Question #314015

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.


1
Expert's answer
2022-03-19T02:37:53-0400

All the samples of size 2:

(2,5),xˉ=3.5(2,7),xˉ=4.5(2,8),xˉ=5(2,9),xˉ=5.5(5,7),xˉ=6(5,8),xˉ=6.5(5,9),xˉ=7(7,8),xˉ=7.5(7,9),xˉ=8(8,9),xˉ=8.5\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(xˉ=3.5)=P(xˉ=4.5)=P(xˉ=5)=P(xˉ=5.5)=P(xˉ=6)==P(xˉ=6.5)=P(xˉ=7)=P(xˉ=7.5)=P(xˉ=8)=P(xˉ=8.5)=0.1P\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(3.5+4.5+5+5.5+6+6.5+7+7.5+8+8.5)=6.2EX2=0.1(3.52+4.52+52+5.52+62+6.52+72+7.52+82+8.52)=40.75DX=EX2(EX)2=40.756.22=2.31σX=DX=2.31=1.51987EX=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


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