Which of the following is not a property of the sampling distribution of the sample proportion?

• A. The mean of the sampling distribution equals the population proportion $p$.
• B. The sampling distribution of the sample proportion is always perfectly normal, regardless of sample size.
• C. The sample proportion is an unbiased estimator of the population proportion $p$.
• D. The standard deviation of the sampling distribution is given by $\sqrt{\frac{p(1-p)}{n}}$.

Answer: (B)

Verified step by step guidance

1. Understand that the sampling distribution of the sample proportion \( \hat{p} \) describes the distribution of sample proportions from all possible samples of size \( n \) drawn from a population with true proportion \( p \).
2. Recall the key properties of the sampling distribution of \( \hat{p} \):
3. 1. The mean of the sampling distribution is equal to the population proportion \( p \), meaning \( E(\hat{p}) = p \). This shows \( \hat{p} \) is an unbiased estimator.
4. 2. The standard deviation (also called the standard error) of the sampling distribution is given by \( \sqrt{\frac{p(1-p)}{n}} \), which depends on the population proportion \( p \) and the sample size \( n \).
5. 3. The shape of the sampling distribution of \( \hat{p} \) approaches a normal distribution as the sample size \( n \) becomes large, according to the Central Limit Theorem. However, it is not always perfectly normal for any sample size; small samples or extreme values of \( p \) can cause the distribution to be skewed.