# Power Analysis and Non-Central t Distribution

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?