Answer to Question #325990 in Statistics and Probability for Lhang

Question

Random samples of size N=2 are drawn from a finite population consisting of the number 5,6,7,8, and 9. Compute for the mean, Variance,and Standard deviation, and also the mean, Variance, and the Standard deviation of the sample mean

Answer 1

The population mean:

$\mu=\cfrac{5+6+7+8+9}{5}=7.$

The population variance:

$\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i),$

$X-\mu=\begin{Bmatrix} 5-7,6-7,7-7, 8-7,9-7 \end{Bmatrix}=$

$=\begin{Bmatrix} -2, -1, 0, 1,2 \end{Bmatrix},$

$\sigma^2=(-2)^2\cdot \cfrac{1}{5}+(-1)^2\cdot \cfrac{1}{5}+0^2\cdot \cfrac{1}{5}+\\ +1^2\cdot \cfrac{1}{5}+2^2\cdot \cfrac{1}{5}=2.$

The population standard deviation:

$\sigma=\sqrt{2}=1.414.$

The mean of the sampling distribution of sample means:

$\mu_{\bar x} =\mu=7.$

The variance of the sampling distribution of sample means:

$\sigma^2_{\bar x}=\cfrac{\sigma^2}{n}=\cfrac{2}{2}=1.$

The standard deviation of the sampling distribution of sample means:

$\sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{\sqrt 2}{\sqrt 2}=1.$