Question

At a confidence level of 95% a confidence interval for a population proportion is determined to be 0.65 to 0.75. If the sample size had been larger and the estimate of the population proportion the same, this 95% confidence interval estimate as compared to the first interval estimate would be

A. the same

B. narrower

C. wider

Answer 1

Option B is correct.

Intuitively this can be justified as:

More sample observations means more information regarding the population parameter, which is the population proportion here. Now as information increases the prediction regarding the parameter value is ought to be better. So the confidence interval with the same confidence coefficient always becomes narrower as the sample size increases.

Mathematically, we know that , confidence interval of population proportion for a given confidence coefficient is of this type(see picture below):

Z^* is constant for the given confidence coefficient. Now as n increases, the right hand term decreases and hence the interval becomes more concentrated around . Thus the interval narrows down as sample size increases.

Hence option B is the only correct option.

Hope the solution helps. Thank you.