Question

Samples of two cards are drawn at random from a population of 6 cards numbered from 1 to 6.

A. How many possible samples can be drawn?

B. Construct the sampling distribution of sample means.

Answer 1

a. $C^2_6=\frac{6!}{2!4!}=15$

b.m(1,2)=(1+2)/2=1.5

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

m(1,4)=(1+4)/2=2.5

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

m(1,6)=(1+6)/2=3.5

m(2,3)=(2+3)/2=2.5

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

m(2,5)=(2+5)/2=3.5

m(2,6)=(2+6)/2=4

m(3,4)=(3+4)/2=3.5

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

m(3,6)=(3+6)/2=4.5

m(4,5)=(4+5)/2=4.5

m(4,6)=(4+6)/2=5

m(5,6)=(5+6)/2=5.5

Frequency F(1.5)=F(2)=F(5)=F(5.5)=1

F(2.5)=F(3)=F(4)=F(4.5)=2

F(3.5)=3

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

P(1.5)=P(2)=P(5)=P(5.5)=1/15

P(2.5)=P(3)=P(4)=P(4.5)=2/15

P(3.5)=3/15

$E(x)=\sum P(x)x=1/15(2+5+1.5+5.5)+2/15(2.5+3+4+4.5)+3/15(3.5)=14/15+28/15+10.5/15=52.5/15=3.5,$

$\sigma^2=1/15(4+25+2.25+30.25)+2/15(6.25+9+16+20.25)+3/15x12.25-12.25=4.1+6.87+2.45-12.25=1.17$