You may ignore the questions $a$, $b$ and $c$ as they are relatively easy. I know how to test hypotheses for a sample data but question $d$ is confusing me. How do I know what type of test statistic I should be using for this and what quantile distribution I should be comparing this with? We've only really dealt with testing hypotheses for sample distributions. Thanks in advance for any help.
If you draw a single observation attempting to test $H_0 : \theta=2$, then the standard test is the same as it usually is for the mean: you reject this null hypothesis with level of significance $1-\alpha$ if your result falls outside of an interval $[2-\delta,2+\delta]$ which has probability $1-\alpha$ when $\theta=2$. The probability of such an interval is $\int_{2-\delta}^{2+\delta} \frac{1}{2} e^{-t/2} dt=e^{-(2+\delta)/2}-e^{-(2-\delta)/2}=2e^{-1}\sinh(\delta/2)$ if $0 \leq \delta<2$ and $\int_0^{2+\delta} \frac{1}{2} e^{-t/2} dt = 1-e^{-(2+\delta)/2}$ if $\delta \geq 2$. If $\alpha=0.05$, you note that $2e^{-1}\sinh(1)<0.95$, so that you look at the second case. This gives the interval as $[0,-2\log(0.05)] \approx [0,6]$. You accept this null hypothesis given a result in this interval and reject it otherwise.