Question

Random sample of size N = 2 are drawn from a finite population consisting of

the numbers 1, 3, 8, 9.

Find the following:

Mean, Variance and Standard deviation of the Population.

Mean, Variance and Standard deviation of the Sample means.

Answer 1

The population mean:

$\mu=\cfrac{1+3+8+9}{4}=5.25.$

The population variance:

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

$X-\mu=\\ =\begin{Bmatrix} 1-5.25,3-5.25,8-5.25,9-5.25 \end{Bmatrix}=$

$=\begin{Bmatrix} -4.25, -2.25, 2.75, 3.75 \end{Bmatrix},$

$\sigma^2=(-4.25)^2\cdot \cfrac{1}{4}+(-2.25)^2\cdot \cfrac{1}{4}+2.75^2\cdot \cfrac{1}{4}+\\ +3.75^2\cdot \cfrac{1}{4}=11.189.$

The population standard deviation:

$\sigma=\sqrt{11.189}=3.345.$

The mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=5.25.$

The variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{11.189}{2}=5.594.$

The standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{3.345}{\sqrt 2}=2.365.$