What is the expected value of the sample mean x̅ denoted as, according to the Central Limit Theorem?

The expected value of the sample mean, denoted as E(x̅), is equal to the population mean μ. This principle is a fundamental concept in statistics, especially within the framework of the Central Limit Theorem (CLT). The theorem states that, regardless of the distribution of the population, the distribution of the sample means will approach a normal distribution as the sample size increases, and the mean of these sample means will converge to the population mean.

Specifically, when taking random samples from a population and calculating the mean of those samples, the average of those sample means will tend to be equal to the population mean. This property allows statisticians to make inferences about the population mean based on the sample mean, reinforcing the importance of E(x̅) = μ in statistical analysis.

Understanding this concept is crucial for performing hypothesis testing and constructing confidence intervals. These tools depend upon the assumption that the sample means are centered around the true population mean, ensuring that our estimates are reliable.