western Slide 9-1 Chapter 9 Hypothesis Testing with Single Samples

© 2002 Thomson / South-Western Slide 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. 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.

© 2002 Thomson / South-Western Slide 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

© 2002 Thomson / South-Western Slide 9-4 Hypothesis Testing A process of testing hypotheses about parameters by setting up null and alternative hypotheses, gathering sample data, computing statistics from the samples, and using statistical techniques to reach conclusions about the hypotheses.

© 2002 Thomson / South-Western Slide 9-5 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.

© 2002 Thomson / South-Western Slide 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.

© 2002 Thomson / South-Western Slide 9-7 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.

© 2002 Thomson / South-Western Slide 9-8 Rejection and Nonrejection Regions  =12 oz Nonrejection Region Rejection Region Critical Value Rejection Region Critical Value

© 2002 Thomson / South-Western Slide 9-9 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 . –Power is the probability of rejecting a false null hypothesis, and equal to 1- 

© 2002 Thomson / South-Western Slide 9-10 Decision Table for Hypothesis Testing ( () Null TrueNull False Fail to reject null Correct Decision Type II error  ) Reject nullType I error  Correct Decision (Power)

© 2002 Thomson / South-Western Slide 9-11 One-tailed Tests One-tailed and Two-tailed Tests Two-tailed Test

© 2002 Thomson / South-Western Slide 9-12 One-tailed Tests  =12 oz Rejection Region Nonrejection Region Critical Value  =12 oz Rejection Region Nonrejection Region Critical Value

© 2002 Thomson / South-Western Slide 9-13 Two-tailed Tests  =12 oz Rejection Region Nonrejection Region Critical Values Rejection Region

© 2002 Thomson / South-Western Slide 9-14 CPA Net Income Example: Two-tailed Test Rejection Region Nonrejection Region  =0 Rejection Region

© 2002 Thomson / South-Western Slide 9-15 CPA Net Income Example: Critical Value Method (Part 1) Rejection Region Nonrejection Region  =0 Rejection Region 72,22377,605

© 2002 Thomson / South-Western Slide 9-16 CPA Net Income Example: Critical Value Method (Part 2) Rejection Region Nonrejection Region  =0 Rejection Region 72,22377,605

© 2002 Thomson / South-Western Slide 9-17 Demonstration Problem 9.1 (Part 1) Rejection Region Nonrejection Region 0  =.05

© 2002 Thomson / South-Western Slide 9-18 Demonstration Problem 9.1 (Part 2) Rejection Region Nonrejection Region 0  =

© 2002 Thomson / South-Western Slide 9-19 Rejection Region Nonrejection Region 0  =.05 Demonstration Problem 9.1 (Part 3)

© 2002 Thomson / South-Western Slide 9-20 Two-tailed Test: Small Sample,  Unknown,  =.05 (Part 1) Weights in Pounds of a Sample of 20 Plates

© 2002 Thomson / South-Western Slide 9-21 Two-tailed Test: Small Sample,  Unknown,  =.05 (Part 2) Critical Values Nonrejection Region Rejection Regions

© 2002 Thomson / South-Western Slide 9-22 Two-tailed Test: Small Sample,  Unknown,  =.05 (Part 3) Critical Values Non Rejection Region Rejection Regions

© 2002 Thomson / South-Western Slide 9-23 Demonstration Problem 9.2 (Part 1) Size in Acres of 23 Farms

© 2002 Thomson / South-Western Slide 9-24 Demonstration Problem 9.2 (Part 2) Critical Value Nonrejection Region Rejection Region

© 2002 Thomson / South-Western Slide 9-25 Demonstration Problem 9.2 (Part 3) Critical Value Nonrejection Region Rejection Region

© 2002 Thomson / South-Western Slide 9-26 Z Test of Population Proportion

© 2002 Thomson / South-Western Slide 9-27 Testing Hypotheses about a Proportion: Manufacturer Example (Part 1) Critical Values Nonrejection Region Rejection Regions

© 2002 Thomson / South-Western Slide 9-28 Testing Hypotheses about a Proportion: Manufacturer Example (Part 2) Critical Values Nonrejection Region Rejection Regions

© 2002 Thomson / South-Western Slide 9-29 Demonstration Problem 9.3 (Part 1) Critical Value Nonrejection Region Rejection Region

© 2002 Thomson / South-Western Slide 9-30 Demonstration Problem 9.3 (Part 2) Critical Value Nonrejection Region Rejection Region