Question

A population consists of three numbers (3,6,9).Consider all possible samples of size n=2 which can be drawn WITHOUT REPLACEMENT from the population.

Questions to be answered:

Population mean

Population variance

Population standard deviation

Mean of the sampling distribution of the means

Variance of the sampling distributionof the means

Standard deviation of the sampling distribution of the means

Answer 1

Population mean:

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

Population variance:

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

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

$=\begin{Bmatrix} -3, 0,3 \end{Bmatrix},$

$\sigma^2=(-3)^2\cdot \cfrac{1}{3}+0^2\cdot \cfrac{1}{3}+3^2\cdot \cfrac{1}{3}=6.$

Population standard deviation:

$\sigma=\sqrt{6}=2.449.$

Mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=6.$

Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{6}{2}=3.$

Standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{\sqrt 6}{\sqrt 2}=1.732.$