Sampling Distributions for Proportions Question 1

Which of the following pairs of sample size $n$ and population proportion $p$ would produce the greatest standard deviation for the sampling distribution of a sample proportion $\hat{p}$ ?

n=1{,}000

and

p

close to

0

n=1{,}000

and

p

close to

\dfrac{1}{2}

n=100

and

p

close to

0

n=100

and

p

close to

\dfrac{1}{2}

Approach

The standard deviation for the sampling distribution of the sample proportion is given by the following formula:

s_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Maximizing this value involves making the numerator as large as possible, and the denominator as small as possible

Having a $\hat{p}$ value of $\frac{1}{2}$ will maximize the numerator, whereas have a value of $n=100$ is the best option to minimize the deonminator.

If you're not confident with this reasoning, simply plug in all of the answer choices into the formula and find the largest result.