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 $Exp(\lambda )$ 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 $P(X<C)|{H}_{0})=1-{e}^{-\lambda x}=1-{e}^{-0.2x}$ So $1-{e}^{-0.2x}=0.05$ so x=0.25 and the critical region is $(-\mathrm{\infty},0.25]$, is it the right method to solve this question?

Let the random variable X be the waiting time until the next student has to go to the toilet. Assume that X has an $Exp(\lambda )$ 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 $P(X<C)|{H}_{0})=1-{e}^{-\lambda x}=1-{e}^{-0.2x}$ So $1-{e}^{-0.2x}=0.05$ so x=0.25 and the critical region is $(-\mathrm{\infty},0.25]$, is it the right method to solve this question?