Which of the following statements regarding the sampling distribution of the sample proportion ($p̂$) is true?

• A. The sampling distribution of the sample proportion ($p̂$) is always perfectly symmetric regardless of sample size.
• B. The standard deviation of the sampling distribution of the sample proportion ($p̂$) does not depend on the sample size.
• C. The mean of the sampling distribution of the sample proportion ($p̂$) equals the population proportion ($p$).
• D. The sampling distribution of the sample proportion ($p̂$) is always normal, even for very small sample sizes.

Answer: (C)

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 a given size \( n \) drawn from a population with true proportion \( p \).
2. Recall that the mean of the sampling distribution of \( \hat{p} \) is equal to the population proportion \( p \). This means \( E(\hat{p}) = p \).
3. Recognize that the standard deviation of the sampling distribution of \( \hat{p} \), often called the standard error, depends on both the population proportion \( p \) and the sample size \( n \), and is given by the formula: \[ \\text{SE} = \sqrt{\frac{p(1-p)}{n}} \]
4. Note that 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. For small sample sizes, the distribution may not be normal and may not be symmetric.
5. Therefore, the true statement is that the mean of the sampling distribution of the sample proportion equals the population proportion, while the other statements about symmetry, standard deviation independence, and always normal shape are incorrect.