Question

Random samples of size 4 are drawn with replacement from a finite population 3,6,9. Find the variance of the sample and the standard deviation of the sample

Answer 1

We have population values $3,6,9$ population size $N=3$ and sample size $n=4.$ Thus, the number of possible samples which can be drawn without replacement is

$$N^n=3^4=81$$

To find variance and standard deviation we should find mean

In sampling with replacement the mean of all sample means equals the mean of the population:

$$\mu_{\bar{X}}=\mu=\dfrac{3+6+9}{3}=6$$

When sampling with replacement the variance of all sample means equals the variance of the population divided by the sample size

$$\sigma^2=\dfrac{1}{3}((3-6)^2+(6-6)^2+(9-6)^2=6$$ $$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{6}{4}=1.5$$ $$\sigma_{\bar{X}}=\sqrt{\sigma_{\bar{X}}^2}=\sqrt{\dfrac{\sigma^2}{n}}=\dfrac{\sigma}{\sqrt{n}}$$ $$=\dfrac{\sqrt{6}}{\sqrt{4}}=\sqrt{1.5}\approx1.224745$$