What is the expected value of p̅ in a sampling distribution?

The expected value of the sample proportion (p̅) in a sampling distribution is precisely equal to the population proportion (p). This concept stems from the properties of sampling distributions, particularly as it relates to the unbiased nature of the sample proportion.

When conducting a study or experiment, if multiple random samples are taken from a population and the sample proportions are calculated, the average of all those sample proportions will converge to the true population proportion as the sample size increases. This is a significant result in statistics known as the Central Limit Theorem, which asserts that regardless of the population's distribution, the sampling distribution of the sample mean (or sample proportion) will approximate a normal distribution as the sample size grows.

Thus, the correct choice highlights that the expected value of the sample proportion, E(p̅), directly reflects the true proportion in the population, p. This property ensures that if you were to take infinitely many samples and compute the average of the sample proportions, you would arrive at the population proportion. Understanding this helps in making inferences about a population based on sample data and underscores the reliability of sample estimates when utilized correctly.