Question

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample. a. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)?

b. What value do the sample means target? That is, what is the mean of all such sample means?

Short answer

a. The sampling distribution of the sample means will be approximately bell-shaped (normal).

b. The sample means target the population mean. The mean of all the samples of annual incomes will be equal to the population mean annual income.

Step-by-step solution

Given information

The population of all the college presidents (4200) of the United States is considered. A sample size of four is considered to extract many samples from the population.

Sampling distribution of sample means

The sampling distribution of the sample means always follows the normal distribution; that is, the distribution of the sample means is bell-shaped

Here, several samples of 40 college presidents are selected, and the mean annual income for each sample is computed.

a.

It is given that the population mean annual income of college presidents is skewed.

Although the population distribution is skewed, the distribution of the sample means will be bell-shaped and not normal or uniform or any other because the sampling distribution of a statistic is normally distributed, and the normal distribution is bell-shaped.

b.

The sample means always target the population mean. The mean value of all the samples of annual incomes of the college presidents will be equal to the population mean annual income.