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 were randomly divided into two groups. One group received​ vaccinations, and the other did not. The following​ year, only 326 of 2457 vaccinated children had ear​ infections, compared to 507 of 2452 unvaccinated children.

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​ infection, p1−p2.

The confidence interval is (___%,___%)

​(Do not round until the final answer. Then round to one decimal place as​ needed.)

Answer 1

p1cap = X1/N1 = 507/2452 = 0.2068
p1cap = X2/N2 = 326/2457 = 0.1327

Here, , n1 = 2452 , n2 = 2457
p1cap = 0.2068 , p2cap = 0.1327

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.2068 * (1-0.2068)/2452 + 0.1327*(1-0.1327)/2457)
SE = 0.0107

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.2068 - 0.1327 - 1.96*0.0107, 0.2068 - 0.1327 + 1.96*0.0107)
CI = (0.053 , 0.095)

The confidence interval is (5.3%,9.5%)