Question

Appendix B.4 is a table of random numbers that are uniformly distributed. Hence, each digit from 0 through (including) 9 has the same likelihood of occurrence. (Round your answers to 2 decimal places.) picture Click here for the Excel Data File Compute the population mean and standard deviation of the uniform distribution of random numbers. Assume that 10 random samples of five values are selected from a table of random numbers. The results follow. Each row represents a random sample. 2 2 8 9 0 5 7 8 7 8 5 1 5 3 9 7 3 3 8 1 9 1 7 3 0 2 2 1 7 4 1 2 3 5 0 5 3 9 6 5 8 6 2 6 0 7 4 6 3 3 Compute the mean of each sample. Compute the mean and standard deviation of the sample means. Compare the values to the population mean and standard deviation.

Sample 1	2	2	8	9	0
Sample 2	5	7	8	7	8
Sample 3	5	1	5	3	9
Sample 4	7	3	3	8	1
Sample 5	9	1	7	3	0
Sample 6	2	2	1	7	4
Sample 7	1	2	3	5	0
Sample 8	5	3	9	6	5
Sample 9	8	6	2	6	0
Sample 10	7	4	6	3	3

Answer 1

Solution:

For the given data, by using excel, the population mean and population standard deviation is given as below:

Population mean = 4.42

Population standard deviation = 2.814612

Now, we have to find the mean of the sample means and standard deviation of the sample means.

Sample means for each sample and calculation table for standard deviation is given as below:

Sample No.	Sample means	(X - mean)^2
1	4.2	0.0484
2	7	6.6564
3	4.6	0.0324
4	4.4	0.0004
5	4	0.1764
6	3.2	1.4884
7	2.2	4.9284
8	5.6	1.3924
9	4.4	0.0004
10	4.6	0.0324
Total	44.2	14.756

Mean of sample means = ∑X/n = 44.2/10 = 4.42

Standard deviation of sample means = Sqrt[∑(X - mean)^2/(n – 1)]

Standard deviation of sample means = Sqrt[14.756/(10 – 1)]

Standard deviation of sample means = Sqrt(1.639555556)

Standard deviation of sample means = 1.280451309

The comparison is given as below:

From above calculations, we get the following conclusions:

The population mean is equal to the mean of the sample means, but population standard deviation is not equal to the standard deviation of the sample means.

Note, if we increase the sample size, then we get population standard deviation equal to the standard deviation of the sample means. For the given scenario, the sample size is very less and that’s why population standard deviation is not equal to the standard deviation of the sample means.