= Solution
The <Poisson deviance> compares the fitted <Poisson regression> with a <saturated statistical model>:
$$
D=2\sum_i\left[y_i\log\frac{y_i}{\widehat\mu_i}-(y_i-\widehat\mu_i)\right],
$$
using $0\log0=0$. If the usual <deviance goodness-of-fit test> approximation is adequate, compare $27.2$ with a <chi-squared distribution> on $29$ <statistical degrees of freedom>. Its upper-tail <p-value> is about $0.561$, and $D/29=0.938$. \b[There is no detected lack of fit or evidence of substantial overdispersion from this deviance.] This neither proves the <Poisson regression> correct nor establishes that the slope is nonzero. Small expected counts, dependence or inadequate covariate specification can undermine the reference approximation, so <deviance residuals> and other <regression diagnostics> still matter.
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