Question

Which of the following statements about the sampling distribution of the sample mean, x-bar, is not true?

• The distribution is normal regardless of the shape of the population distribution, as long as the sample size,n, is large enough.
• The distribution is normal regardless of the sample size, as long as the population distribution is normal.
• The distribution's mean is the same as the population mean.
• The distribution's standard deviation is smaller than the population standard deviation.
• All of the above statements are correct.

Answer 1

Statement 1 is true, due to the assumption of normality.

Statement 2 is true by Central Limit Theorem.

Statement 3 is true, since the sample mean is an unbiased estimator of the population mean.

Statement 4 is true, since, E(sample sd) < population sd.

Hence, all of the above statements are true.