Question

Consider all samples of size 2 from the population 2,5,6,9,11 and 13. Compute the mean, the variance of the sampling distribution of the sample mean

Answer 1

$\mu =EX=\frac{2+5+6+9+11+13}{6}=\frac{23}{3}\\EX^2=\frac{2^2+5^2+6^2+9^2+11^2+13^2}{6}=\frac{218}{3}\\\sigma ^2=DX=EX^2-\left( EX \right) ^2=\frac{218}{3}-\left( \frac{23}{3} \right) ^2=\frac{125}{9}\\\mu _{\bar{x}}=\mu =\frac{23}{3}\\{\sigma ^2}_{\bar{x}}=\frac{\sigma ^2}{n}=\frac{125}{18}$