Question

Consider a population consisting of the values (1, 3, 8), n= 2 with replacement.

1. mean of sampling distribution of means
2. variance of the sampling distribution of means
3. standard deviation of the sampling distribution

Answer 1

Solution:

Possible sample of size 2 with replacement$=\{(1,3)(3,8)(1,8),(1,1),(3,3),(8,8)\}$

Sample means for these samples$=\{2,5.5,4.5,1,3,8\}$

(1): Mean of sample means$=(2+5.5+4.5+1+3+8)/6=4$

Population mean$=(1+3+8)/3=4$

Population variance$,\sigma^2=\dfrac{(1-4)^2+(3-4)^2+(8-4)^2}{3}=\dfrac{26}3$

(2): The variance of the sampling distribution of means$,\sigma_x^2=\dfrac{\sigma^2(N-n)}{n(N-1)}$

$=\dfrac{\dfrac{26}3(3-2)}{2(3-1)}=\dfrac{13}6$

(3): The standard deviation of the sampling distribution$,\sigma_x=\sqrt{\dfrac{13}6}$