# Determine the critical region for this test at significance level &#x03B1;<!-- α --> = 0.

Determine the critical region for this test at significance level $\alpha =0.05$ .
Let the random variable X be the waiting time until the next student has to go to the toilet. Assume that X has an $Exp\left(\lambda \right)$ distribution with unknown $\lambda$. We test ${H}_{0}:\lambda =0.2$ against ${H}_{1}:\lambda <0.2$, where we use X as test statistic. Determine the critical region for this test at significance level $\alpha =0.05$.
In my opinion I should calculate $P\left(X So $1-{e}^{-0.2x}=0.05$ so x=0.25 and the critical region is $\left(-\mathrm{\infty },0.25\right]$, is it the right method to solve this question?
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Melina Richard
In fact, any region $R\subset \mathbb{R}$ such that
${P}_{\lambda =0.2}\left(X\in R\right)=0.05$
is a correct answer, in theory. But some of these answers would be absurd in practice.
Since the alternative hypothesis is that $\lambda <0.2$, and since this is equivalent to say that $E\left(X\right)>\frac{1}{0.2}=5$, we see that it is only reasonable to reject ${H}_{0}$ when X takes values sensibly greater than 5. That is, the critical region should be
$\left(C,\mathrm{\infty }\right)$
for $C\in \mathbb{R}$ such that
$P\left(X\in \left(C,\mathrm{\infty }\right)\right)=P\left(X>C\right)=1-\left(1-{e}^{-0.2C}\right)={e}^{-0.2C}=0.05.$
So $C=-5\mathrm{ln}\left(0.05\right)\approx 14.98$ and the critical region is
$\left(14.98,+\mathrm{\infty }\right).$.