Question

in a randomized controlled trial, insecticides-treated bednets were tested as a way to reduce malaria. among 302 infants using bednets, 14 developed malaria. among 275 infants not using bednets, 31 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 correct hypotheses below
oa. ( h_0:p_1
eq p_2 )
( h_1:p_1 = p_2 )
ob. ( h_0:p_1 = p_2 )
( h_1:p_1
eq p_2 )
oc. ( h_0:p_1 = p_2 )
( h_1:p_1lt p_2 )
od. ( h_0:p_1leq p_2 )
( h_1:p_1gt p_2 )
oe. ( h_0:p_1 = p_2 )
( h_1:p_1gt p_2 )
of. ( h_0:p_1geq p_2 )
( h_1:p_1lt p_2 )
identify the test statistic
(round to two decimal places as needed)

Explanation:

Step1: Calculate sample proportions

For sample 1 (using bednets): \(n_1 = 302\), \(x_1=14\), so \(\hat{p}_1=\frac{x_1}{n_1}=\frac{14}{302}\approx0.0464\)
For sample 2 (not using bednets): \(n_2 = 275\), \(x_2 = 31\), so \(\hat{p}_2=\frac{x_2}{n_2}=\frac{31}{275}\approx0.1127\)
The pooled proportion \(\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{14+31}{302 + 275}=\frac{45}{577}\approx0.078\)

Step2: Calculate the test statistic

The formula for the test statistic \(z\) in a two - proportion z - test is \(z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\)
Substitute the values:

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Answer:

The test statistic is approximately \(-2.96\)