1. Ekstrom Math 115b Mathematics for Business Decisions, part II Sample Mean Math 115b

2. Ekstrom Math 115b Sample Mean  Samples give information about random variables  Note that and  Also note that and Sample Mean

3. Ekstrom Math 115b Sample Mean  Ex. If X is a finite R.V. that only assumes the values 0 and 1 with and, find and.  Soln: Sample Mean

4. Ekstrom Math 115b Sample Mean  Soln:  The Excel file Sample Means.xls shows sets of 1000 samples taken, verifying that samples closely predict the behavior of the random variable

5. Ekstrom Math 115b Sample Mean  The shape of the approximate p.m.f. graphs change slightly for different sample sizes  The differences can be removed by plotting the standardization  The graph of the standardized values appear to be approximately as follows

6. Ekstrom Math 115b Sample Mean Sample Mean: Finite R.V.

7. Ekstrom Math 115b Sample Mean Sample Mean: Finite R.V.  Ex. Let X be a finite R.V. that only assumes the values 0 and 1 with and. Consider the case when samples of size 4 are taken. Use BINOMDIST to find the values of the p.m.f. for B where B is the finite random variable that counts the number of 1’s.

8. Ekstrom Math 115b Sample Mean  Soln: Note that the possible outcomes are: {(0000), (0001), (0010), (0011), (0100), (0101), (0110), (0111), (1000), (1001), (1010), (1011), (1100), (1101), (1110), (1111)} b 00.0016 10.0256 20.1536 30.4096 4

9. Ekstrom Math 115b Sample Mean  Ex. Find the variance and standard deviation of the finite random variable B from the previous example Sample Mean b 00.0016-3.210.240.016384 10.0256-2.24.840.123904 20.1536-1.21.440.221184 30.4096-0.20.040.016384 40.40960.80.640.262144

10. Ekstrom Math 115b Sample Mean  So, the mean is 3.2, variance is 0.64, and standard deviation is 0.8 Sample Mean b 00.0016-3.210.240.016384 10.0256-2.24.840.123904 20.1536-1.21.440.221184 30.4096-0.20.040.016384 40.40960.80.640.262144

11. Ekstrom Math 115b Sample Mean  Ex. Let S be the standardization of the finite random variable B from the previous example. Find the mean and standard deviation of S.  Note that where,  This means that the b-values must be adjusted Sample Mean

12. Ekstrom Math 115b Sample Mean  Ex. Let S be the standardization of the finite random variable B from the previous example. Find the mean and standard deviation of S. Sample Mean s -40.0016-4160.0256 -2.750.0256-2.757.56250.1936 -1.50.1536-1.52.250.3456 -0.250.4096-0.250.06250.0256 10.409611

13. Ekstrom Math 115b Sample Mean  So, the mean is 0 and the standard deviation is 1. (This MUST be true for all standardized variables) Sample Mean s -40.0016-4160.0256 -2.750.0256-2.757.56250.1936 -1.50.1536-1.52.250.3456 -0.250.4096-0.250.06250.0256 10.409611

14. Ekstrom Math 115b Sample Mean  Ex. Let X be a finite R.V. that only assumes the values 0 and 1 with and. Consider the case when samples of size 4 are taken. Use BINOMDIST to find the values of the p.m.f. for Sample Mean

15. Ekstrom Math 115b Sample Mean  Soln: We are finding the AVERAGE value in each sample Note that the possible outcomes are: {(0000), (0001), (0010), (0011), (0100), (0101), (0110), (0111), (1000), (1001), (1010), (1011), (1100), (1101), (1110), (1111)} 0.000.0016 0.250.0256 0.500.1536 0.750.4096 1.000.4096

16. Ekstrom Math 115b Sample Mean  Ex. Standardize the values for from the previous example and find the mean and standard deviation of the standardized values. Sample Mean

17. Ekstrom Math 115b Sample Mean  To standardize the values, we must first find the mean and standard deviation  The mean is 0.8 and the standard deviation is 0.2 Sample Mean x 0.000.0016-0.800.640.001024 0.250.0256-0.550.30250.007744 0.500.1536-0.300.090.013824 0.750.4096-0.050.00250.001024 1.000.40960.200.040.016384

18. Ekstrom Math 115b Sample Mean So, the mean is 0 and the standard deviation is 1. (This MUST be true for all standardized variables) Sample Mean: Continuous R.V. s -40.0016-4160.0256 -2.750.0256-2.757.56250.1936 -1.50.1536-1.52.250.3456 -0.250.4096-0.250.06250.0256 10.409611