Solution (source code)

= Solution

The coefficient labelled `yr2` identifies year as a factor. With $z_i=0$ in the first year and $z_i=1$ in the second, the <Poisson regression> assumes independent daily counts conditional on year,
$$
Y_i\sim\operatorname{Pois}(\mu_i),\qquad
\log\mu_i=\beta_0+\beta_1z_i.
$$
The unknown mean <statistical parameters> are the first-year log daily rate $\beta_0$ and the second-year log rate ratio $\beta_1$; the <dispersion parameter> is fixed at one. Approximate 95% <Wald confidence intervals> are
$$
\begin{aligned}
\widehat\beta_0&=1.78810,&\quad \beta_0&\in1.78810\pm1.96(0.02141)=(1.74614,1.83006),\\
\widehat\beta_1&=-0.09816,&\quad \beta_1&\in-0.09816\pm1.96(0.03105)=(-0.15902,-0.03730).
\end{aligned}
$$
Exponentiating yields the first-year fitted daily mean $5.978$, with <confidence interval> $(5.732,6.234)$, and
$$
\boxed{\frac{\widehat\mu_2}{\widehat\mu_1}=0.9065,\qquad
\frac{\mu_2}{\mu_1}\in(0.8530,0.9634).}
$$
The second-year fitted daily mean is $e^{1.78810-0.09816}=5.419$. Thus this <Poisson regression> estimates a 9.35% fall, with an approximate interval for the percentage fall from 3.66% to 14.70%. These <confidence intervals> rely on the <Poisson distribution> and independence assumptions.