Question

Consider a population with values 1, 2, 3, 5, 7, 11

a. Find the population mean

b. Find the population variance

c. Find the population standard deviation.

d. Find all possible samples of size 4 which can be drawn with replacement from this

population

e. Find the mean of the sampling distribution.

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

g. Find the standard deviation of the sampling distribution of means

Answer 1

We have population values 1,2,3,5,7,11, population size N=6 and sample size n=4.

a.Mean of population $(\mu)$ = $\dfrac{1+2+3+5+7+11}{6}=\dfrac{29}{6}$

b. Variance of population 

$$\sigma^2=\dfrac{\Sigma(x_i-\bar{x})^2}{N}=\dfrac{1}{6}((1-\dfrac{29}{6})^2+(2-\dfrac{29}{6})^2$$

$$+(3-\dfrac{29}{6})^2+(5-\dfrac{29}{6})^2+(7-\dfrac{29}{6})^2$$

$$+(11-\dfrac{29}{6})^2)=\dfrac{413}{36}\approx11.4722$$

c.

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{413}{36}}=\dfrac{\sqrt{413}}{6}\approx3.3871$$

d. The number of possible samples which can be drawn with replacement is $N^n=6^4=1296.$

e. Mean of sampling distribution 

$$\mu_{\bar{X}}=E(\bar{X})=\mu=\dfrac{29}{6}$$

f. The variance of sampling distribution 

$$Var(\bar{X})=\sigma^2_{\bar{X}}=\dfrac{\sigma^2}{n}=\dfrac{413}{36(4)}=\dfrac{413}{144}$$

g.

$$\sigma_{\bar{X}}=\sqrt{\sigma_{\bar{X}}^2}=\sqrt{\dfrac{413}{144}}=\dfrac{\sqrt{413}}{12}\approx1.6935$$