= Solution
The <mean> ratio is $R=(\lambda e^\beta)/\lambda=e^\beta$, so its logarithm is exactly $\beta$; the baseline coefficient $\alpha$ cancels. Under the stated asymptotic <normal approximation>, the <Wald confidence interval> for $\beta$ is $0.69\pm1.96(0.15)=(0.396,0.984)$. Exponentiation is monotone and therefore preserves the interval's coverage event. Hence
$$
\boxed{\widehat R=e^{0.69}\simeq1.99,\qquad
R\text{ has approximate }95\%\text{ CI }(1.49,2.68).}
$$
The <standard error> of $\alpha$ is not needed for this <confidence interval>: the <Poisson regression> already supplies the <standard error> of the contrast $\beta$. It would be incorrect to combine the two reported <standard errors> as if the <mean> ratio depended on both coefficients independently.
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