Question

Given a population consists of three numbers (2,5,9). 

Consider all possible samples of size 2 which can be drawn from the population. 

Find the variance of the sampling distribution of the sample means.

Answer 1

All possible samples:{(2,5), (2,9), (5,9)}.

Sample means for these samples: 3.5, 5.5, 7.

Mean of the sample means: (3.5+5.5+7)/3=5.33

Population mean = (2+5+9)/3=5.33

So, the sample mean is an unbiased estimate of the population mean.

Population variance: $\sigma^2=\frac{(2-5.33)^2+(5-5.33)^2+(9-5.33)^2}{3}=8.22$

The variance of the sampling distribution of means: $\sigma_x^2=\frac{\sigma^2(N-n)}{n(N-1)}=\frac{8.22(3-2)}{2(3-1)}=2.055$