= Solution
The null hypothesis is that mortality is independent of carbolic-acid use, equivalently that the two mortality probabilities are equal. The two-sided alternative is that they differ.
Use the <Chi-squared test of independence>, equivalently the likelihood-ratio statistic, with Pearson's statistic
$$
X^2=\sum_{i,j}\frac{(O_{ij}-E_{ij})^2}{E_{ij}},
$$
where $E_{ij}$ are the fitted counts under independence using the observed row and column totals. Under the null, $X^2$ is asymptotically $\chi^2_1$. A size-$\alpha$ test rejects when
$$
\boxed{X^2>\chi^2_{1,1-\alpha}.}
$$
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