Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 2 / 8H (Statistics)

A $100\gamma \%$ confidence interval is a pair of statistics $L(X)\le U(X)$ whose random interval satisfiesAn exact confidence interval has equality throughout. The probability refers to repeated sampling at a fixed unknown parameter; the endpoints are random before observation. It is not a posterior probability assigned to the parameter after seeing the data. Here $0<\gamma <1$.

The sample mean $\overline{X}$ has normal distribution $N(\mu ,1/n)$ because the observations are independent with known variance 1. The pivotal quantity $Z=\sqrt{n}(\overline{X}-\mu )$ has the standard normal distribution. Consequentlyand inversion gives the confidence intervalReplacing 1.96 by the exact $0.975$ normal quantile gives exact 95% coverage; the printed numerical value gives the usual rounded interval. No variance estimation or Student distribution is needed.

Let $U_{(1)}=\min (U_1,U_2)$ and $U_{(2)}=\max (U_1,U_2)$ be the order statistics. For the symmetric uniform distribution, each observation is above or below $\theta$ with probability $1/2$, independently. ThusSo $\boxed{[U_{(1)},U_{(2)}]}$ is an exact 50% confidence interval.

The observed interval $[10,11.5]$ contains values impossible under the sampling model. Both observations must lie within one unit of the parameter, so the interval of all compatible values is $[U_{(2)}-1,U_{(1)}+1]=[10.5,11]$. It contains the true parameter almost surely. A support-restricted confidence interval for a uniform location parameter can therefore reportMore precisely, the procedure $[U_{(1)},U_{(2)}]\cap [U_{(2)}-1,U_{(1)}+1]$ retains exact 50% unconditional coverage, since the second interval contains $\theta$ almost surely. The intersection is a nonempty interval for every sample in the model and equals $[10.5,11]$ for these data. Reporting the full compatible interval alone is also valid and has coverage one. On this sample's large-spread event it lies wholly between the observations; this does not turn the unconditional 50% procedure into a generally 100% procedure.