Question

A population consists of the values {1, 2, 3, 4, 5). Consider samples of size 3 that can be drawn

from this population.

a. List down all the possible samples and corresponding sample mean

Sample

Sample Means

5

b. Construct the sampling distribution of the sample means.

Sample Means Frequency

Probability P(x)

c. Draw a histogram of the sampling distribution of the sample mean.

Answer 1

m(1,2,3)=(1+2+3)/3=2

m(1,2,4)=(1+2+4)/3=2.3

m(1,2,5)=(1+2+5)/3=2.7

m(1,3,4)=(1+3+4)/3=2.7

m(1,3,5)=(1+3+5)/3=3

m(1,4,5)=(1+4+5)/3=3.3

m(2,3,4)=(2+3+4)=3

m(2,3,5)=(2+3+5)/3=3.3

m(3,4,5)=(3+4+5)/3=4

m(2,4,5)=(2+4+5)/3=3.7

Frequency

F(2)=F(2.3)=F(4)=F(3.7)=1

F(2.7)=F(3)=F(3.3)=2

Probabilitys

$P(x)=F(x)/\sum F(x)$

P(2)=P(2.3)=P(4)=P(3.7)=0.1

P(2.7)=P(3)=P(3.3)=0.2

$E(x)=\sum P(x)x=0.1(2+2.3+4+3.7)+0.2(2.7+3+3.3)=1.2+1.8=3$

$\sigma^2=\sum P(x)x^2-(\sum P(x)x)^2=0.1(4+5.29+16+13.69)+0.2(7.29+9+10.89)-9=3.898+5.436-9=0.334$