Question

According to the most recent census, the average total yearly expenses of households in Wake County is $38,500. It is also known that the distribution of household expenses in Wake County is strongly skewed to the right with a standard deviation of $10,500. A researcher is going to randomly select a sample of 5 households from Wake County. Which of the following is true? Select all that apply.

Select one or more:

a. The sampling distribution of the sample mean will have a smaller standard deviation than the population.

b. The sampling distribution of the sample mean will have a different standard deviation than the population.

c. The sampling distribution of the sample mean will be skewed right.

d. The sampling distribution of the sample mean will have a larger standard deviation than the population.

e. We cannot tell what the shape of the sampling distribution of the sample mean will look like.

f. The shape of the sampling distribution of the sample mean will be approximately symmetric.

Answer 1

Solution:

Given that ,

= 38,500

= 10,500

A sample of size n = 5 is taken from this population.

Let be the mean of sample.

The sampling distribution of the has mean and standard deviation as

Mean() = =   38,500

SD() =    = 10,500/​5 = 4695.7427

The sampling distribution will be approximately normal in following 2 conditions.

If population distribution is known to be normal.

If not so , but if sample size is greater than 30.

In given example , we the population distribution is not normal. And n = 5 is a small sample size.

Answer : Correct options are

a. The sampling distribution of the sample mean will have a smaller standard deviation than the population..

b. The sampling distribution of the sample mean will have a different standard deviation than the population.

e. We cannot tell what the shape of the sampling distribution of the sample mean will look like.