Question

The number of undergraduates at Johns Hopkins University is approximately 2000 , while the number at Ohio State University is approximately 60,000. At both schools, a simple random sample of about $3\%$of the undergraduates is taken. Each sample is used to estimate the proportion p of all students at that university who own an iPod. Suppose that, in fact, p=0.80 at both schools. Which of the following is the best conclusion?

a. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it comes from a smaller population.

b. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it is based on a smaller sample size.

c. We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it comes from a larger population.

d. We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it is based on a larger sample size.

e. We expect that the estimate from Johns Hopkins will be about the same distance from the truth as the estimate from Ohio State because both samples are 3 % of their populations.

Short answer

(d) We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it is based on a larger sample size.

Step-by-step solution

Given Information

The following options are

a. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it comes from a smaller population.

b. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it is based on a smaller sample size.

c. We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it comes from a larger population.

d. We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it is based on a larger sample size.

e. We expect that the estimate from Johns Hopkins will be about the same distance from the truth as the estimate from Ohio State because both samples are 3 % of their populations.

Explanation for correct option

Consider that,

Johns Hopkins $3\%\times 2000=0.03\times 2000=60$

Ohio State $3\%\times 60,000=0.03\times 60,000=1800$

The sample size of Ohio State is substantially bigger than the sample size of Johns Hopkins, implying that Ohio State's population estimates are much more accurate than Johns Hopkins' population estimations.

Because of the greater sample size, it is projected that the computed from Ohio State will be far closer to the truth than the calculated from Johns Hopkins.

As a result, the best solution is (d)

Explanation for incorrect option

a. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it comes from a smaller population is not the answer.

b. We expect that the estimate from Johns Hopkins will be closer to the truth than the estimate from Ohio State because it is based on a smaller sample size is not the answer.

c. We expect that the estimate from Ohio State will be closer to the truth than the estimate from Johns Hopkins because it comes from a larger population is not the answer.

e. We expect that the estimate from Johns Hopkins will be about the same distance from the truth as the estimate from Ohio State because both samples are 3 % of their populations is not the answer.