Question

Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large ​(ngreater than​30) random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is​ large? Why or why​ not? Choose the correct answer below.

(A) Yes; the sample size is over 30, so the sample of incomes will be normally distributed
(B) No; unless more than 30 samples are collected, the sample of incomes will not be normally distributed
(C) No; the population of incomes is not normally distributed, so the sample means will not be normally distributed for any sample size.
(D) No, the sample means will be normally distributed, but the sample of incomes will be skewed to the right.

Answer 1

(D) No, the sample means will be normally distributed, but the sample of incomes will be skewed to the right.

Explanation:

We use the Central Limit Theorem to solve this question.

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, a sample size of 30 or higher can be approximated to a normal distribution with mean

In this problem, we have that:

We are selecting a sample of incomes, not finding a sample of sample means, for which the Central Limit Theorem is valid.

So the correct answer to this question is:

(D) No, the sample means will be normally distributed, but the sample of incomes will be skewed to the right.