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 \(\hat{p}\) ?

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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.