Question

1. Samples of eight cards are drawn at random from a population of nine cards numbered from 1 to 9.

a. How many possible samples can be drawn?

b. Construct the sampling distribution of sample means.

Answer 1

a$C^8_9=\frac{9!}{8!1!}=9$

m(1,2,3,4,5,6,7,8)=(1+2+3+4+5+6+7+8)/8=4.5

m(1,2,3,4,5,6,7,9)=(1+2+3+4+5+6+7+9)/8=4.625

m(1,2,3,4,5,6,8,9)=(1+2+3+4+5+6+8;9)/8=4.75

m(1,2,3,4,5,7,8,9)=(1+2+3+4+5+7+8+9)/8=4.875

m(1,2,3,4,6,7,8,9)=(1+2+3+4+6+7+8+9)/8=5

m(1,2,3,5,6,7,8,9)=(1+2+3+5+6+7+8+9)/8=5.125

m(1,2,4,5,6,7,8,9)=(1+2+4+5+6+7+8+9)/8=5.25

m(1,3,4,5,6,7,8,9)=(1+3+4+5+6+7+8+9)/8=5.375

m(2,3,4,5,6,7,8,9)=5.5

f(4.5)=f(4.625)=f(4.75)=f(5)=f(5.125)=f(5.25)=f(5.375)=f(5.5)=1/9

$E(x)=\sum fx=1/9(4.5+4.625+4.75+4.875+5+5.125+5.25+5.375+5.5)=45/9=5$

$\sigma^2=\sum fx^2-(\sum fx)^2=1/9(20.25+21.39+22.56+23.77+25+26.27+27.56+28.89+30.25)-25=0.1$