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I'm a bit confused about power analysis and the non-central t-distribution.

If I am in a situation where $H_0$ is $\mu = 0$, and $H_1$ is $\mu > 0$, I calculate for what value $\mu$ would have a probability of 95% or less of occurring. I do this by saying that, if $\mu$ really were 0, then $\frac{\bar{X}}{s(X)}$ has a t-distribution.

Now for a power analysis, I suppose that $\mu$ is something like 1.5 and then I calculate the probability of correctly rejecting $H_0$. It seems then that $\frac{\bar{X} - 1.5}{s(X)}$ should be a t-distribution also. I can then calculate the probability of exceeding the critical value (that would be obtained by assuming $H_0$) I have using this distribution.

Why would the distribution of the test-statistic change based on what I consider $H_0$ to be? I must be missing something.

Update: It looks like the answer may be that (the way I'm being taught) is that you use the same test statistic as you would under ${H_0}$, but it now has a non-central t-distribution because of the new assumed value of $\mu$. However, instead of using the non-central t distribution -- you could do what I described in my third paragraph. Is that correct?

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