Understanding the Power of a t-Test

[Mogarrr]

I'm looking over my notes, and I'm puzzled by this non-central t-distribution, and why it is the alternative hypothesis.

Non-central t-distribution: If $X \sim (\delta, 1)$ and $Y \sim \chi^2_{r}$, in addition if X and Y are independent random variables, then $\frac {X}{\sqrt{\frac {Y}{r}}}$ has a t-distribution with non-centrality parameter, $\delta$

$\frac {\frac {\bar{X} - \mu}{\frac {\sigma}{\sqrt{n}}}}{\sqrt{\frac {(n-1)s^2}{\sigma^2 (n-1)}}} = \frac {\bar{X} - \mu_0}{\frac {s}{\sqrt{n}}} \sim t_{\delta = \frac {\bar{X} - \mu}{\frac {\sigma}{\sqrt{n}}}}$

$H_0 : \mu = \mu_0$, $H_1 : \mu \neq \mu_0$

$1 - \beta = P(\frac {|\bar{X} - \mu_0|}{\frac {s}{\sqrt{n}}} > t_{1 - \frac {\alpha}2} | \delta = \frac {\mu - \mu_0}{\frac {\sigma}{\sqrt{n}}}, n-1)$

I do see that the general form for the power of a test is P{null is rejected | alternative is true}, but why is it that the alternative hypothesis is this crazy looking distribution?

 

[DrDu]

The noncentral t distribution arises as you are testing a special alternative, namely that X is distributed normally around delta, and not around 0 as in the null hypothesis.