Question

A population consists of the four numbers 3, 4, 2, 5. Consider all possible 

distinct samples (without replacement) of size two and verify that the population mean 

is equal to the mean of sample means

Answer 1

$\mu=(3+4+2+5)/4=3.5$

$\mu(3,5)=4$

$\mu(3,2)=2.5$

$\mu(3,4)=3.5$

$\mu(4,2)=3$

$\mu(4,5)=4.5$

$\mu(2,5) =3.5$

$f(4)=1$/6

f(2.5)=1/6

f(3.5)=2/6

f(3)=1/6

f(4.5)=1/6

$E(x)=\sum f(x)x=4/6+(2.5)/6+7/6+0.5+(4.5)/6=3.5$

$E(x)=\mu$