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? Question 4 (0/1) Let the infants using bednets be sample 1 and let the infants not using bednets be sample 2. Choose the correct hypotheses below. A. \[ \begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1}p_{2} \end{array} \] D. \[ \begin{array}{l} H_{0}: p_{1} \geq p_{2} \\ H_{1}: p_{1}p_{2} \end{array} \] F. \[ \begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array} \] Identify the test statistic. -2.01 (Round to two decimal places as needed.) Identify the P-value. (Round to three decimal places as needed.)

Step 1 ：Define the null and alternative hypotheses. The null hypothesis is that the two 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. The correct hypotheses are: \[H_{0}: p_{1} \geq p_{2}\] \[H_{1}: p_{1}

Step 2 ：Calculate the sample sizes and the number of successes (developing malaria) for each group. We have: \[n1 = 312\] \[n2 = 266\] \[x1 = 15\] \[x2 = 24\]

Step 3 ：Calculate the sample proportions. We get: \[p1 = 0.04807692307692308\] \[p2 = 0.09022556390977443\]

Step 4 ：Calculate the pooled proportion. We get: \[p = 0.06747404844290658\]

Step 5 ：Calculate the standard error. We get: \[se = 0.020933635195508546\]

Step 6 ：Calculate the test statistic using the formula for the test statistic for two proportions. We get: \[z = -2.0134410693224765\]

Step 7 ：Find the P-value by looking up the test statistic in a standard normal (Z) distribution table. We get: \[p_{value} = 0.02203412435391147\]

Step 8 ：Compare the P-value with the significance level. The P-value is greater than the significance level of 0.01, so we do not reject the null hypothesis. This means that we do not have enough evidence to support the claim that the incidence of malaria is lower for infants using bednets.

Step 9 ：The test statistic is approximately \(-2.01\) and the P-value is approximately \(0.022\). The final answer is: \[\boxed{-2.01}\] and \[\boxed{0.022}\]