Question

annual incomes of statistics students are known to have a distribution that is skewed to the right instead of being normally distributed. assume that we collect a random sample of annual incomes of 50 statistics students. 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 50, so the sample of incomes will be normally distributed

b. no, the population of incomes is not normally distributed, so the sample means will not be normally distributed for any sample size

c. no, unless more than 50 samples are collected, the sample of incomes will not be normally distributed

d. no, the sample means will be normally distributed, but the sample of incomes will be skewed to the right

Explanation:

Step1: Recall central - limit theorem

The central - limit theorem states that the sampling distribution of the sample mean will be approximately normal for a large enough sample size (usually \(n\geq30\)) regardless of the shape of the population distribution. But it is about the sampling distribution of the sample mean, not the distribution of the individual data points in the sample.

Step2: Analyze the problem

The population of annual incomes of statistics students is skewed to the right. We have a sample of \(n = 50\) students' annual incomes. The distribution of individual incomes in the sample will still follow the shape of the population distribution (skewed to the right), while the sampling distribution of the sample mean will be approximately normal due to the central - limit theorem.

Answer:

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