Question

A new vaccine was tested to see if it could prevent the ear infections that many infants suffer from. Babies about a year old are randomly divided into two groups. One group received vaccinations, and the their did not. The following year, only 337 of 2460 vaccinated children had ear infections, compared to 491 of 2446 unvaccinated children. A.) Let P1 be the sample proportion of success in the unvaccinated group and let P2 be the sample proportion of success in the vaccinated group. Find the 95% confidence interval for the difference in rates of ear infections, p1 - p2 B.) Use your confidence interval to explain whether you think the vaccine is effective.

Answer 1

We need to construct the 95% confidence interval for the difference between population proportions p1​−p2​. We have been provided with the following information about the sample proportions:

Favorable Cases 1 (X1​) =	337
Sample Size 1 (N1​) =	2460
Favorable Cases 2 (X2​) =	491
Sample Size 2 (N2​) =	2446

The sample proportion 1 is computed as follows, based on the sample size N1​=2460 and the number of favorable cases X1​=337:

The sample proportion 2 is computed as follows, based on the sample size N2​=2446 and the number of favorable cases X2​=491:

The critical value for α=0.05 is . The corresponding confidence interval is computed as shown below:

b)

Since 0 does not lie in the above confidence interval, and the difference in the proportions p1-p2 is negative we can conclude that the vaccine is effective.

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