= Solution
Model deaths in disjoint equal-duration periods as independent observations with <Poisson distributions> with comparable population exposure and ascertainment. The expected before and after counts are $\mu_0=400$ and $\mu_1=320$. Their difference has
$$
\mathbb E(D_0-D_1)=80,\qquad\operatorname{Var}(D_0-D_1)=400+320=720.
$$
The expected signal relative to its standard deviation is therefore
$$
\boxed{80/\sqrt{720}\simeq2.98.}
$$
Under the null of equal rates, condition on $N=D_0+D_1$: the first-period count has a <binomial distribution> with parameters $N$ and $1/2$. At a total near 720, the two-sided 5% rejection boundary for $D_0-D_1$ is approximately $1.96\sqrt{720}$. Under the proposed alternative, conditional on that total, its mean is $N/9$ and its variance is $80N/81$. Thus at $N=720$ the <normal approximation> gives power approximately
$$
\boxed{\Phi\left(\frac{80-1.96\sqrt{720}}{\sqrt{720\cdot80/81}}\right)\simeq0.85.}
$$
The lower-tail rejection probability is negligible at this alternative; allowing the Poisson total to fluctuate and using discrete rejection cutoffs changes the approximation slightly: an exact two-sided conditional test has about $83.8\%$ power under these Poisson means. \b[Two years on each side should have adequate power for a 20% reduction under this model], though a particular realization can still fail to detect it.
This conclusion needs stable recording of drug-related deaths, comparable exposure denominators and no substantial <overdispersion> or temporal dependence. A change in population size, certification practice or other drug use can affect the comparison. Detecting a decline does not establish that <mephedrone> caused it: attributing displacement requires a credible comparison of <potential outcomes>, using an otherwise expected trend or other controls for changes occurring at the same time.
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