m > 3
Suppose that we want to test the hypothesis that m 5. Then we can think of our opponent suggesting that m = 5. We call the opponent's hypothesis the null hypothesis and write:
H0: m = 5
and our hypothesis the alternative hypothesis and write
H1: m 5
For the null hypothesis we always use equality, since we are comparing m with a previously determined mean.
For the alternative hypothesis, we have the choices: < , > , or .
Procedures in Hypothesis Testing
When we test a hypothesis we proceed as follows:
Errors in Hypothesis Tests
We define a type I error as the event of rejecting the null hypothesis when the null hypothesis was true. The probability of a type I error (a) is called the significance level.
We define a type II error (with probability b) as the event of failing to reject the null hypothesis when the null hypothesis was false.
Suppose that you are a lawyer that is trying to establish that a company has been unfair to minorities with regard to salary increases. Suppose the mean salary increase per year is 8%.
You set the null hypothesis to be
H0: m = .08
H1: m < .08
Q. What is a type I error?
A. We put sanctions on the company, when they were not being discriminatory.
Q. What is a type II error?
A. We allow the company to go about its discriminatory ways.
Note: Larger a results in a smaller b, and smaller a results in a larger b.