Question

1. Three randomly 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 selected with replacement from the population of

3, 4, and 11. Listed below are the nine 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

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.

s2	Probability

b. Compare the population variance to 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?

2. Assume a population of 1, 4, and 10. Assume that samples of size n=2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below.

1,1        1,4        1,10      4,1        4,4        4,10      10,1      10,4      10,10

a. Find the value of the population standard deviation.

b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution.

c. Find the mean of the sampling distribution of the sample standard deviations. The mean of the sample standard deviations is the sum of the nine sample standard deviations divided by the number of samples.

d. Do the sample standard deviations target the value of the population standard deviation? Ingeneral, do sample standard deviations make good estimators of population standard deviations? Why or why not?

3. Three randomly 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 selected with replacement from the population of 3, 4, and 11.Construct a probability distribution table that describes the sampling distribution of the proportion of odd numbers when samples of sizes n=2 are randomly selected. Does the mean of the sample proportions equal the proportion of odd numbers in the population? Do the sample proportions target the value of the populationproportion? Does the sample proportion make a good estimator of the population proportion? Listed below are the nine possible samples.

3,3  3,4 3,11 4,3 4,4 4,11 11,3 11,4  11,11

  

Construct the probability distribution table.

Sample ProportionProbability

4. The assets (in billions of dollars) of the four wealthiest people in a particular country are 43, 28, 25, 16. Assume that samples of size n=2 are randomly selected with replacement from this population of four values.

a. After identifying the 16 different possible samples and finding the mean of each sample, construct a table representing the sampling distribution of the sample mean. In the table, values of the sample mean that are the same have been combined.

b. Compare the mean of the population to the mean of the sampling distribution of the sample mean.

c. Do the sample means target the value of the population mean? In general, do sample means make good estimates of population means? Why or why not?

Answer 1

2a)

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

The variance of each sample is

Var \(=\frac{(x 1-\text { mean })^{2}+(x 2-\text { mean })^{2}}{2-1}=(x 1-\text { mean })^{2}+(x 2-\text { mean })^{2}\)

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

\(E\left(s^{2}\right)=\sum s^{2} P\left(s^{2}\right)=12.6654\)

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

The population variance is equal to mean of sample variances.

c)

The sample variances target the value of the population​ variance. In​ general, Sample variances make good estimators of population​ variances.