A variable of a population is normally distributed with mean

mu

and standard deviation

sigma

. Answer parts (a) through (d) below.

a. Identify the distribution of

x overbar

x. Choose the correct answer below.

A.

Normal with mean

StartFraction mu Over StartRoot n EndRoot EndFraction

n

and standard deviation

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

B.

Normal with mean

mu

and standard deviation

sigma

C.

Normal with mean

mu

and standard deviation

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

D.

Normal with mean

StartFraction mu Over StartRoot n EndRoot EndFraction

n

and standard deviation

sigma

b. Does your answer to part (a) depend of the sample size? Explain your answer.

A.

No, because the variable under consideration is normally distributed.

B.

Yes, because the variable under consideration is normally distributed.

C.

Yes, because the distribution of

x overbar

x is normal only if the sample size is large.

D.

No, because the shape of a sampling distribution does not depend on sample size.

c. Identify the mean and standard deviation of

x overbar

x. Choose the correct answer below.

A.

StartFraction mu Over StartRoot n EndRoot EndFraction

n

and

sigma

B.

mu

and

sigma

C.

StartFraction mu Over StartRoot n EndRoot EndFraction

n

and

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

D.

mu

and

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

d. Does your answer to part (c) depend on the assumption that the variable under consideration is normally distributed? Why or why not?

A.

No, for any variable, the mean of

x overbar

x equals

mu

, and the standard deviation of

x overbar

x is at least approximately equal to

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

.

B.

No, for any variable, the mean of

x overbar

x is at least approximately equal to

mu

, and the standard deviation of

x overbar

x exactly equals

StartFraction sigma Over StartRoot n EndRoot EndFraction

n

.

C.

Yes, if the distribution of x is skewed, the mean of

x overbar

x will be different from

mu

in the direction that the distribution of x is skewed.

D.

Yes, for some distributions of x, the mean of

x overbar

x is

StartFraction mu Over StartRoot n EndRoot EndFraction

n

, and for other distributions of x, the standard deviation of

x overbar

x is

sigma

.