Question

Using the numbers 2, 5, 7, 8, and 9 as the elements of the population, do the following:

Find the mean of the samples of size 2 (n=2)

Construct the sampling distribution of the sample means (SDSM).

Create a graph of the histogram of the SDSM. 

Compute the Mean, Variance, and Standard Deviation of the SDSM.

Answer 1

All the samples of size 2:

$\left( 2,5 \right) ,\bar{x}=3.5\\\left( 2,7 \right) ,\bar{x}=4.5\\\left( 2,8 \right) ,\bar{x}=5\\\left( 2,9 \right) ,\bar{x}=5.5\\\left( 5,7 \right) ,\bar{x}=6\\\left( 5,8 \right) ,\bar{x}=6.5\\\left( 5,9 \right) ,\bar{x}=7\\\left( 7,8 \right) ,\bar{x}=7.5\\\left( 7,9 \right) ,\bar{x}=8\\\left( 8,9 \right) ,\bar{x}=8.5$

The sampling distribution

$P\left( \bar{x}=3.5 \right) =P\left( \bar{x}=4.5 \right) =P\left( \bar{x}=5 \right) =P\left( \bar{x}=5.5 \right) =P\left( \bar{x}=6 \right) =\\=P\left( \bar{x}=6.5 \right) =P\left( \bar{x}=7 \right) =P\left( \bar{x}=7.5 \right) =P\left( \bar{x}=8 \right) =P\left( \bar{x}=8.5 \right) =0.1$

Histogram:

$EX=0.1\left( 3.5+4.5+5+5.5+6+6.5+7+7.5+8+8.5 \right) =6.2\\EX^2=0.1\left( 3.5^2+4.5^2+5^2+5.5^2+6^2+6.5^2+7^2+7.5^2+8^2+8.5^2 \right) =40.75\\DX=EX^2-\left( EX \right) ^2=40.75-6.2^2=2.31\\\sigma X=\sqrt{DX}=\sqrt{2.31}=1.51987$