Question

# Three randomly selected households are surveyed. The numbers of people in the households are 11​, 22​,...

Three randomly selected households are surveyed. The numbers of people in the households are

11​,

22​,

and

1212.

Assume that samples of size

nequals=2

are randomly selected with replacement from the population of

11​,

22​,

and

1212.

Listed below are the nine different samples. Complete parts​ (a) through​ (c).

11​,11

11​,22

11​,1212

22​,11

22​,22

22​,1212

1212​,11

1212​,22

1212​,1212

a. Find the variance of each of the nine​ samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution of the distinct variance values.

s2

Probability

0

0.3

1

0

nothing

1

25

1

0.5

nothing

25

50

100

nothing

60.5

30.3

121

nothing

let first sample is x1 and 2nd sample is x2

 x1 x2 s2 1 1 0 2 1 0.5 12 1 60.5 1 2 0.5 2 2 0 12 2 50 1 12 60.5 2 12 50 12 12 0

for 1st sample (1,1) , mean =(1+1)/2=1 ; therefore variance =(x-xbar)2/(n-1)

=((1-1)^2+(1-1)^2)/(2-1)=0

for 2nd sample (2,1) mean =(2+1)/2=1.5 ; therefore variance =(x-xbar)2/(n-1)

=((2-1.5)^2+(1-1.5)^2)/(2-1)=0.5

all other sample variance can be calculated in above manner:

therefore"

 s2 probability 0 1/3 0.5 2/9 50 2/9 60.5 2/9