Which of the following statements about the sampling distribution of the sample proportion is incorrect?

• A. The sampling distribution of the sample proportion becomes approximately normal as the sample size increases, provided $np$ and $n(1-p)$ are both at least 10.
• B. The standard deviation of the sampling distribution of the sample proportion is given by $\sqrt{\frac{p(1-p)}{n}}$ where $p$ is the population proportion and $n$ is the sample size.
• C. The mean of the sampling distribution of the sample proportion equals the population proportion.
• D. The sampling distribution of the sample proportion is always exactly normal, regardless of the sample size or population proportion.

Answer: (D)

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 proportion \( p \).
2. Recall the conditions for the sampling distribution of \( \hat{p} \) to be approximately normal: both \( n p \) and \( n (1 - p) \) must be at least 10. This ensures the sample size is large enough for the Central Limit Theorem to apply.
3. Know that the mean of the sampling distribution of \( \hat{p} \) is equal to the population proportion \( p \), i.e., \( \mu_{\hat{p}} = p \).
4. Recognize that the standard deviation (also called the standard error) of the sampling distribution of \( \hat{p} \) is given by the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion and \( n \) is the sample size.
5. Identify that the incorrect statement is the one claiming the sampling distribution of \( \hat{p} \) is always exactly normal regardless of sample size or population proportion, because normality is only approximate and depends on the sample size and proportion conditions.