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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, 116 of 674 subjects in the experimental group (group 1) experienced drowsiness as a side effect. After the second dose, 73 of 555 subjects in the control group (group 2) experienced drowsiness as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced drowsiness as a side effect than subjects in group 2 at the [tex]$\alpha=0.05$[/tex] level of significance?

Verify the model requirements. Select all that apply.

A. [tex]n _1 \hat{ p }_1\left(1-\hat{ p }_1\right) \geq 10[/tex] and [tex]n _2 \hat{ p }_2\left(1-\hat{ p }_2\right) \geq 10[/tex]

B. The sample size is less than [tex]5 \%[/tex] of the population size for each sample.

C. The samples are dependent.

D. The samples are independent.

E. The sample size is more than [tex]5 \%[/tex] of the population size for each sample.

F. The data come from a population that is normally distributed.

Sagot :

GinnyAnsweridn
To determine if the evidence suggests that a higher proportion of subjects in group 1 experienced drowsiness as a side effect than subjects in group 2 at the [tex]\(\alpha = 0.05\)[/tex] level of significance, we need to verify certain model requirements. We will examine each requirement one by one:

A. [tex]\(n_1 \hat{p}_1(1-\hat{p}_1) \geq 10\)[/tex] and [tex]\(n_2 \hat{p}_2(1-\hat{p}_2) \geq 10\)[/tex]:
- We need to check if the sample sizes and proportions meet the necessary condition for the normal approximation.
- This requirement is true.

B. The sample size is less than [tex]\(5 \%\)[/tex] of the population size for each sample:
- This means that the sample is a small fraction of the entire population.
- This requirement is true.

C. The samples are dependent:
- For the samples to be dependent, the selection or outcome in one group would have to affect the other.
- This requirement is false.

D. The samples are independent:
- If selecting or observing a subject in one group does not affect the selection or outcome in the other group, then the samples are independent.
- This requirement is true.

E. The sample size is more than [tex]\(5 \%\)[/tex] of the population size for each sample:
- This implies that the sample is a significant portion of the population, which can affect the assumption of independence and the sampling distribution.
- This requirement is false.

F. The data come from a population that is normally distributed:
- This model requirement pertains to the distribution of the population from which samples are taken.
- This requirement is false.

Given these considerations, the verified model requirements are:
- A. [tex]\(n_1 \hat{p}_1(1-\hat{p}_1) \geq 10\)[/tex] and [tex]\(n_2 \hat{p}_2(1-\hat{p}_2) \geq 10\)[/tex]
- B. The sample size is less than [tex]\(5 \%\)[/tex] of the population size for each sample.
- D. The samples are independent.

So, in summary, the model requirements that apply are:

- A. [tex]\(n_1 \hat{p}_1(1-\hat{p}_1) \geq 10\)[/tex] and [tex]\(n_2 \hat{p}_2(1-\hat{p}_2) \geq 10\)[/tex]
- B. The sample size is less than [tex]\(5 \%\)[/tex] of the population size for each sample.
- D. The samples are independent.
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