The drug Prevnar is a vaccine meant to prevent certain types of bacterial meningitis.  It is typically administered to infants starting around 2 months of age.  In randomized double-blind clinical trials of Prevnar, infants were randomly divided into two groups.  Subjects in group 1 received Prevnar, while subjects in group 2 received a control vaccine.  After receiving two doses of the treatment, 138 of 482 subjects in the experimental group (group 1) experienced drowsiness as a side effect, while 41 of 103 subjects in the control group (group 2) experienced drowsiness as a side effect.  Does the evidence suggest that a lower proportion of subjects taking Prevnar experience drowsiness as a side effect than subjects taking the control, at the α = 0.01 level of significance.

Let p1 denote the proportion of people taking Prevnar who experience drowsiness, and p2 the proportion of people taking the control who experience drowsiness.
(Use sample proportions rounded to 3 decimal places in the calculations below.)
np1^(1-p1^) = 

np2^(1-p2^) =

To conduct the hypothesis test comparing the proportions, we first need to calculate the sample proportions and then verify if we can use the normal approximation for the test.

Let's start with calculating the sample proportions \( \hat{p}_1 \) and \( \hat{p}_2 \):

- For group 1 (Prevnar):
  - Number of subjects experiencing drowsiness: \( x_1 = 138 \)
  - Total number of subjects: \( n_1 = 482 \)
  - Sample proportion: 
    \[
    \hat{p}_1 = \frac{x_1}{n_1} = \frac{138}{482}
    \]

- For group 2 (Control):
  - Number of subjects experiencing drowsiness: \( x_2 = 41 \)
  - Total number of subjects: \( n_2 = 103 \)
  - Sample proportion: 
    \[
    \hat{p}_2 = \frac{x_2}{n_2} = \frac{41}{103}
    \]

Let's calculate these sample proportions and then compute \( n_1 \hat{p}_1 (1 - \hat{p}_1) \) and \( n_2 \hat{p}_2 (1 - \hat{p}_2) \).The calculated sample proportions and values for \( np(1-p) \) are as follows:

- Sample proportion for group 1 (Prevnar), \( \hat{p}_1 \): 0.286 (rounded to 3 decimal places)
- Sample proportion for group 2 (Control), \( \hat{p}_2 \): 0.398 (rounded to 3 decimal places)

The values for \( n_1 \hat{p}_1 (1 - \hat{p}_1) \) and \( n_2 \hat{p}_2 (1 - \hat{p}_2) \) are:

- \( n_1 \hat{p}_1 (1 - \hat{p}_1) \): 98.49
- \( n_2 \hat{p}_2 (1 - \hat{p}_2) \): 24.68

These values are both greater than 5, so the normal approximation is appropriate for this test.

Do you want to proceed with the hypothesis test to determine if there is a statistically significant difference at the \( \alpha = 0.01 \) level?

Here are some follow-up questions to consider:
1. Do you want to state the null and alternative hypotheses for this test?
2. Would you like to calculate the test statistic and p-value for this hypothesis test?
3. Are you interested in constructing a confidence interval for the difference in proportions?
4. Should we interpret the results in the context of this study?
5. Do you need an explanation of how to use the normal approximation in hypothesis testing?

**Tip:** When conducting a hypothesis test for proportions, always check if the sample size is large enough to justify using the normal approximation.

The drug Prevnar is a vaccine meant to prevent certain types of bacterial meningitis. It is typically administered to infants starting around 2 months of age. In randomized double-blind clinical trials of Prevnar, infants were randomly divided into two groups. Subjects in group 1 received Prevnar, while subjects in group 2 received a control vaccine. After receiving two doses of the treatment, 138 of 482 subjects in the experimental group (group 1) experienced drowsiness as a side effect, while 41 of 103 subjects in the control group (group 2) experienced drowsiness as a side effect. Does the evidence suggest that a lower proportion of subjects taking Prevnar experience drowsiness as a side effect than subjects taking the control, at the α = 0.01 level of significance?

Explore how to conduct a hypothesis test comparing the proportions of subjects experiencing drowsiness as a side effect in a Prevnar vs. control group study. This problem applies key statistical concepts such as hypothesis testing, normal approximation, and the test statistic for proportions. Suitable for undergraduate statistics students.

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