Question

in randomized, double - blind clinical trials of a new vaccine, infants were randomly divided into two groups. subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. after the second dose, 111 of 714 subjects in the experimental group (group 1) experienced fever as a side effect. after the second dose, 68 of 595 of the subjects in the control group (group 2) experienced fever as a side effect. does the evidence suggest that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the \\( \alpha=0.05 \\) level of significance?
a. the data come from a population that is normally distributed
b. \\( n_{1} \hat{p}_{1}\left(1 - \hat{p}_{1}\
ight)\geq10 \\) and \\( n_{2} \hat{p}_{2}\left(1 - \hat{p}_{2}\
ight)\geq10 \\)
c. the sample size is less than 5% of the population size for each sample
d. the sample size is more than 5% of the population size for each sample
e. the samples are dependent
f. the samples are independent
determine the null and alternative hypotheses.
\\( h_{0}: p_{1}=p_{2} \\)
\\( h_{1}: p_{1}>p_{2} \\)
find the test statistic for this hypothesis test.
(round to two decimal places as needed.)

Explanation:

Step1: Calculate sample proportions

For group 1: \(n_1 = 714\), \(x_1=111\), \(\hat{p}_1=\frac{x_1}{n_1}=\frac{111}{714}\approx0.1555\)
For group 2: \(n_2 = 595\), \(x_2 = 68\), \(\hat{p}_2=\frac{x_2}{n_2}=\frac{68}{595}\approx0.1143\)

Step2: Calculate the pooled proportion

\(\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{111+68}{714 + 595}=\frac{179}{1309}\approx0.1367\)

Step3: 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:

\(2.16\)