Question

The assets​ (in billions of​ dollars) of the four wealthiest people in a particular country are 37, 35, 31, 18. 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.

_

x probability

37

36

35

34

33

31

27.5

26.5

24.5

18

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

Mean of population is? Sample mean is?

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?

The sample means target/do not target the population mean. In​ general, sample means do/do not make good estimates of population means because the mean is an unbiased/ a biased estimator.

Answer 1

Since all 16 samples are equally likely so probability of selecting each sample is 1/16 = 0.25

Following table shows the all possible 16 samples, there mean and corresponding probabilities;

Samples		xbar	P(xbar)
37	37	37	0.0625
37	35	36	0.0625
37	31	34	0.0625
37	18	27.5	0.0625
35	37	36	0.0625
35	35	35	0.0625
35	31	33	0.0625
35	18	26.5	0.0625
31	37	34	0.0625
31	35	33	0.0625
31	31	31	0.0625
31	18	24.5	0.0625
18	37	27.5	0.0625
18	35	26.5	0.0625
18	31	24.5	0.0625
18	18	18	0.0625

Following table shows the probability distribution of sample means;

xbar	P(xbar)
18	0.0625
24.5	0.125
26.5	0.125
27.5	0.125
31	0.0625
33	0.125
34	0.125
35	0.0625
36	0.125
37	0.0625
Total	1

(b)

Following table shows the calculations:

xbar	P(xbar)	xbar*P(xbar)
18	0.0625	1.125
24.5	0.125	3.0625
26.5	0.125	3.3125
27.5	0.125	3.4375
31	0.0625	1.9375
33	0.125	4.125
34	0.125	4.25
35	0.0625	2.1875
36	0.125	4.5
37	0.0625	2.3125
Total	1	30.25

So,

The population mean is

The mean of the population 30.25 is equal to the mean of the sample means 30.25.

(C)

The sample means target population mean.

In​ general, sample means do make good estimates of population means because the mean is an unbiased estimator.