Question

A population of a certain ethnic group is consisting of three numbers (3,4,5,6,7). if samples of size 2 will be taken without replacement, find..

a. population mean

b. population variance

c. population standard deviation

d. mean of the sampling distribution of the sample mean

e. variance of the sampling distribution of the sample mean

f. standard deviation of the sampling distribution of the sample mean

Answer 1

a. Population mean:

$\mu=\cfrac{3+4+5+6+7}{5}=5.$

b. Population variance:

$\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i),$

$X-\mu=\begin{Bmatrix} 3-5,4-5, 5-5, 6-5,7-5 \end{Bmatrix}=$

$=\begin{Bmatrix} -2, -1,0,1, 2 \end{Bmatrix},$

$\sigma^2=(-2)^2\cdot \cfrac{1}{5}+(-1)^2\cdot \cfrac{1}{5}+0^2\cdot \cfrac{1}{5}+\\ +1^2\cdot \cfrac{1}{5}+2^2\cdot \cfrac{1}{5}=2.$

c. Population standard deviation:

$\sigma=\sqrt{2}=1.414.$

d. Mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=5.$

e. Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{2}{2}=1.$

f. Standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{\sqrt 2}{\sqrt 2}=1.$