= Solution
{c}
A <confidence interval> is a pair of statistics $L(X),U(X)$ such that, for each parameter value $\mu$,
$$
\mathbb P_\mu\{L(X)\leq\mu\leq U(X)\}\geq1-\alpha.
$$
In repeated sampling, at least the stated fraction of intervals constructed by this rule cover the fixed unknown parameter. It is not a posterior <probability> statement about the parameter after one interval has been observed. Discrete exact constructions can have coverage greater than the nominal value.
For independent observations from a <Poisson distribution>, $S=\sum_iX_i$ has a <Poisson distribution> with mean $\lambda=n\mu$. Invert equal-tail tests of that total mean. At an observed integer $s>0$, the lower endpoint solves
$$
\mathbb P_{\lambda_L}(S\geq s)=\alpha/2
\iff F_{\lambda_L}(s-1)=1-\alpha/2,
$$
and the upper endpoint solves
$$
\mathbb P_{\lambda_U}(S\leq s)=\alpha/2
\iff F_{\lambda_U}(s)=\alpha/2.
$$
Then divide both endpoints by $n$. If $s=0$, use lower endpoint zero. Each inversion allocates at most $\alpha/2$ noncoverage <probability> to one side, giving at least $1-\alpha$ total coverage.
Here the single total count is 39. The supplied values with the correct adjacent count indices are $F_{27.7}(38)=0.975$ for the lower endpoint and $F_{53.3}(39)=0.025$ for the upper one. Thus
$$
\boxed{\mu\in[27.7,\ 53.3]\quad\text{at the stated 95\% level}.}
$$
The values 28.6 and 52.2 use the other count index and are not the endpoints of this central interval. This is <adjacent count tails in an exact Poisson confidence interval>.
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