Examples of Sampling Distribution
Example:
Draw all possible samples of size 2 without replacement from a population consisting of 3, 6, 9, 12, 15. Form the sampling 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 are drawn without replacement from the population consisting of four numbers 4, 5, 5, 7. Find the sample mean for each sample and make a sampling distribution of . Calculate the mean and standard deviation of this sampling distribution. Compare your calculations with the 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:

4

5

5

7



16

25

25

49


and
Hence and