Question

Let X¯X¯ be a random variable whose values are the possible means of size n for the random variable X. Which of the following statements are true?

(i) If X is normal, then the central limit theorem states that X¯X¯ is normal.

(ii) If X is normal, then X¯X¯ is normal for any value of n.

(iii) If X is not normal, then X¯X¯ is approximately normal for large values of n (n≥30n≥30).

(iv) If X is a random variable with mean μμ and standard deviation σσ, then X¯X¯ is normally distributed with mean μμ and standard deviation σ/nσ/n.

Answer 1

The correct statements are:

(i) If X is normal, then the central limit theorem states that X¯X¯ is normal.

(ii) If X is normal, then X¯X¯ is normal for any value of n.

(iii) If X is not normal, then X¯X¯ is approximately normal for large values of n (n≥30).