Question

1.   A population consisting of the values (1, 3, 5, 7). Consider all possible samples of size 2 which can be drawn with replacement from the population. Find the mean and variance of the sampling distribution of the sample mean.

Answer 1

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

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

Population mean:

$\mu=\cfrac{1+3+5+7}{4}=4.$

Population variance:

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

$X-\mu=\begin{Bmatrix} 1-4,3-4, 5-4,7-4 \end{Bmatrix}=$

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

$\sigma^2=(-3)^2\cdot \cfrac{1}{4}+(-1)^2\cdot \cfrac{1}{4}+\\ +1^2\cdot \cfrac{1}{4}+3^2\cdot \cfrac{1}{4}=5.$

Mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=4.$

Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{5}{2}=2.5.$