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By going through the method in which we reject a null hypothesis: any observed sample mean value which falls in the rejection region; that is when test statistic > Z(1-alpha) (consider > alternate hypothesis). The definition of Power is the probability of getting an extreme (=alpha) or even more than that. I infer that both refers to the same region and hence their area is nothing different and thus Power is the probability of rejecting a null hypothesis when it is FALSE. Please let me know if I am missing anything. Help much appreciated.

PS: I dont have a statistical background

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Power is the probability of rejecting a null hypothesis when it is FALSE

This is correct; specifically at some particular point in the alternative (each has its own power).

By contrast, the rejection region different; indeed it is not even a probability, it's a subset of the values that the test statistic can take (in particular, the subset for which you will reject the null hypothesis).

Power is the probability of getting a test statistic in the rejection region, at some particular alternative (some point in the space of the alternative). If the alternatives are indexed by a parameter you can find the power as a function of it (obtaining a power curve). The rejection region is the same at each point but the power would be different.

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