Question

A population consists of the four number (2,3,6,9).

Consider all possible samples of size 2 that can be drawn with replacement from this population. Answer the following:

a. List all possible samples of size 2 which can be drawn with replacement from this population.

b. Compute the Population mean.

c. Compute the Population Standard deviation.

d. Find the mean of the sampling distribution.

Answer 1

a. The number of possible samples which can be selected with replacement is

$N^n=4^2=16.$

All the possible samples of sizes n=2 wich can be drawn with replacement from the population:

$\{ (2,2), (2,3),(2,6),(2,9),\\ (3,2),(3,3),(3,6),(3,9),\\ (6,2), (6,3),(6,6),(6,9),\\ (9,2),(9,3),(9,6),(9,9)\}.$

b. The population mean:

$\mu=\cfrac{2+3+6+9}{4}=5.$

c. The population variance:

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

$X-\mu=\begin{Bmatrix} 2-5,3-5,6-5,9-5 \end{Bmatrix}=$

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

$\sigma^2=(-3)^2\cdot \cfrac{1}{4}+(-2)^2\cdot \cfrac{1}{4}+\\ +1^2\cdot \cfrac{1}{4}+4^2\cdot \cfrac{1}{4}=7.5.$

The population standard deviation:

$\sigma=\sqrt{7.5}=2.739.$

d. The mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=5.$