= Solution
Treat each release as an independent Bernoulli observation with a two-week event <probability>, giving two independent <binomial distributions>. Under the specified effect, the <probabilities> are $p_0=0.0016$ and $p_1=0.0008$ with $n=15000$ in each period. The expected event counts are therefore 24 and 12. For a conventional two-sided 5% comparison of proportions, let $D=\widehat p_0-\widehat p_1$, $\delta=p_0-p_1=0.0008$ and $\bar p=(p_0+p_1)/2=0.0012$. Use
$$
s_0=\sqrt{\frac{2\bar p(1-\bar p)}n}=0.00039976,\qquad
s_1=\sqrt{\frac{p_0(1-p_0)+p_1(1-p_1)}n}=0.00039973.
$$
The approximate null rejection region is $|D|>1.96s_0$, and under the alternative $D$ is approximately $N(\delta,s_1^2)$. Thus the <power of a two-sample rare-event comparison> is
$$
\begin{aligned}
\operatorname{Power}
&\approx1-\Phi\left(\frac{1.96s_0-\delta}{s_1}\right)
+\Phi\left(\frac{-1.96s_0-\delta}{s_1}\right)\\
&\approx0.5165.
\end{aligned}
$$
Here $\Phi$ is the <standard normal distribution function>. Hence the normal approximation gives $\boxed{\text{two-sided power approximately }52\%}$: detecting the proposed halving is far from assured.
For the small counts, a discrete calculation is also appropriate. Approximate the two counts by independent <Poisson distributions> with means 24 and 12. Conditional on their total $K$, the pre-intervention count is $\operatorname{Binomial}(K,1/2)$ under equal rates and $\operatorname{Binomial}(K,2/3)$ under the halving alternative; $K\sim\operatorname{Poisson}(36)$ under that alternative. If $\mathcal R_k$ is the rejection set of the symmetric exact two-sided 5% binomial test, its power is
$$
\sum_{k=0}^{\infty}e^{-36}\frac{36^k}{k!}
\sum_{j\in\mathcal R_k}\binom kj(2/3)^j(1/3)^{k-j}\approx0.454.
$$
This conservative discrete test has about 45% power, lower than the normal approximation because its achieved level can be below 5%. If a decrease-only one-sided 5% test had been specified in advance, replace 1.96 by 1.645; the normal-approximation power is about 64%. The tail convention and the chosen test must accompany any quoted <statistical power>.
Back to article page
