Answer to Question #262828 in Statistics and Probability for Andeng

Question

A population consists of three numbers (3, 6, 9). Consider all possible samples of size 2

which can be drawn without replacement from the population. Find the following:

a. Population mean

b. Population variance

c. Populations standard deviation

d. Mean of the sampling distribution of the means

e. Variance of the sampling distribution of the means

f. Standard deviation of the sampling distributions of the means

Answer 1

a.

Population mean:

$\mu=\frac{\sum x_i}{n}=\frac{3+6+9}{3}=6$

b.

Population variance:

$s^2=\frac{\sum (x_i-\mu)^2}{n}=6$

c.

Population standard deviation:

$s=\sqrt{\frac{\sum (x_i-\mu)^2}{n}}=\sqrt6=2.45$

d.

possible samples of size 2 mean

(3,6) 4.5

(3,9) 6

(6,9) 7.5

Mean of the sampling distribution of the means:

$\frac{4.5+6+7.5}{3}=6$

e.

Variance of the sampling distribution of the means:

$\frac{(4.5-6)^2+(6-6)^2+(7.5-6)^2}{3}=1.5$

f.

Standard deviation of the sampling distributions of the means:

$\sqrt{\frac{(4.5-6)^2+(6-6)^2+(7.5-6)^2}{3}}=\sqrt{1.5}=1.22$