Question

Draw all the possible samples of 2 without replacement from a population consisting of 4, 9, 12, 15, 20. Contruct the sampling distribution of sample mean and verify

Answer 1

m(4,9)=(4+9)/2=6.5

m(4,12)=(4+12)/2=8

m(4,15)=(4+15)/2=9.5

m(4,20)=(4+20)/2=12

m(9,12)=(9+12)/2=10.5

m(9,15)=(9+15)/2=12

m(9,20)=(9+20)/2=14.5

m(12,15)=(12+15)/2=13.5

m(12,20)=(12+20)/2=16

m(15,20)=(15+20)/2=17.5

Frequency

F(6.5)=F(8)=F(9.5)=F(10.5)=F(14.5)=F(13.5)=F(16)=F(17.5)=1

F(12)=2

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

P(6.5)=P(8)=P(9.5)=P(10.5)=P(14.5)=P(13.5)=P(16)=P(17.5)=0.1

P(12)=0.2

$E(x)=\sum P(x)x=0.1(6.5+8+9.5+10.5+14.5+13.5+16+17.5)+0.2x12=9.6+2.4=12$

$\sigma^2=\sum P(x)x^2-(\sum P(x)x)^2=0.1(42.25+64+90.25+110.25+210.25+182.25+256+306.25)+0.2x144-144=10.95$