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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 the first dose, 107 of 710 subjects in the experimental group (Group 1) experienced fever as a side effect. After the first dose, 67 of 611 of the subjects in the control group (Group 2 ) experienced fever as a side effect. (a) 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? (b) Construct a \(90 \%\) confidence interval for the difference between the two population proportions, \(p_{1}-p_{2}\)

Step by step solution

01

State the Hypotheses

We need to state the null and alternative hypotheses. Let \( p_1 \) be the proportion of subjects in Group 1 who experienced fever, and \( p_2 \) be the proportion of subjects in Group 2 who experienced fever.Null hypothesis: \( H_0: p_1 = p_2 \)Alternative hypothesis: \( H_1: p_1 > p_2 \)

02

Determine the Test Statistic

Calculate the sample proportions:\( \hat{p}_1 = \frac{107}{710} \text{ and } \hat{p}_2 = \frac{67}{611} \)Compute pooled proportion \( \hat{p} = \frac{107 + 67}{710 + 611} \)Next, use the pooled proportion to calculate the standard error:\( SE = \sqrt{\hat{p}\left(1 - \hat{p}\right)\left(\frac{1}{710} + \frac{1}{611}\right)} \)The test statistic \( z \) will be:\( z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \)

03

Find the Critical Value

For a right-tailed test at \(\alpha = 0.05\) significance level, refer to the standard normal distribution table to find the critical value \( z_{0.05} \). The critical value is approximately 1.645.

04

Compare Test Statistic with Critical Value

Compare the calculated test statistic \( z \) from Step 2 with the critical value from Step 3. If \( z \) is greater than 1.645, reject the null hypothesis.

05

Construct the Confidence Interval

Calculate the difference between sample proportions:\( \hat{p}_1 - \hat{p}_2 \)Find the standard error for the difference of proportions:\( SE_{diff} = \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{710} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{611}} \)Construct the 90% confidence interval for \( p_1 - p_2 \):\(CI = \left(\hat{p}_1 - \hat{p}_2 \right) \pm Z_{\alpha/2} \times SE_{diff} \)Where \( Z_{\alpha/2} \) is 1.645.

06

Interpretation

Interpret the results of the hypothesis test and confidence interval. If the null hypothesis is rejected, then it can be concluded that there is significant evidence at the 0.05 level to suggest that the proportion of subjects experiencing fever in Group 1 is greater than in Group 2. The confidence interval will give the range in which the true difference of proportions lies with 90% confidence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, denoted as \(H_0\), is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference. In the given exercise, the null hypothesis is: \(H_0: p_1 = p_2\). This means that initially, we assume there is no difference in the proportion of subjects experiencing fever between Group 1 (those who received Prevnar) and Group 2 (those who received the control vaccine). The null hypothesis serves as the default or baseline assumption that the study aims to test against. Rejecting the null hypothesis would imply evidence of a statistically significant difference between the groups.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_1\), is what researchers hope to prove. It represents a statement that there is an effect or a difference. In this exercise, the alternative hypothesis is: \(H_1: p_1 > p_2\). This hypothesis suggests that a higher proportion of subjects in Group 1 experience fever compared to Group 2. The alternative hypothesis is critical because it frames the direction of the test (one-tailed in this case), and the statistical evidence must be strong enough to reject the null hypothesis in favor of the alternative.

Test Statistic

The test statistic is a single measure calculated from the sample data, used to evaluate the plausibility of the null hypothesis. In this context, it involves the following steps:

• Calculate sample proportions: \(\hat{p}_1 = \frac{107}{710} \) and \( \hat{p}_2 = \frac{67}{611} \).
• Compute the pooled proportion: \(\hat{p} = \frac{107 + 67}{710 + 611} \).
• Calculate the standard error: \(SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{710} + \frac{1}{611}\right)} \).
• Calculate the test statistic \(z\): \(z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \).

The test statistic helps in determining whether to reject the null hypothesis by comparing it with a critical value.

Confidence Interval

A confidence interval provides a range of values that likely contain the true difference between the population proportions, given a certain level of confidence (90% in this case). To construct this interval:

• Calculate the difference between sample proportions: \( \hat{p}_1 - \hat{p}_2 \).
• Find the standard error for the difference of proportions: \( SE_{diff} = \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{710} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{611}} \).
• Use the critical value for a 90% confidence interval (1.645).
• Construct the interval: \(CI = (\hat{p}_1 - \hat{p}_2) \pm 1.645 \times SE_{diff} \).

This interval offers insight into the range within which the true difference of proportions is expected to lie.

Standard Error

The standard error (SE) measures the variability or dispersion of the sampling distribution of a statistic. It essentially quantifies the precision of an estimate. For this exercise, the standard error is involved in two main calculations:

• The standard error of the pooled proportion: \(SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{710} + \frac{1}{611}\right)} \).
• The standard error for the difference of proportions: \( SE_{diff} = \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{710} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{611}} \).

These calculations help in determining how much the sample proportions can be expected to vary from the true population proportions, which is crucial for hypothesis testing and constructing confidence intervals.