= Solution
{c}
The total $S$ under $H_0$ has <Poisson distribution> with mean $12$, and the observed total is $16$. Its one-sided <p-value> is therefore
$$
\boxed{\mathbb P_{H_0}(S\geq16)=\sum_{k=16}^\infty e^{-12}\frac{12^k}{k!}\approx0.1556.}
$$
The supplied $\phi_{12}(17)\approx0.1$ also shows that the observed count does not enter the approximate 10% upper-tail rejection region. Thus \b[there is insufficient evidence to reject $\mu=1$ at either the 10% or 5% level]. The question specifies no testing level, so the p-value gives the general conclusion: rejection is justified only at levels at least about $0.1556$.
The printed $\phi_1(16)\approx0$ concerns a Poisson variable with mean one, not the twelve-month total. It is therefore not the null tail for this data; substituting it would incorrectly treat the yearly count as one month's count.
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