business statistic

Question

a)       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 5, 7, and 9. Consider the values of 5, 7, and 9 to be a population. Assume that samples of size n = 2 are randomly selected with replacements from the population of 5, 7, and 9. The nine different samples are as follows:

(5, 5), (5, 7), (5, 9), (7, 5), (7, 7), (7, 9), (9, 5), (9, 7), and (9, 9).

a) 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

b) Compare the population mean to the mean of the sample means

c) Do the sample means target the value of the population mean?

Answer 1

a) Here is a table with the samples and the sample means:

5,5	5
5,7	6
5,9	7
7,5	6
7,7	7
7,9	8
9,5	7
9,7	8
9,9	9

Here is the probability distribution for the 9 sample means:

X	5	6	7	8	9
P(X)	1/9	2/9	3/9	2/9	1/9

b) the population mean is (5+7+9)/3 = 21/3 = 7

The mean of all the sample means is (5 + 2x6 + 3x7 + 2x8 + 9)/9 = 63/9 = 7

Clearly the population mean is equal to the mean of the sampling distribution of sample means. Because this is true, the sample mean is considered an unbiased estimator of the population mean.

c) I assume that the question is actually asking if the sample mean is an unbiased estimator of the population mean and the answer is yes - because the mean of the sampling distribution of sample means is equal to the mean of the population, the sample mean is an unbiased estimator of the population mean.