When the level of confidence and sample size remain the same,a confidence interval for a population proportion p will be ______________ when $\hat { p } ( 1 - \hat { p } )$
Is larger than when $\hat { p } ( 1 - \hat { p } )$
Is smaller.
A)Wider
B)Narrower
C)Neither Wider nor Narrower,they will be the same.

Question

Quizplus · Accepted Answer

The formula for the standard error of a proportion is SE = sqrt((p*(1-p))/n), where p is the sample proportion and n is the sample size. The confidence interval for a proportion can be calculated as p +/- z*(SE), where z is the z-score that corresponds to the desired level of confidence.

Since the standard error involves the term p*(1-p), we can say that if this value is larger, the standard error will be larger as well. This means that the confidence interval will be wider, as there is more uncertainty in the estimate of the population proportion. Similarly, if p*(1-p) is smaller, the standard error will be smaller, and the confidence interval will be narrower.

Therefore, the correct answer is A, wider.

When the Level of Confidence and Sample Size Remain the Same,a