Final answer:
The question asks for a hypothesis test for the difference in disease proportions between vaccinated and placebo groups in a large clinical trial. The statistical analysis would involve setting up null and alternative hypotheses, calculating the test statistic, and making a decision based on the p-value and the significance level of 0.01.
Step-by-step explanation:
The question involves a large clinical trial to test the efficacy of a vaccine in preventing a certain disease. The trial involved 399,356 children who were randomly assigned to two groups: a treatment group that received the vaccine, and a control group that received a placebo. Only 37 children in the treatment group and 128 children in the control group developed the disease.
Based on the information provided:
a. To test the claim that the proportions of children who develop the disease after vaccination (p1) and after placebo (p2) are different, we can conduct a hypothesis test for the difference between two proportions.
b. The null hypothesis (H0) would be that there is no difference in the proportions (H0: p1 = p2), and the alternative hypothesis (Ha) would be that there is a difference (Ha: p1 != p2).
c. The test statistic would be calculated using the formula for the difference between two proportions.
d. After computing the test statistic, we would compare it against a critical value from the Z-distribution or use the computed p-value to make our decision at the 0.01 significance level. If the p-value is less than 0.01, we reject the null hypothesis and conclude that the vaccine has an effect on the disease rate. If it is greater, we fail to reject the null hypothesis.