Consider the following four statements: 1.The distribution of the mean of a random sample approaches the normal distribution as the sample size increases, only if the parent population is normally distributed. 2-The distribution of the mean of a random sample taken from a normal population is normal, irrespective of the sample size. 3-The distribution of the mean of a random sample taken from practically any population approaches the normal distribution as the sample size increases. 4-The distribution of the mean of a random sample taken from practically any population is normal, irrespective of the sample size. Which of the above statements is/are true?
A)Only 1
B)2 and 3
C)2 and 4
D)Only 3
E)None of the above

Question

Quizplus · Accepted Answer

Statement 2 is true: the distribution of the mean of a random sample taken from a normal population is normal, irrespective of the sample size. This is a well-known property of the normal distribution.

Statement 3 is also true: the distribution of the mean of a random sample taken from practically any population approaches the normal distribution as the sample size increases. This is a consequence of the central limit theorem.

Statements 1 and 4 are false. Statement 1 is a common misconception. The central limit theorem guarantees that the distribution of the sample mean approaches normality as the sample size increases, regardless of the shape of the parent population (as long as the sample size is sufficiently large). Statement 4 is also false, as the distribution of the sample mean may not be normal for small sample sizes, even if the parent population is normal.

Consider the Following Four Statements: 1