Q.5 - The Practice of Statistics for AP

Question

The number of undergraduates at Johns Hopkins University is approximately $2000$, while the number at Ohio State University is approximately $40,000$. 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) The estimate from Johns Hopkins has less sampling variability than that from Ohio State. 

(b) The estimate from Johns Hopkins has more sampling variability than that from Ohio State. 

(c) The two estimates have about the same amount of sampling variability. 

(d) It is impossible to make any statement about the sampling variability of the two estimates since the students surveyed were different. 

(e) None of the above

Step-by-Step Solution

Answer

The correct answer is (b) The estimate from Johns Hopkins has more sampling variability than that from Ohio State. 

Step 1 Given Information

Given the number of undergraduates at Johns Hopkins University $=2000$

Number of undergraduates at Ohio State University $=40,000$

Percent of undergraduates for a sample $=3\%$

Proportion of students who own an iPod, $p=0.80$

Given that each university has $3\%$ students who own iPod.

Step 2 Explanation

Consider that the sample of students from Johns Hopkins University be $n_{\text{Јн }}$ and the sample of students from Ohio State University be $n\_\{0 S\}.$

Since the number of undergraduates of John Hopkins University is less than the number of undergraduates of Ohio State University; so, $n_{\mathrm{JH}}<n_{\mathrm{OS}}$

Know that if the sample size is larger, then it gives more information about the population and it has less sampling variability. This means the estimate from Johns Hopkins University has more sampling variability than that from Ohio State.

Hence, the correct answer is (b).