Step 1 :Define the null and alternative hypotheses. The null hypothesis is that the proportions are equal, and the alternative hypothesis is that the proportion of infants using bednets who developed malaria is less than the proportion of infants not using bednets who developed malaria. So, the null and alternative hypotheses are \(H_{0}: p_{1}=p_{2}\) and \(H_{1}: p_{1} Step 2 :Calculate the sample proportions. For the infants using bednets, \(\hat{p}_1 = \frac{x_1}{n_1} = \frac{15}{312} = 0.0481\). For the infants not using bednets, \(\hat{p}_2 = \frac{x_2}{n_2} = \frac{24}{266} = 0.0902\). Step 3 :Calculate the combined sample proportion. \(\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{15 + 24}{312 + 266} = 0.0675\). Step 4 :Calculate the test statistic using the formula \(Z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\). Substituting the calculated values, we get \(Z = \frac{(0.0481 - 0.0902) - 0}{\sqrt{0.0675(1-0.0675)(\frac{1}{312} + \frac{1}{266})}} = -2.01\). Step 5 :Find the critical value at the 0.01 significance level. The critical value is approximately -2.33. Step 6 :Compare the test statistic with the critical value. Since the test statistic of -2.01 is greater than the critical value of -2.33, we do not reject the null hypothesis at the 0.01 significance level. Step 7 :Conclude the hypothesis test. We do not have sufficient evidence to support the claim that the incidence of malaria is lower for infants using bednets. Therefore, the final answer is \(\boxed{-2.01}\).