For a large sample from a population with proportion p, the distribution of p-hat is approximately Normal with mean and variance given. Which is the correct pair for the mean and variance?

• A. Mean = p; Variance = p(1-p)
• B. Mean = p; Variance = p(1-p)/n
• C. Mean = np; Variance = p(1-p)/n
• D. Mean = p/n; Variance = p(1-p)/n^2

Answer: (B)

The sample proportion p-hat is the average of n independent Bernoulli trials with success probability p. Each trial has mean p, so the average has mean p. The variance of a single trial is p(1-p); the variance of the sum of n trials is n p(1-p). Since p-hat is the sum divided by n, its variance is (n p(1-p)) / n^2 = p(1-p)/n. For large n, the Central Limit Theorem makes p-hat approximately Normal with this mean and variance. So the correct pairing is mean p and variance p(1-p)/n. The other options mix up the mean or omit the 1/n scaling, which is why they don’t fit.