Suppose you are reading about a statistically significant result * x* (

*just*at level α) from a one-sided test T+ of the mean of a Normal distribution with

*n*iid samples, and (for simplicity) known σ:

*H*

_{0}: µ ≤

_{ }0 against

*H*

_{1}: µ >

_{ }0.

I have heard some people say [0]:

A. If the test’s power to detect alternative µ’ is very low, then the statistically significant

isxpoorevidence of a discrepancy (from the null) corresponding to µ’. (i.e., there’s poor evidence that µ > µ’ ).◊See point on language in notes.They will generally also hold that if POW(µ’) is reasonably high (at least .5), then the inference to µ > µ’ is warranted, or at least not problematic.

I have heard other people say:

B. If the test’s power to detect alternative µ’ is very low, then the statistically significant

isxgoodevidence of a discrepancy (from the null) corresponding to µ’ (i.e., there’s good evidence that µ > µ’).They will generally also hold that if POW(µ’) is reasonably high (at least .5), then the inference to µ > µ’ is

unwarranted.

**Which is correct, from the perspective of the (error statistical) philosophy, within which power and associated tests are defined?**

Allow the test assumptions are adequately met. I have often said on this blog, and I repeat, the most misunderstood and abused (or unused) concept from frequentist statistics is that of a test’s power to reject the null hypothesis under the assumption alternative µ’ is true: POW(µ’). I deliberately write it in this correct manner because it is faulty to speak of the power of a test without specifying against what alternative it’s to be computed. It will also get you into trouble if you define power as in the first premise in a recent post: Continue reading

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