Question

A population consists of three numbers 2, 5, 8. Enumerate all possible samples of size 2
which can be drawn without replacement from this population. Verify that the sample mean is an unbiased estimate of the population mean. Calculate the standard error of the sample
mean.

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: (3.5+5+6.5)/3=5.

Population mean: (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: $\sigma_x=\sqrt{\frac{3}{2}}$