Question

random samples of size 4 are drawn with replacement from a finite population 3,6,9

1. How many possible samples of size 4 are possible?
2. Find the mean of the sample means.
3. Find the variance of the sample.
4. Find the standard deviation of the sample.

Answer 1

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

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

2.

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

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

3.

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

$$\sigma^2=\dfrac{1}{3}((3-6)^2+(6-6)^2+(9-6)^2=6$$

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}=\dfrac{6}{4}=1.5$$

4.

$$\sigma_{\bar{X}}=\sqrt{\sigma_{\bar{X}}^2}=\sqrt{\dfrac{\sigma^2}{n}}=\dfrac{\sigma}{\sqrt{n}}$$

$$=\dfrac{\sqrt{6}}{\sqrt{4}}=\sqrt{1.5}\approx1.224745$$