Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 3 / 8H / Solution

A 95 percent confidence set is a measurable data-dependent set $C(X_1,\dots ,X_n)$ whose repeated-sampling coverage satisfies ${\mathbb{P}}_{\theta }\{\theta \in C(X_1,\dots ,X_n)\}\ge 0.95$ for every parameter value, with equality for an exact 95 percent procedure. The parameter is fixed; the set is random. This is not a posterior probability statement about the parameter after the data have been observed.

For independent normal random variables of known variance, $\overline{X}\sim N(\mu ,{\sigma }^2/n)$, so $(\overline{X}-\mu )/(\sigma /\sqrt{n})$ has the standard normal distribution. The supplied quantile gives the exact confidence intervalIts deterministic length is $3.920\sigma /\sqrt{n}$. Requiring this to be at most $\epsilon$ givesAmong fixed-length intervals based equivariantly on the sample mean, the centred interval has greatest coverage: an interval of a given length captures most mass of a symmetric unimodal normal distribution when centred on its mean. Thus the width calculation is sharp for the usual fixed-length normal-mean construction.