Question

Consider the population described by the probability distribution shown in the table. The random variable x is observed twice. If these observations are independent, all the different samples of size 2 and their probabilities are shown in the accompanying table. Complete parts a through e below.

x	1	2	3	4	5
p(x)	0.4	0.1	0.2	0.2	0.1

x (bar over x)	1.0	1.5	2.0	2.5	3	3.5	4	4.5	5
p(x) (bar over x)	0.16	0.08	0.17	0.2	0.16	0.1	0.08	0.04	0.01

Sample	Mean	Probability	Sample	Mean	Probability
1, 1	1.0	0.16	3, 4	3.5	0.04
1, 2	1.5	0.04	3, 5	4.0	0.02
1, 3	2.0	0.08	4, 1	2.5	0.08
1, 4	2.5	0.08	4, 2	3.0	0.02
1, 5	3.0	0.04	4, 3	3.5	0.04
2, 1	1.5	0.04	4, 4	4.0	0.04
2, 2	2.0	0.01	4, 5	4.5	0.02
2, 3	2.5	0.02	5, 1	3.0	0.04
2, 4	3.0	0.02	5, 2	3.5	0.01
2, 5	3.5	0.01	5, 3	4.0	0.02
3, 1	2.0	0.08	5, 4	4.5	0.02
3, 2	2.5	0.02	5, 5	5.0	0.01
3, 3	3.0	0.04

a.) Find the sampling distribution of s^2. Type the answers in ascending order for s^2

s^2
P(s^2)

(type as integers or decimals)

b. Find the population variance:

c.) Find the sampling distribution of the sample standard deviation.

s
P(s)

Answer 1

Variance for each sample will be calculated using excel formula "=var()". Following table shows the sample variances and corresponding probabiltes:

Samples		Probabilites	Sample variances, s^2	P(s^2)
1	1	0.16	0	0.16
1	2	0.04	0.5	0.04
1	3	0.08	2	0.08
1	4	0.08	4.5	0.08
1	5	0.04	8	0.04
2	1	0.04	0.5	0.04
2	2	0.01	0	0.01
2	3	0.02	0.5	0.02
2	4	0.02	2	0.02
2	5	0.01	4.5	0.01
3	1	0.08	2	0.08
3	2	0.02	0.5	0.02
3	3	0.04	0	0.04
3	4	0.04	0.5	0.04
3	5	0.02	2	0.02
4	1	0.08	4.5	0.08
4	2	0.02	2	0.02
4	3	0.04	0.5	0.04
4	4	0.04	0	0.04
4	5	0.02	0.5	0.02
5	1	0.04	8	0.04
5	2	0.01	4.5	0.01
5	3	0.02	2	0.02
5	4	0.02	0.5	0.02
5	5	0.01	0	0.01

Following table shows the sampling distribution of sample variances:

s^2	P(s^2)
0	0.26
0.5	0.24
2	0.24
4.5	0.18
8	0.08

(b)

Following table shows the calculations for population variances:

x	p(x)	xp(x)	x^2*p(x)
1	0.4	0.4	0.4
2	0.1	0.2	0.4
3	0.2	0.6	1.8
4	0.2	0.8	3.2
5	0.1	0.5	2.5
Total		2.5	8.3

The population variance is:

c)

For sampligng distribution of SD we need to square root of variances. Following table shows the same:

s	P(s)
0	0.26
0.7071	0.24
1.4142	0.24
2.1213	0.18
2.8284	0.08