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

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) =$\alpha$. Oct 16, 2018 at 15:29
• @a_statistician brief though that is why not post it as an answer? Oct 16, 2018 at 15:56
• @mdewey Yes. Because it is simple and common knowledge in statistics. Oct 16, 2018 at 16:06

1 Answer

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.

• Well, power is usually seen as a function of parameter value. Then the power at the null is simply $\alpha$, the significance level. Jul 13, 2019 at 18:21