If the size of a sample randomly selected sample from a population is increased from n = 100 to n = 400, then the standard deviation of $\hat{p}$ will
A) remain the same.
B) increase by a factor of 4.
C) decrease by a factor of 4.
D) decrease by a factor of 2.

Question

Quizplus · Accepted Answer

The standard deviation of $\hat{p}$, the sample proportion, is given by $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size. When the sample size increases, the denominator of the fraction under the square root increases, which decreases the value under the square root. Specifically, increasing the sample size by a factor of 4 (from 100 to 400) decreases the standard deviation of $\hat{p}$ by a factor of 2, because the square root of 4 is 2.

If the Size of a Sample Randomly Selected Sample from a Population