 Solution

A frequentist confidence interval is a pair of data-dependent endpoints whose random interval covers the fixed parameter with probability at least $1-\alpha$ under every allowed parameter value. In repeated uses of the same sampling procedure, at least the advertised fraction of intervals cover the true value. It is not a posterior probability assigned to the parameter after observing the data.

For a sample of size $n$ from a Poisson law with mean $\mu$, the sum $S$ is Poisson with mean $n\mu$. At observed $s>0$, define the lower and upper limits byAt $s=0$ set $L=0$ and use the same upper equation. This inverts the two exact one-sided tail tests. Discreteness means the resulting central interval is generally conservative: its coverage is at least the nominal value, not identically equal to it.

Here one observes a four-year total of 39, so the lower endpoint satisfies ${\varphi }_L(38)=0.975$ and the upper satisfies ${\varphi }_U(39)=0.025$. Selecting the corresponding entries supplied in the source givesThis interval concerns the expected four-year total, not an annual rate. The other two supplied numbers correspond to the wrong inclusive tail for one of these endpoints.