Solution (source code)

= Solution

Define the difference as control minus intervention, positive for fewer deaths under intervention assignment. The observed difference per thousand is
$$
\widehat\Delta=1000\left(\frac{148}{30000}-\frac{123}{30000}\right)=\frac{25}{30}\simeq0.833.
$$
For independent <binomial distributions>, its unpooled standard error per thousand is
$$
\widehat{\rm SE}=1000\sqrt{\frac{\widehat p_C(1-\widehat p_C)}{30000}+\frac{\widehat p_I(1-\widehat p_I)}{30000}}\simeq0.547.
$$
The approximate 95% <confidence interval> is therefore
$$
\boxed{0.833\pm1.96(0.547)\simeq(-0.24,\ 1.91)\text{ fewer deaths per thousand}.}
$$
The rare-event Poisson standard error $\sqrt{148+123}/30$ gives almost the same interval. \b[A reduction of 1.5 per thousand is consistent with the data], since it lies inside the interval. Zero also lies inside, so a nonzero benefit is not established at the two-sided 5% significance level. This does not establish equivalence or prove the proposed effect size, and cannot by itself establish <treatment contamination>.