Answer to Question #314015 in Math for Azrael

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:

"\\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"


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