Question 11 • Probability and Statistics with Applications Honors Sem-2-FL-1210300 (2024-2025)
A principal of a large high school wants to estimate the true proportion of high school students who use the community’s public library. To do so, he selects a random sample of 50 students and asks them if they use the community’s public library. The 95% confidence interval for the true proportion of all students who use the community’s public library is 0.25 to 0.34. If the principal had used a sample size of 200 students rather than 50 students, how would the length of the second interval compare to the original interval?
Answer
A
It would be half as wide.
B
It would be twice as wide.
C
It would be one-fourth as wide.
D
It would be four times as wide.
Explanation
For the same confidence level and estimated proportion, a confidence interval’s width is approximately inversely proportional to the square root of the sample size. Increasing the sample size from 50 to 200 multiplies it by 4, so the interval’s width is multiplied by 1/√4 = 1/2. The original interval is 0.34 − 0.25 = 0.09 wide, so the new interval would be about 0.045 wide: half as wide.
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