Data on salaries in the public school system are published annually by a teachers' association. The mean annual salary of (public) classroom teachers is

$56.6

thousand. Assume a standard deviation of

$7.6

thousand. Complete parts (a) through (e) below.

a. Determine the sampling distribution of the sample mean for samples of size 64.

The mean of the sample mean is

x=$enter your response here.

(Type an integer or a decimal. Do not round.)

The standard deviation of the sample mean is

x=$enter your response here.

(Type an integer or a decimal. Do not round.)

b. Determine the sampling distribution of the sample mean for samples of size 256.

The mean of the sample mean is

x=$enter your response here.

(Type an integer or a decimal. Do not round.)

The standard deviation of the sample mean is

x=$enter your response here.

(Type an integer or a decimal. Do not round.)

c. Do you need to assume that classroom teacher salaries are normally distributed to answer parts (a) and (b)? Explain your answer.

A.

No, because if

x

is normally distributed, then x must be normally distributed.

B.

Yes, because the sample sizes are not sufficiently large so that

x

will be approximately normally distributed, regardless of the distribution of x.

C.

Yes, because

x

is only normally distributed if x is normally distributed.

D.

No, because the sample sizes are sufficiently large so that

x

will be approximately normally distributed, regardless of the distribution of x.

d. What is the probability that the sampling error made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 64 classroom teachers will be at most $1000?

enter your response here

(Round to three decimal places as needed.)

e. What is the probability that the sampling error made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 256 classroom teachers will be at most $1000?

enter your response here

(Round to three decimal places as needed.)