Question

Suppose samples of size 2 are drawn from the population. List all the possible samples and the corresponding means.

Answer 1

All possible samples:{(2,5), (2,8), (5,8)}.

Sample means for these samples: 3.5, 5, 6.5.

Mean of the sample means: $\dfrac{(3.5+5+6.5)}{3}=5.$

Population mean:$\dfrac{(2+5+8)}{3}=5.$

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

Population variance: $\sigma^2=\frac{(2-5)^2+(5-5)^2+(8-5)^2}{3}=6.$

The variance of the sampling distribution of means: $\sigma_x^2=\frac{\sigma^2(N-n)}{n(N-1)}=\frac{6(3-2)}{2(3-1)}=\frac{3}{2}$

The standard deviation of the sampling distribution of means: $\sigma_x=\sqrt{\frac{3}{2}}$