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Example:
Draw all possible samples of size 2 without replacement from a population consisting of 3, 6, 9, 12, 15. Form the sample distribution of sample means and verify the results.
(i)  (ii) 
Solution:
We have population values 3, 6, 9, 12, 15, population size  and sample size  Thus, the number of possible samples which can be drawn without replacement is
|
Sample No.
|
Sample Values
|
Sample Mean
|
Sample No.
|
Sample Values
|
Sample Mean
|
|
1
|
3, 6
|
4.5
|
6
|
6, 12
|
9.0
|
|
2
|
3, 9
|
6.0
|
7
|
6, 15
|
10.5
|
|
3
|
3, 12
|
7.5
|
8
|
9, 12
|
10.5
|
|
4
|
3, 15
|
9.0
|
9
|
9, 15
|
12.0
|
|
5
|
6, 9
|
7.5
|
10
|
12, 15
|
13.5
|
The sampling distribution of the sample mean  and its mean and standard deviation are:
|
|
|
|
|
|
|
4.5
|
1
|
1/10
|
4.5/10
|
20.25/10
|
|
6.0
|
1
|
1/10
|
6.0/10
|
36.00/10
|
|
7.5
|
2
|
2/10
|
15.0/10
|
112.50/10
|
|
9.0
|
2
|
2/10
|
18.0/10
|
162.00/10
|
|
10.5
|
2
|
2/10
|
21.0/10
|
220.50/10
|
|
12.0
|
1
|
1/10
|
12.0/10
|
144.00/10
|
|
13.5
|
1
|
1/10
|
13.5/10
|
182.25/10
|
|
Total
|
10
|
1
|
90/10
|
877.5/10
|
The mean and variance of the population are:
|
|
3
|
6
|
9
|
12
|
15
|
|
|
|
9
|
36
|
81
|
144
|
225
|
|
 and 
Verification:
(i)  (ii) 
Example:
If random samples of size three and drawn without replacement from the population consisting of four numbers 4, 5, 5, 7. Find sample mean  for each sample and make sampling distribution of  . Calculate the mean and standard deviation of this sampling distribution. Compare your calculations with population parameters.
Solution:
We have population values 4, 5, 5, 7, population size  and sample size  . Thus, the number of possible samples which can be drawn without replacement is 
|
Sample No.
|
Sample Values
|
Sample Mean 
|
|
1
|
4, 5, 5
|
14/3
|
|
2
|
4, 5, 7
|
16/3
|
|
3
|
4, 5, 7
|
16/3
|
|
4
|
5, 5, 7
|
17/3
|
The sampling distribution of the sample mean  and its mean and standard deviation are:
|
|
|
|
|
|
|
14/3
|
1
|
1/4
|
14/12
|
196/36
|
|
16/3
|
2
|
2/4
|
32/12
|
512/36
|
|
17/3
|
1
|
1/4
|
17/12
|
289/36
|
|
Total
|
4
|
1
|
63/12
|
997/36
|
The mean and standard deviation of the population are:
 and 
Hence  and 
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