Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 38 / 5 / b / Solution

If $D\sim \mathrm{Bin}(n,p_0)$, the observed binomial proportion $\hat{p}=D/n$ has variance $p_0(1-p_0)/n$. Consequently its standard error is proportional to $n^{-1/2}$. A common target therefore produces broad uncertainty at low volume and narrower uncertainty at high volume, forming the funnel-shaped envelope around the horizontal target. The narrowing represents sampling precision, not a predicted change of mortality with volume.