Suppose $X_1, X_2, \ldots, X_n$ is a random sample from an exponential distribution with probability density function
\[
f(x \ ; \ \lambda) =
\begin{cases}
\lambda e^{-\lambda x},&\text{if } x > 0, \\
0,&\text{otherwise}.
\end{cases}
\]
Consider the hypotheses $H_0 : \lambda = \lambda_0$ and $H_1 : \lambda = \lambda_1$ and a significance level $\alpha$.
Show that if $\lambda_0 < \lambda_1$, then the test given by the Neyman-Pearson lemma is equivalent to rejecting $H_0$ if $\bar{X} \le k$, where $k$ is chosen so that $\alpha = P(\bar{X} \le k \ | \ H_0)$.
Show that if $\lambda_0 > \lambda_1$, then the test given by the Neyman-Pearson lemma is equivalent to rejecting $H_0$ if $\bar{X} \ge k$, where $k$ is chosen so that $\alpha = P(\bar{X} \ge k \ | \ H_0)$.