Question

A population consist of 2,4,5,9 and 10, with sample size of 3. Compute the mean and variance of the

sampling distribution of the sample mean.

Answer 1

The mean of the sampling distribution of the sample means is the mean of the population from which the scores were sampled:

$\mu_{\bar x} =\mu=\\ =\cfrac{2+4+5+9+10} {5} =6.$

Population variance:

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

$X-\mu=\\ =\begin{Bmatrix} 2-6, 4-6, 5-6, 9-6, 10-6 \end{Bmatrix}=$

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

$\sigma^2=(-4)^2\cdot \cfrac{1}{5}+(-2)^2\cdot \cfrac{1}{5}+(-1)^2\cdot \cfrac{1}{5}+\\ +3^2\cdot \cfrac{1}{5}+4^2\cdot \cfrac{1}{5}=9.2.$

Variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{9.2}{3}=3.07.$