Question

The incomes in a large population of college teachers have an extremely right-skewed distribution with mean $55,000 in standard deviation $5,000. Random sample of five teachers will be selected from this population to serve on a salary review committee. Can we validly use the standard Normal table to calculate the probability that their mean income will be greater than $60,000?

A. Yes, because the sampling distribution of X will be approximately normal
B. Yes, because using the standard Normal table is always appropriate
C. No, because the population standard deviation is not known
D. No, because the central limit theorem does not apply

Answer 1

We cannot use the standard Normal table to calculate the probability that the mean income of a very small sample of college teachers will be greater than $60,000, because the central limit theorem does not apply to small samples from non-normal distributions.

Step-by-step explanation:

The question revolves around whether we can use the standard Normal table to calculate the probability that the mean income of a sample of five college teachers will be greater than $60,000. Given that the distribution of incomes is extremely right-skewed, we must consider the Central Limit Theorem (CLT). CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough, typically if the sample size is greater than or equal to 30. Since we are dealing with a sample size of only five, the CLT does not apply, and the sampling distribution of the mean may not be normal.

Therefore, the correct answer is D. No, because the central limit theorem does not apply to such a small sample size from a population with a non-normal distribution.