Sample Mean

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

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

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

4. Sample Mean 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 Ekstrom Math 115b

5. Sample Mean 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 Ekstrom Math 115b

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

7. 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. Ekstrom Math 115b

8. Sample Mean 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)} Ekstrom Math 115b

9. Sample Mean Sample Mean • Ex. Find the variance and standard deviation of the finite random variable B from the previous example Ekstrom Math 115b

10. Sample Mean Sample Mean • So, the mean is 3.2, variance is 0.64, and standard deviation is 0.8 Ekstrom Math 115b

11. Sample Mean 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 Ekstrom Math 115b

12. Sample Mean 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. Ekstrom Math 115b

13. Sample Mean Sample Mean • So, the mean is 0 and the standard deviation is 1. (This MUST be true for all standardized variables) Ekstrom Math 115b

14. Sample Mean 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 Ekstrom Math 115b

15. Sample Mean 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)} Ekstrom Math 115b

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

17. Sample Mean 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 Ekstrom Math 115b

18. Sample Mean Sample Mean: Continuous R.V. So, the mean is 0 and the standard deviation is 1. (This MUST be true for all standardized variables) Ekstrom Math 115b