 Solution

Test the null hypothesis $H_0:{\theta }_m={\theta }_f=\theta$ against the alternative hypothesis that the two rate parameters differ. Under $H_0$, the pooled maximum-likelihood estimate is ${\hat{\theta }}_0=16/36.8\simeq 0.434783$. Under the alternative, the estimates are ${\hat{\theta }}_m=8/22\simeq 0.363636$ and ${\hat{\theta }}_f=8/14.8\simeq 0.540541$.

In the likelihood-ratio test statistic, the data-only terms cancel, and both fitted models have total $\hat{\theta }S=16$. Consequently,The alternative hypothesis adds one parameter, so Wilks theorem gives an approximate chi-squared distribution with one degree of freedom. Do not reject the common-rate model at the 5% level, since $T<3.841$. The corresponding approximate p-value is $0.43$. The fitted expected values are $5.5$ and $3.7$ days, but four observations per group supply little evidence that the underlying expected values differ; failure to reject does not establish equality. The chi-squared distribution calibration is asymptotic and should be treated cautiously for these small samples.