Question

A population consists of the numbers 1, 2, 3, and 4 with the sample size of 2. Compute the variance of the sampling distribution of means.

Answer 1

Population mean:

$\mu=\cfrac{1+2+3+4} {4} =2.5.$

Population variance:

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

$X-\mu=\begin{Bmatrix} 1-2.5, 2-2.5, 3-2.5, 4-2.5 \end{Bmatrix}=$

$=\begin{Bmatrix} -1.5, -0.5, 0.5, 1.5 \end{Bmatrix},$

$\sigma^2=(-1.5)^2\cdot \cfrac{1}{4}+(-0.5)^2\cdot \cfrac{1}{4}+\\+0.5^2\cdot \cfrac{1}{4}+1.5^2\cdot \cfrac{1}{4}=1.25.$

Variance of the sampling distribution of means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{1.25}{2}=0.625.$