Question

selected households are surveyed. The numbers of people in the households are 3, 4, and 11. Assume that samples of size n -2 are randomly Three randomly selected with replacement from th different samples. Complete parts (a) through c) 3,3 3,4 3,11 4,3 4,4 4,11 11,3 11,4 11,11 b.Compare the population variance to the mean of the sample variances. Choose the correct answer below O A. The po O B. The population variance is equal to the mean of the sample variances O C. The population variance is equal to the square root 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? 8 variance is equal to the square of the mean of the sample variances y or wh not? O A. The sample variances do not target the population variance, therefore, sample variances do not make good estimators of population variances. O B. The sample variances do not target the population variance, therefore, sample variances make good estimators of population variances O C. The sample variances target the population variance, therefore, sample variances do not make good estimators of population variances O D. The sample variances target the population variances, therefore, sample variances 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

$Var=\frac{(x1-mean)^{2}+(x2-mean)^{2}}{2-1}=(x1-mean)^{2}+(x2-mean)^{2}$

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

Samples		Variances,s^2	P(s^2)
3	3	0	0.11111111
3	4	0.5	0.11111111
3	11	32	0.11111111
4	3	0.5	0.11111111
4	4	0	0.11111111
4	11	24.5	0.11111111
11	3	32	0.11111111
11	4	24.5	0.11111111
11	11	0	0.11111111

Following table shows the probability distribution of sample variances:

Variances,s^2	P(s^2)
0	0.3333
0.5	0.2222
24.5	0.2222
32	0.2222

b)

Following table shows the calculations for mean of population variances:

Variances,s^2	P(s^2)	s^2P(s^2)
0	0.3333	0
0.5	0.2222	0.1111
24.5	0.2222	5.4439
32	0.2222	7.1104
Total		12.6654

So mean of sample variances is

$E(s^2)=\sum s^2P(s^2)=12.6654$

The population variance is $\sigma^{2}=12.6667$

Correct option is B.

c)

Correct option is D.

----------------------------------