Question

For which of the following combinations of sample size and population proportion would the standard deviation of \(\hat{p}\) be smallest? $$ \begin{array}{ll} n=40 & p=0.3 \\ n=60 & p=0.4 \\ n=100 & p=0.5 \end{array} $$

Short answer

The short answer would be deterministic after manual comparison of the standard deviations calculated in steps 1-3. For example, if the standard deviation obtained from Step 1 is found to be the smallest, the answer would be the combination \(n=40\) and \(p=0.3\).

Step-by-step solution

Compute the standard deviation for the first pair

For \(n=40\) and \(p=0.3\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.3(1-0.3)}{40}}\).

Compute the standard deviation for the second pair

For \(n=60\) and \(p=0.4\), again calculate the standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.4(1-0.4)}{60}}\).

Compute the standard deviation for the third pair

For \(n=100\) and \(p=0.5\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.5(1-0.5)}{100}}\).

Compare the standard deviations

The sample size and population proportion that gives the smallest standard deviation is the answer. After comparing the three results, select the one with the least standard deviation.