Chapter 9 Hypothesis Testing

Introduction Chapters 7 and 8 show how a sample could be used to develop a point and interval estimate of population parameters. Now we’ll use many of the same techniques to test hypotheses (statements) about the value of a population parameter.

I.Null and Alternative Hypotheses Begin with a tentative hypothesis about a parameter. This is the Null hypothesis, Ho. The opposite of what’s stated in the Null is the Alternative hypothesis, Ha. Gather data in a sample to test Ho.

A. An Example A Criminal Trial is a good example of a hypothesis test. A person is “Innocent until proven guilty.”Criminal Trial Null Ho: Innocent. Alternative Ha: Guilty A test is conducted (trial), data are gathered (evidence). If data are inconsistent with “Innocent”, we reject Ho and presume Ha, “guilt”.

B. 3 Business Examples There are 3 common situations where hypothesis tests are used. 1. Testing a research hypothesis. Ex. A new carburetor is designed to increase gas mileage above the current 24 mpg. Ho:   24 mpg. Ha:  >24 (the research hypothesis)

2. Testing the validity of a claim. Ex. A soft drink label claims at least 67.6 fluid ounces of drink. Ho:   67.6 Ha:  < 67.6 If you reject Ho, penalties may be enforced.

3. Decision making, where action is taken in either case. Ex. A small part is machined to precise specifications. One of those specs is that it must be 2” wide. Ho:  = 2 inches Ha:   2 inches

II. Type I and II Errors Ideally, a hypothesis test should lead to acceptance of Ho when the null is true, and the rejection of Ho when the alternative is true. Regrettably, there are times when our sample will cause us to make a mistake.

A. An Illustration

B. Consequences Let  be the probability of making a type I error. Let  be the probability of making a type II error. Recall the carburetor example. Ho:   24 mpg. Ha:  >24 mpg.

If we reject Ho, we can conclude that the new design does in fact increase mpg above 24. But what about the possibility of a type I error? Type I: Researchers conclude that the new design increases mpg when in fact it’s no better than the old model. Consequences?

Type II: Researchers conclude that the new design is no better than the current model when in fact it does increase mpg. Consequences?

C. What to do? Type I: Specify a maximum allowable probability of making the type I error. This is called a level of significance for the test. Usually  =.10,.05, or.01.

Don’t forget about type II! Type II: This is more difficult to account for, so you qualify your conclusion statement. Example. If you decide to accept Ho, it is usually stated as “failure to reject Ho”, or “we do not reject Ho”, instead of a more definitive “we accept Ho.” By qualifying your statement, you acknowledge the probability of a type II error.