Question

Consider the set of five numbers {0, 2, 4, 6, 8}.

1) Make a list of all possible samples of size 2 that can be drawn from this set of integers. (Sample without replacements, that is, once a number is selected, you don’t put it back in the sample set.)

2) Make a list of all possible sample means for samples of size 2 selected from this set.

3) List the distribution of the sample means and construct a histogram of this distribution.

Answer 1

Answer 1 :

List of all possible samples of size 2 -

• (0 , 2)
• (0 , 4)
• (0 , 6)
• (0 , 8)
• (2 , 4)
• (2 , 6)
• (2 , 8)
• (4 , 6)
• (4 , 8)
• (6 , 8)

The samples are drawn using WITHOUT REPLACEMENT procedure . There are a total of 10(= 5C2) different samples of size 2

Answer 2 :

The following table shows the Sample and its mean -

Sample	Mean
(0 , 2)	1
(0 , 4)	2
(0 , 6)	3
(0 , 8)	4
(2 , 4)	3
(2 , 6)	4
(2 , 8)	5
(4 , 6)	5
(4 , 8)	6
(6 , 8)	7

List of all possible sample means {1 , 2 , 3 , 4 , 5 , 6 , 7}

Answer 3 :

The following table shows the calculations -

Mean value (xi)	Probability , P(xi)	xi P(xi)	xi2 P(xi)
1	1/10 = 0.1	0.1	0.1
2	1/10 = 0.1	0.2	0.4
3	2/10 = 0.2	0.6	1.8
4	2/10 = 0.2	0.8	3.2
5	2/10 = 0.2	1.0	5.0
6	1/10 = 0.1	0.6	3.6
7	1/10 = 0.1	0.7	4.9
Total	1.0	4.0	19

Mean , Expectation of x , E(x) = xi P(xi) = 4.0

E(x2) = xi2 P(xi) = 19

Variance of x , Var. (x) = E(x2) - (E(x))2 = 19 - 16 = 3

Standard deviation , S.D. (x) = (Var. (x))1/2 = 1.732

So , the distribution of sample means have mean = 4 and standard deviation = 1.732

The following shows the Histogram -

(where the Horizontal Axis represents the Mean values and the Vertical Axis represents the Probabilities )