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I'm wondering how do we decide the "shape" of region of rejection in a hypothesis testing problem. Let's take a two-tailed z-test as an example. What if we choose the weird-looking region of rejection on the left instead of the more widely-used one on the right?

While I intuitively believe choosing the classic one is a better idea, the significance level of both tests are 0.05, so what makes one superior to the other?

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what makes one superior to the other?

The power of the left one is much lower than that of the right. To quote the Wikipedia entry:

The power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H0) when a specific alternative hypothesis (H1) is true.

Talking about significance without power doesn't have much meaning. You could simply ignore the observations, generate a random number between 0 and 1, and reject if less than something predetermined, for example.

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  • $\begingroup$ This answer looks convincing but I just cannot wrap my mind around this: is power related to the power function? Also since the power function is continuous, we can say $\alpha + \beta = 1$ at the “juncture” of H0 and H1. Thus, increasing power means reducing $\beta$, which would also increase $\alpha$, and that doesn’t seem good to me.., $\endgroup$
    – nalzok
    Jun 3, 2018 at 3:00

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