Sampling Distributions and Central Limit Theorem

Question

The sampling distribution of sample means is from a highly skewed population with µ = 4.47 and σ = 1.40. For repeated random samples of size 100 from this population:

a) Find the mean and standard deviation of the sampling distribution of the sample mean.

b) Explain why the sampling distribution of the sample mean is bell-shaped, even though the population was highly skewed.

Answer 1

a) The mean of the sampling distribution is equal to the mean of the sample µ = 4.47. The standard deviation of the sampling distribution is σx = σ⁄√n. So σx = 1.4 ⁄√100 = 1.4/10 = .14

b) The central limit theorem states that the sampling distribution of the mean will always be normally distributed as long as the sample size is large enough (usually n ≥ 30). So since the sample size of 100 is large, the sampling distribution will be approximately normal.