A simple random sample of 450 residents in the state of New York is taken to estimate the proportion of people who live within 1 mile of a hazardous waste site.If a 95% confidence interval were calculated for a scenario where p = 0.3 and another scenario where p = 0.7,how would the two confidence intervals compare?
A)Since the sample proportion in the first scenario is smaller,its confidence interval would be narrower.
B)Even though the sample proportions are different,the widths of the confidence intervals would be equal.
C)Since the sample proportion in the second scenario is larger,its confidence interval would be shifted to the right and narrower than the one calculated from the first scenario.
D)This cannot be determined from the information given.

Question

Quizplus · Accepted Answer

The width of a confidence interval for a proportion is determined by the formula $ \sqrt{\frac{p(1-p)}{n}} $ times the critical value from the z-distribution for the desired confidence level. Since $ p(1-p) $ is maximized at $ p = 0.5 $, both $ p = 0.3 $ and $ p = 0.7 $ yield the same product $ 0.3(0.7) = 0.21 $, resulting in confidence intervals of equal width.

A Simple Random Sample of 450 Residents in the State