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Use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield,...

Use the population of ages {56, 49, 58, 46} of the four U.S. presidents (Lincoln, Garfield, McKinley, Kennedy) when they were assassinated in office. Assume that random samples of size n = 2 are selected with replacement.

1. List the 16 different samples. For example, the samples for age 56 would be

56, 56

56, 49

56, 58

56, 46.

2. After listing all 16 samples, find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same.

3. Compare the mean of the population {56, 49, 58, 46} to the mean of the sampling distribution of the sample mean.

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

Solutions

Expert Solution

1. The list of 16 differnt sample is

Sample
56 56
56 49
56 58
56 46
49 56
49 49
49 58
49 46
58 56
58 49
58 58
58 46
46 56
46 49
46 58
46 46

2.   The mean of each sample, and the sampling distribution of the sample mean is

Sr.No Sample Sample mean Probability
1 56 56 56 1/16
2 56 49 52.5 1/16
3 56 58 57 1/16
4 56 46 51 1/16
5 49 56 52.5 1/16
6 49 49 49 1/16
7 49 58 53.5 1/16
8 49 46 47.5 1/16
9 58 56 57 1/16
10 58 49 53.5 1/16
11 58 58 58 1/16
12 58 46 52 1/16
13 46 56 51 1/16
14 46 49 47.5 1/16
15 46 58 52 1/16
16 46 46 46 1/16

the sampling distribution of the sample mean is

Sr.No Sample Sample mean Probability
1 56 56 56 1/16
2 56 49 52.5 2/16
3 56 58 57 2/16
4 56 46 51 2/16
5 49 49 49 1/16
6 49 58 53.5 2/16
7 49 46 47.5 2/16
8 58 58 58 1/16
9 58 46 52 2/16
10 46 46 46 1/16

3) Using above Table mean of sample means i.e. mean of the sampling distribution of the sample mean is 52.25.

The mean of the population {56, 49, 58, 46} is 52.25.

Thus ,mean of the sampling distribution of the sample mean is equal to the population mean.

4) Since Sample means=52.5=Population mean,we conclude that  the sample means do target the value of the population mean.

4.) In general, the sample means do make good estimators of population means since they target the ppopulation mean.

milcah answered 2 months ago
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