Question

random samples of size n=2 are drawn from a finite population consisting of the numbers 5,6,7,8,9. compute fot the mean, variance and standard deviation of the population, and the mean, variance and standard deviation of the mean

Answer 1

The population values are $5, 6, 7, 8$ $and$ $9$.

The sample size $n=2$ .

$Population\:mean=\frac{5+6+7+8+9}{5}$

$Population\:mean=7$

$\:Variance=\frac{\left(5-7\right)^2+\left(6-7\right)^2+\left(7-7\right)^2+\left(8-7\right)^2+\left(9-7\right)^2}{5}$

$Variance=\frac{4+1+0+1+4}{5}$

$Variance=2$

$Standard\:deviation=\sqrt{2}$ $=1.41$

The table for sampling distribution is shown below:

$The\:variance\:of\:sampling\:distribution=\frac{\left(5.5-7\right)+\left(6-7\right)+2\left(6.5-7\right)+2\left(7-7\right)+2\left(7.5-7\right)+\left(8-7\right)+\left(8.5-7\right)}{10}$

$Variance\:of\:sampling\:distribution\:is\:=\frac{2.25+1+0.5+0+0.5+1+2.25}{10}$

$Variance\:of\:sampling\:distribution\:is\:=0.75$

$Sample\:standard\:deviation=\sqrt{0.75}$

$Sample\:standard\:deviation=0.86$