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?
verify the model requirements. select all that apply
\\( \square \\) a. the data come from a population that is normally distributed.
\\( \square \\) 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 \\)
\\( \square \\) c. the sample size is less than 5% of the population size for each sample.
\\( \square \\) d. the sample size is more than 5% of the population size for each sample.
\\( \square \\) e. the samples are dependent.
\\( \square \\) f. the samples are independent.
determine the null and alternative hypotheses.
\\( h_{0}: p_{1}\frac{\
abla}{\
abla} p_{2} \\)
\\( h_{1}: p_{1} \
abla p_{2} \\)

Explanation:

Step1: Verify model requirements

• For two - proportion z - test:
• The samples must be independent (F). Since infants are randomly divided into two groups, the outcome of one group does not affect the other.
• \(n_1\hat{p}_1(1 - \hat{p}_1)\geq10\) and \(n_2\hat{p}_2(1 - \hat{p}_2)\geq10\) (B). Calculate \(\hat{p}_1=\frac{111}{714}\approx0.1555\), \(n_1\hat{p}_1(1 - \hat{p}_1)=714\times0.1555\times(1 - 0.1555)\approx714\times0.1555\times0.8445\approx94.7\geq10\). Calculate \(\hat{p}_2=\frac{68}{595}\approx0.1143\), \(n_2\hat{p}_2(1 - \hat{p}_2)=595\times0.1143\times(1 - 0.1143)\approx595\times0.1143\times0.8857\approx59.5\geq10\).
• The sample size is less than 5% of the population size for each sample (C). In most cases (assuming the population of infants is large), \(n_1 = 714\) and \(n_2=595\) are likely less than 5% of the population.
• The data does not need to come from a normally distributed population (A is wrong). The samples are independent, not dependent (E is wrong).

Step2: Determine null and alternative hypotheses

• The null hypothesis \(H_0:p_1 = p_2\) (equality of proportions).
• The alternative hypothesis \(H_1:p_1>p_2\) (since we want to test if group 1 has a higher proportion of fever as a side - effect).

Answer:

• Model requirements: B, C, F
• \(H_0:p_1 = p_2\), \(H_1:p_1>p_2\)