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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 statisticwhere $E_{ij}$ are the fitted counts under independence using the observed row and column totals. Under the null, $X^2$ is asymptotically ${\chi }_1^2$. A size-$\alpha$ test rejects when