In a randomized controlled trial, insecticide-treated bednets were tested as a way to reduce malaria. Among 312 infants using bednets, 15 developed malaria. Among 266 infants not using bednets, 24 developed malaria. Use a 0.01 significance level to test the claim that the incidence of malaria is lower for infants using bednets.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Based on the results, do the bednets appear to be effective?
a. What are the null and alternative hypotheses?
Let the infants using bednets be sample 1 and let the infants not using bednets be sample 2 . Choose the correet hypotheses below.
A. $H_{0}: p_{1}=p_{2}$
$H_{1}: p_{1}< p_{2}$
D. $H_{0}: P_{1} \geq p_{2}$
$\mathrm{H}_{1}: \mathrm{p}_{1}< \mathrm{p}_{2}$
B. $H_{0}: p_{1} \neq p_{2}$
$H_{1}: p_{1}=p_{2}$
E. $H_{0}: p_{1}=p_{2}$
$H_{1}: p_{1}> p_{2}$
C. $H_{0}: p_{1} \leq p_{2}$
$H_{1}: p_{1}> p_{2}$
F. $H_{0}: p_{1}=p_{2}$
$H_{1}: p_{1} \neq p_{2}$
Identify the-test statistic.
(Round to two decimal places as needed.)
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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 \(\boxed{-2.01}\).