### Compute a confidence interval for the proportion of children

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A 2003 article in the Journal of the American Medical Association 1 documents the results of a randomized clinical trial designed to evaluate whether the influenza vaccine is effective in reducing the occurrence of acute otitis media (AOM) in young children.

Acute otitis media is an infection that causes inflammation of the middle ear canal.

In the study, children were randomized to receive either the influenza vaccine or a placebo. (randomization was done in a 2 to 1 ratio, meaning that two times as many children were randomized to the vaccine treatment as were randomized to the placebo group).

The children were followed for one year after randomization, and monitored for AOM during this period. 262 children were randomized to the vaccine group, and 150 of these children experienced at least one incident of AOM during the follow-up period. 134 children were randomized to the placebo group, and 83 of these children experience at least one incident of AOM during the follow-up period

(a) Compute a 95% confidence interval for the proportion of children experiencing at least one incident of AOM during the follow-up period in each of the randomization groups. How do these 95% CI's compare? (similar range of values? Overlap?)

(b) Compute a 95% confidence interval for the difference in proportions of children experiencing AOM over the follow-up period. (you may choose the direction of comparison - ie: placebo to vaccine or vice-versa) . Intepret the confidence interval in a sentence.

(c) State the null and alternative hypotheses associated with testing for an association between the influenza vaccine and AOM. What is the p-value for testing these hypotheses? Is this p-value consistent with your result in part b?

(d) Give an estimate of the relative risk of AOM (in the follow-up period) for children in the vaccine group compared to those in placebo. Interpret this.

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