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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?

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  • $\begingroup$ Power given the null hypothesis is true = Prob(Type I error) =$\alpha$. $\endgroup$
    – user158565
    Oct 16, 2018 at 15:29
  • $\begingroup$ @a_statistician brief though that is why not post it as an answer? $\endgroup$
    – mdewey
    Oct 16, 2018 at 15:56
  • $\begingroup$ @mdewey Yes. Because it is simple and common knowledge in statistics. $\endgroup$
    – user158565
    Oct 16, 2018 at 16:06

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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.

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

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