Question

consider tje population consisting of the value 1,3,4 list all the possible size2 that can be drawn from the population with replacement , compute the mean, variance and standard deviation of the sampling distribution of the means , construct the probability histogram of the sample mean with replacement when n=2

Answer 1

List of all possible samples of size 2 with replacement, and their means:

{1, 1}, mean = (1+ 1) / 2 = 1

{1, 3}, mean = (1 + 3) / 2 = 2

{1, 4}, mean = (1 + 4) / 2 = 2.5

{3, 3}, mean = (3 + 3) / 2 = 3

{3, 4}, mean = (3 + 4) / 2 = 3.5

{4, 4}, mean = (4 + 4) / 2 = 4

Number of samples is n = 6

Mean of the sampling distribution of the means:

$\mu=\frac{1+2+2.5+3+3.5+4}{6}\approx2.67$

Variance:

$\sigma^2=\frac{1}{n-1}\sum_{k=1}^{n}(\mu_k-\mu)^2=\\ =\frac{1}{5}((1-2.67)^2+(2-2.67)^2+(2.5-2.67)^2+\\ +(3-2.67)^2+(3.5-2.67)^2+(4-2.67)^2)\approx\\ \approx1.16668\approx1.17$

Standard deviation:

$\sigma=\sqrt{\sigma^2}\approx1.08$

Since each mean appears exactly 1 time, probability of each mean is 1 / 6 = 0.1667

Probability histogram: