Question

Three randomly selected households are surveyed. The numbers of people in the households are

1​,

3​,

and

8.

Assume that samples of size

nequals=2

are randomly selected with replacement from the population of

1​,

3​,

and

8.

Listed below are the nine different samples. Complete parts​ (a) through​ (c).

1,1​,   

1​,3   

1​,8   

3​,1   

3,3   

3​,8   

8​,1   

8​,3   

8​,8

a. Find the variance of each of the nine​ samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution of the distinct variance values.(Type an integer or a fraction. Use ascending order of the sample​ variances.) s2 and probability

​

b. Compare the population variance to the mean of the sample variances. Choose the correct answer below.

A.

The population variance is equal to the mean of the sample variances.

B.

The population variance is equal to the square root of the mean of the sample variances.

C.

The population variance is equal to the square of the mean of the sample variances.

c. Do the sample variances target the value of the population​ variance? In​ general, do sample variances make good estimators of population​ variances? Why or why​ not?

A.

The sample variances target the population​ variances, therefore, sample variances make good estimators of population variances.

B.

The sample variances do not target the population​ variance, therefore, sample variances make good estimators of population variances.

C.

The sample variances do not target the population​ variance, therefore, sample variances do not make good estimators of population variances.

D.

The sample variances target the population​ variance, therefore, sample variances do not make good estimators of population variances.

Answer 1

a)

Since each sample is equally likely so probability of getting any sample is 1/9.

The variance of each sample is

Here X1 and X2 shows the observations of the samples. Following table shows the variances of samples and corresponding probabilities:

Samples		Mean	Variances,s^2	P(s^2)
1	1	1	0	0.11111111
1	3	2	2	0.11111111
1	8	4.5	24.5	0.11111111
3	1	2	2	0.11111111
3	3	3	0	0.11111111
3	8	5.5	12.5	0.11111111
8	1	4.5	24.5	0.11111111
8	3	5.5	12.5	0.11111111
8	8	8	0	0.11111111

Following table shows the probability distribution of sample variances:

Variances,s^2	P(s^2)
0	0.3333
2	0.2222
12.5	0.2222
24.5	0.2222

b)

Following table shows the calculations for mean of population variances:

Variances,s^2	P(s^2)	s^2*P(s^2)
0	0.3333	0
2	0.2222	0.4444
12.5	0.2222	2.7775
24.5	0.2222	5.4439
Total		8.6658

So mean of sample variances is

The population variance is

Correct option:

A. The population variance is equal to the mean of the sample variances.

c)

Correct option:

A. The sample variances target the population​ variances, therefore, sample variances make good estimators of population variances.