Question

Random samples of size 3 are taken from a population of the numbers 3, 4, 5, 6 , 7 and 8. Construct the sampling distribution of the sample means and determine its standard deviation.

Answer 1

All the possible samples of sizes n=3 wich can be drawn without replacement from the population:

$\{ (3,4,5), (3,4,6),(3,4,7),(3,4,8),(3,5,6),\\ (3,5,7),(3,5,8),(3,6,7),(3,6,8),(3,7,8),\\ (4,5,6),(4,5,7),(4,5,8),(4,6,7),(4,6,8),\\ (4,7,8),(5,6,7),(5,6,8),(5,7,8),(6,7,8)\}.$

Population mean:

$\mu=\cfrac{3+4+5+6+7+8}{6}=5.5.$

Population variance:

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

$X-\mu=\begin{Bmatrix} 3-5.5,4-5.5, 5-5.5, 6-5.5,7-5.5,8-5.5 \end{Bmatrix}=$

$=\begin{Bmatrix} -2.5, -1.5, -0.5,0.5, 1.5,2.5 \end{Bmatrix},$

$\sigma^2=(-2.5)^2\cdot \cfrac{1}{6}+(-1.5)^2\cdot \cfrac{1}{6}+(-0.5)^2\cdot \cfrac{1}{6}+\\ +0.5^2\cdot \cfrac{1}{6}+1.5^2\cdot \cfrac{1}{6}+2.5^2\cdot \cfrac{1}{6}=2.917.$

Population standard deviation:

$\sigma=\sqrt{2.917}=1.708.$

The standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{1.708}{\sqrt 3}=0.986.$