I am working on a problem and I would like to have some advice. The following is the given information.
1), $X$ is exponential with mean $\frac{1}{\theta}$.
2), $H_0: \theta =5$ vs $H_1: \theta<5$
3), $\alpha=.05$
and our goal is
a), Find the rejection region.
b), Find the power function and determine where it is maximized.
What I have tried so far is this.
Edited:
$$ Pr[X > k | \theta = 5] = \alpha$$ so
$$e^{-k(5)} = \alpha$$ and $$\begin{align} k & = -\frac{1}{5}\ln (\alpha) \\ & \sim .599 \end{align}$$
So I am thinking that the rejection region is when the sample is greater than or equal to $.599 \space$ .
Moving on,
$$\begin{align} \gamma(\theta) & = Pr[X > .599|\theta < 5] \\ & = e^{-.599 \space \theta} \end{align}$$
I am more used to using concrete numbers rather than theory, so I am not too confident if I am doing this right.
I would really appreciate any help.