If you repeatedly take random samples from a population and compute their means, the distribution of those sample means will

If you repeatedly take random samples from a population and compute their means, the distribution of those sample means will

• A. It will equal the population mean exactly in every sample
• B. It will be centered around the population mean
• C. It will be unrelated to the population mean
• D. It will always be greater than the population mean

Answer: (B)

The central idea here is the sampling distribution of the sample mean. When you repeatedly draw random samples of the same size from a population and compute each sample mean, those means form a distribution whose center lines up with the population mean μ. This happens because the expected value of the sample mean equals the population mean, so, across many samples, the average of the sample means equals the true mean. In other words, the distribution is centered around the population mean, though individual sample means vary due to sampling error. The spread of that distribution depends on the population standard deviation and the sample size, specifically shrinking as n grows (the standard error is σ/√n). The Central Limit Theorem further tells us that with a reasonably large sample size, the distribution of the sample mean is approximately normal, making inference about μ more straightforward. The other statements aren’t accurate: you don’t expect every sample mean to equal μ, the distribution is related to μ, and it isn’t always greater than μ.

The central idea here is the sampling distribution of the sample mean. When you repeatedly draw random samples of the same size from a population and compute each sample mean, those means form a distribution whose center lines up with the population mean μ. This happens because the expected value of the sample mean equals the population mean, so, across many samples, the average of the sample means equals the true mean. In other words, the distribution is centered around the population mean, though individual sample means vary due to sampling error. The spread of that distribution depends on the population standard deviation and the sample size, specifically shrinking as n grows (the standard error is σ/√n). The Central Limit Theorem further tells us that with a reasonably large sample size, the distribution of the sample mean is approximately normal, making inference about μ more straightforward. The other statements aren’t accurate: you don’t expect every sample mean to equal μ, the distribution is related to μ, and it isn’t always greater than μ.