Question

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 4, and 10. Consider the values of 2, 4, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 4, and 10. The nine different samples are as follows: (2, 2), (2, 4), (2, 10), (4, 2), (4, 4), (4, 10), (10, 2), (10, 4), and (10, 10). (i) Find the mean of each of the nine samples, then summarize the sampling distribution of the means in the format of a table representing the probability distribution. (ii) Compare the population mean to the mean of the sample means. (iii) Do the sample means target the value of the population mean? In general, do means make good estimators of population means? Why or why not?

Answer 1

Solution:

Given:The numbers of people in the households are 2, 4, and 10.

Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 4, and 10.

The nine different samples are as follows: (2, 2), (2, 4), (2, 10), (4, 2), (4, 4), (4, 10), (10, 2), (10, 4), and (10, 10).

i) Find the mean of each of the nine samples,

Samples of size 2		Sample means
2	2	2
2	4	3
2	10	6
4	2	3
4	4	4
4	10	7
10	2	6
10	4	7
10	10	10

summarize the sampling distribution of the means in the format of a table representing the probability distribution.

Sample means	Frequency	Probability
2	1	0.11111
3	2	0.22222
4	1	0.11111
6	2	0.22222
7	2	0.22222
10	1	0.11111
	N=9

To get probability we divide each frequency by 9.

(ii) Compare the population mean to the mean of the sample means.

Find mean by using probability distribution obtained above:

Sample means	Frequency	Probability	Sample Means X Probability
2	1	0.11111	0.22222
3	2	0.22222	0.66667
4	1	0.11111	0.44444
6	2	0.22222	1.33333
7	2	0.22222	1.55556
10	1	0.11111	1.11111
	9		5.33333

Mean:

Mean = Sample means X Probability

Mean = 5.33333

Now find population mean:

Population values = 2,4,10

Thus Population mean is:
Mean = ( 2+4+10)/3

Mean = 16 /3

Mean = 5.33333

Thus population mean is equal to mean of all sample means.

(iii) Do the sample means target the value of the population mean?

In general, do means make good estimators of population means?

Yes , since mean of all sample means is equal to population mean.