Which of the following is NOT listed as one of the three characteristics of the sampling distribution of the sample means under the central limit theorem?

Which of the following is NOT listed as one of the three characteristics of the sampling distribution of the sample means under the central limit theorem?

• A. The sample mean is near the population mean.
• B. The population from which the samples are drawn is normal.
• C. The number of scores in each sample should be at least 30.
• D. The samples are independent.

Answer: (D)

The central limit theorem describes three key features of the sampling distribution of the sample mean: its center, its shape for large samples, and its spread. The sampling distribution is centered at the population mean, so the sample means tend to cluster near the true mean. As the sample size grows, the distribution of those sample means becomes approximately normal, even if the population distribution isn’t normal. The spread of this distribution is the standard error, which equals the population standard deviation divided by the square root of the sample size, meaning the dispersion shrinks as n grows. Independence of the samples is a crucial underlying condition for these results, but it isn’t typically listed as one of the three characteristics of the sampling distribution itself. The three commonly cited characteristics are the mean alignment with the population mean, the approximate normal shape for large n, and the reduced spread given by the standard error. Therefore, the statement about samples being independent is not one of the three characteristics.

The central limit theorem describes three key features of the sampling distribution of the sample mean: its center, its shape for large samples, and its spread. The sampling distribution is centered at the population mean, so the sample means tend to cluster near the true mean. As the sample size grows, the distribution of those sample means becomes approximately normal, even if the population distribution isn’t normal. The spread of this distribution is the standard error, which equals the population standard deviation divided by the square root of the sample size, meaning the dispersion shrinks as n grows.

Independence of the samples is a crucial underlying condition for these results, but it isn’t typically listed as one of the three characteristics of the sampling distribution itself. The three commonly cited characteristics are the mean alignment with the population mean, the approximate normal shape for large n, and the reduced spread given by the standard error. Therefore, the statement about samples being independent is not one of the three characteristics.