Question

A population consists of three housing units, where the number of rooms in each unit is 1, 3 and 4.
a) What is the population distribution?
b) What is the mean of this population distribution?
c) If a random sample without replacement of 2 housing units will be picked from the population, what is the sampling distribution of the sample mean number of rooms?
d) If a random sample with replacement of 2 housing units will be picked from the population, what is the sampling distribution of the sample mean number of rooms?
e) What is the mean of the sample means from all possible samples in d)?
f) Compare your answers in b), and e).

Answer 1

a.

Distribution of Population
Numbe of rooms	P(X)
1	0.333333
3	0.333333
4	0.333333

c.

Unit 1	Unit 2	mean
3	1	2	Mean	P(mean)
4	1	2.5	2	2/6=0.333333
1	3	2	2.5	2/6=0.333333
4	3	3.5	3.5	0.333333
1	4	2.5
3	4	3.5

d.

Unit 1	Unit 2	mean
1	1	1	Mean	P(mean)
3	1	2	1	1/9=0.111111
4	1	2.5	2	2/9=0.222222
1	3	2	2.5	1/9=0.222222
3	3	3	3	0.111111
4	3	3.5	3.5	0.222222
1	4	2.5	4	0.111111
3	4	3.5	Total =	1
4	4	4

e.

Mean	P(mean)	mean*P(mean)
1	0.111111	0.111111
2	0.222222	0.444444
2.5	0.222222	0.555556
3	0.111111	0.333333
3.5	0.222222	0.777778
4	0.111111	0.444444
Total =	1	2.666667

Mean of sample means = 2.67

F .answer of d and e are same