Question

Suppose a population consists of the values 6, 7, 8 and 9 and the sample size two are drawn from this population.

a. List all possible sample size two, with replacement, and compute the sample mean in each case.

b. Construct the sampling distribution of the sample mean.

c. Construct the probability histogram of the sampling distribution of the sample mean.

Answer 1

a. m(6,7)=(6+7)/2=6.5

m(6,8)=(6+8)/2=7

m(6,9)=(6+9)/2=7.5

m(7,8)=(7+8)/2=7.5

m(7,9)=(7+9)/2=8

m(8,9)=(8+9)/2=8.5

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

m(7,7)=(7+7)/2=7

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

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

b. Frequency

F(6.5)=F(8.5)=F(6)=F(9)=1

F(7)=F(7.5)=F(8)=2

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

P(6.5)=P(8.5)=P(6)=P(9)=0.1

P(7)=P(7.5)=P(8)=0.2

$E(x)=\sum P(x)x=0.1(6.5+8.5+6+9)+0.2(7+7.5+8)=3+4.5=7.5$

$\sigma_x^2=\sum P(x)x^2-(\sum P(x)x)^2=0.1(42.25+72.25+36+81)+0.2(49+56.25+64)-56.25=23.15+33.85-56.25=0.75$

c.