Show that [ begin{aligned} & P_{H_{1}}left(frac{bar{X}-mu_{0}}{frac{S}{sqrt{n}}} geq t_{n-1,1-alpha / 2} ight)+P_{H_{1}}left(frac{bar{X}-mu_{0}}{frac{S}{sqrt{n}}} Let X1, ..., X, be i.i.d.

Show that
\[
\begin{aligned}
& P_{H_{1}}\left(\frac{\bar{X}-\mu_{0}}{\frac{S}{\sqrt{n}}} \geq t_{n-1,1-\alpha / 2}\right)+P_{H_{1}}\left(\frac{\bar{X}-\mu_{0}}{\frac{S}{\sqrt{n}}}<-t_{n-1,1-\alpha / 2}\right) \\
= & P\left(t_{n-1, \gamma} \geq t_{n-1,1-\alpha / 2}\right)+P\left(t_{n-1, \gamma}<-t_{n-1,1-\alpha / 2}\right) .
\end{aligned}
\]

The right-hand side is the final expression for the power of the t-test in Section 4.4 .

Data from Section 4.4 

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Transcribed Image Text:

Let X1, ..., X, be i.i.d. N(, o). A test of the hypothesis : Ho o versus Hppo will reject at level a if and only if X-Ho >$n-11-a/2 SIn where tn-1,p is the pth quantile of the tn-1. It can be shown that (x - 20 ] = S Z+Y V/(n-1)