Answer to Question #184315 in Statistics and Probability for Felicity Sy

Question

Consider a population with the values (1, 3, 4, 8).

a. Find the population mean.

b. Find the population variance.

c. Find the population standard deviation.

d. Find the mean of the sampling distribution of means.

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

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

Answer 1

a. Population mean $\mu=\dfrac{1+3+4+8}{4}=4$

b. Population variance $\sigma^2=\dfrac{\sum (x_i-\mu)^2}{N}=\dfrac{9+1+0+16}{4}=6.5$

c. Population Standard Deviation $= \sqrt{Population \ Variance}=\sqrt{6.5}=2.55$

d. Total number of samples of size 2 = $^4C_2=6$

$(1,3),(1,4),(1,8),(3,4),(3,8),(4,8)$

Mean of sampling distribution of means $E(\bar X)=\mu=4$

e. Variance of sampling distribution of means $Var(\bar X)=\dfrac{\sigma^2}{n}(\dfrac{N-n}{N-1})$

$Var(\bar X)=\dfrac{6.5}{2}(\dfrac{2}{3})=2.167$

f. Standard deviation of sampling distribution of means $=\sqrt{Var(\bar X)}=\sqrt{2.167}=1.472$