What is the power of the test if the null hypothesis is true?

Question

E.g. the null hypothesis is $\beta = 0$, while the true value of $\beta$ is 0. What would be the power of the hypothesis test?

My way of thinking is: Since the power of the test is 1- Prob(Type II error) (Type II error is the error to accept the null hypothesis when it is false). However, since the null hypothesis is true here, Prob(Type II error)=0. So the power of the test is 1. Am I correct?

Power given the null hypothesis is true = Prob(Type I error) =$\alpha$. — user158565, Oct 16 '18 at 15:29
@a_statistician brief though that is why not post it as an answer? — mdewey, Oct 16 '18 at 15:56
@mdewey Yes. Because it is simple and common knowledge in statistics. — user158565, Oct 16 '18 at 16:06

score 5 · Accepted Answer · answered Oct 16 '18 at 15:23

5

If the null hypothesis is true, the concept of power doesn't make sense. Power is the probability of drawing a sample that causes you to reject the null hypothesis when the null hypothesis is false. It has no meaning when the null hypothesis is true.

answered Oct 16 '18 at 15:23

Noah

33,180
3
47
105

2

Well, power is usually seen as a function of parameter value. Then the power at the null is simply $\alpha$, the significance level. – kjetil b halvorsen Jul 13 '19 at 18:21

What is the power of the test if the null hypothesis is true?

1 Answers1

Linked