Question

Random samples of size 4 are drawn with replacement form finite population 3, 6, 9, 20. What is the variance of the sample?

Answer 1

Population mean:

$\mu=\cfrac{3+6+9+20}{4}=9.5.$

Population variance:

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

$X-\mu=\\ =\begin{Bmatrix} 3-9.5,6-9.5,9-9.5,20-9.5 \end{Bmatrix}=$

$=\begin{Bmatrix} -6.5, - 3.5,-0.5,10.5 \end{Bmatrix},$

$\sigma^2=(-6.5)^2\cdot \cfrac{1}{4}+(-3.5)^2\cdot \cfrac{1}{4}+\\ +(-0.5)^2\cdot \cfrac{1}{4}+10.5^2\cdot \cfrac{1}{4}=41.25.$

Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{41.25}{4}=10.3125.$