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 331 of 2454 vaccinated children had ear​ infections, compared to 507 of 2453 unvaccinated children. Complete parts​ a) through​ c) below.

​a) Are the conditions for inference​ satisfied?

A. No. The groups were not independent.

B. No. More than​ 10% of the population was sampled.

C. No. It was not a random sample.

D. Yes. The data were generated by a randomized​ experiment, less than​ 10% of the population was​ sampled, the groups were​ independent, and there were more than 10 successes and failures in each group.

​b) Let Modifying Above 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.)

​c) Use your confidence interval to explain whether you think the vaccine is effective.

A.No. We are​ 95% confident that the rate of infection of vaccinated babies could be as much as 5.1​% higher compared to unvaccinated babies.

B.Yes. We are​ 95% confident that the rate of infection is 5.1 to 9.3​% lower. This is a meaningful​ reduction, considering the​ 20% infection rate among unvaccinated babies.

C. No. No conclusion can be made based on the confidence interval.

D.Yes. We are​ 95% confident that about 9.3​% of unvaccinated babies will get an ear​ infection, while only 5.1% of vaccinated babies will. This is a meaningful reduction.

Answer 1

a)

D. Yes. The data were generated by a randomized​ experiment, less than​ 10% of the population was​ sampled, the groups were​independent, and there were more than 10 successes and failures in each group.

b)

	A		B
x1                =	507	x2                =	331
p̂1=x1/n1 =	0.2067	p̂2=x2/n2 =	0.1349
n1                       =	2453	n2                       =	2454
estimated difference in proportion   =p̂1-p̂2   =			0.0718
std error Se =√(p̂1*(1-p̂1)/n1+p̂2*(1-p̂2)/n2) =			0.0107
for 95 % CI value of z=		1.960
margin of error E=z*std error =		0.0210
lower bound=(p̂1-p̂2)-E=		0.0508
Upper bound=(p̂1-p̂2)+E=		0.0928
from above 95% confidence interval for difference in population proportion =(5.08% to 9.28%)

B.Yes. We are​ 95% confident that the rate of infection is 5.1 to 9.3​% lower. This is a meaningful​ reduction, considering the​20% infection rate among unvaccinated babies.