= Solution
Let $Y_i$ be the number of maintenance jobs, $t_i$ average temperature and $p_i$ average precipitation. The fitted <Poisson regression> assumes independent conditional responses with
$$
Y_i\mid t_i,p_i\sim\operatorname{Poisson}(\mu_i),\qquad
\log\mu_i=\beta_0+\beta_Tt_i+\beta_Pp_i.
$$
The logarithm is the <Poisson canonical link>, so fitted means are always positive. The estimated <regression coefficients> are
$$
\boxed{(\widehat\beta_0,\widehat\beta_T,\widehat\beta_P)=(4.079374,-0.006162,-0.002922).}
$$
At fixed precipitation, one unit of temperature multiplies the fitted mean by $e^{-0.006162}\simeq0.99386$, a decrease of about $0.614\%$. The fitted precipitation multiplier is $e^{-0.002922}\simeq0.99708$ per unit, but its large <p-value> gives little evidence for that effect. These are conditional associations, not established causal effects.
The intercept-only model has $n-1=23$ residual degrees of freedom, and the full model has $n-3=21$. Both imply \b[$n=24$ months].
For an approximate <deviance goodness-of-fit test> of the full <Poisson regression>, compare residual deviance $23.527$ with $\chi^2_{21}$. Its upper-tail <p-value> $0.3165361$ supplies \b[no evidence of lack of fit]. The ratio $23.527/21\simeq1.120$ also gives no striking indication of <overdispersion>, though deviance per degree of freedom is only a rough dispersion check. Fitted count means around fifty make the usual chi-squared approximation plausible. Independence between months, the conditional variance-equals-mean assumption and the absence of residual structure still need diagnostics. Failure to reject is not proof that the model is correct.
The null deviance $37.969$ on $23$ degrees of freedom has $p=0.02566753$, suggesting the constant-mean model is inadequate. In the sequential <analysis of deviance for nested generalized linear models>, adding temperature to that model reduces deviance by $14.204$ on one degree of freedom, giving $p=0.000164$. Adding precipitation after temperature reduces it by only $0.238$, with $p=0.625677$. Therefore \b[retain temperature; these data give no reason to retain precipitation after temperature]. The temperature-only residual deviance is $23.765$ on $22$ degrees of freedom. This prediction-oriented reduced model should be refitted before reporting its coefficients: the printed temperature coefficient belongs to the full model. The <Wald test> results, $p=0.000833$ and $0.625582$, broadly support the same conclusion, while the sequential deviance tests answer the stated nested-model questions.
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