In large samples, the sampling distribution of the sample mean is

In large samples, the sampling distribution of the sample mean is

• A. Exactly normal for any distribution.
• B. Approximately normal with mean equal to the population mean.
• C. Skewed to the right.
• D. Unpredictable.

Answer: (B)

The key idea is how the sample mean behaves as you take larger samples, which is described by the Central Limit Theorem. When you repeatedly draw samples and compute their means, the distribution of those means becomes bell-shaped and centers around the true population mean as the sample size grows. This means the mean of the sampling distribution equals the population mean. The spread is the standard error, sigma divided by the square root of the sample size, so the distribution tightens as n increases. So, for large samples, the sampling distribution of the sample mean is approximately normal with mean equal to the population mean. It isn’t exactly normal for every underlying distribution, and it isn’t skewed or unpredictable despite the original data’s shape.

The key idea is how the sample mean behaves as you take larger samples, which is described by the Central Limit Theorem. When you repeatedly draw samples and compute their means, the distribution of those means becomes bell-shaped and centers around the true population mean as the sample size grows. This means the mean of the sampling distribution equals the population mean. The spread is the standard error, sigma divided by the square root of the sample size, so the distribution tightens as n increases.

So, for large samples, the sampling distribution of the sample mean is approximately normal with mean equal to the population mean. It isn’t exactly normal for every underlying distribution, and it isn’t skewed or unpredictable despite the original data’s shape.