# 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) =$\alpha$. – 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