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.
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.

Explanation:

• For a two - proportion z - test (which is likely used here to compare the proportions of subjects with fever in the two groups), the following requirements are relevant:
• The samples must be independent. Since infants are randomly divided into two groups, group 1 and group 2 are independent. So, option F is correct.
• We need \(n_1\hat{p}_1(1 - \hat{p}_1)\geq10\) and \(n_2\hat{p}_2(1 - \hat{p}_2)\geq10\) for the normal approximation to the binomial (which is used in the two - proportion z - test). So, option B is correct.
• Also, the sample size should be less than 5% of the population size for each sample (to ensure that the sampling is approximately independent). So, option C is correct.
• The data does not need to come from a normally distributed population (because we are dealing with proportions and using the normal approximation to the binomial, not assuming normality of the original data). So, option A is incorrect.
• The sample size being more than 5% of the population size (option D) is not a requirement. In fact, we want the sample size to be less than 5% of the population size.
• The samples are not dependent (option E). If they were dependent (e.g., matched pairs), a different test (like a paired - test) would be appropriate.

Answer:

B. \(n_1\hat{p}_1(1-\hat{p}_1)\geq10\) and \(n_2\hat{p}_2(1 - \hat{p}_2)\geq10\), C. The sample size is less than 5% of the population size for each sample, F. The samples are independent.