1. 14-1 SLIDES PREPARED SLIDES PREPARED By By Lloyd R. Jaisingh Ph.D. Lloyd R. Jaisingh Ph.D. Morehead State University Morehead State University Morehead KY Morehead KY SLIDES PREPARED SLIDES PREPARED By By Lloyd R. Jaisingh Ph.D. Lloyd R. Jaisingh Ph.D. Morehead State University Morehead State University Morehead KY Morehead KY STATISTICS for the Utterly Confused, 2 nd ed.

2. 14-2 14-2 The Chi-Square test for Goodness of Fit Hence one may intuitively conclude in this case that the sample data did not come from the population to which it is compared because of the large deviations from the sample distribution to the population distribution.Hence one may intuitively conclude in this case that the sample data did not come from the population to which it is compared because of the large deviations from the sample distribution to the population distribution.

3. 14-3 14-4 Benford’s Law Frank Benford, in the 1930s, noticed that logarithm tables (these were used by scientists long before the common use of computers and calculators) tended to be worn out on the early pages where the numbers started with the digit 1.Frank Benford, in the 1930s, noticed that logarithm tables (these were used by scientists long before the common use of computers and calculators) tended to be worn out on the early pages where the numbers started with the digit 1.

4. 14-4 14-4 Benford’s Law Based on this observation and many others, he discovered that more numbers in the real world started with the digit 1 rather than with 2, and that more started with the digit 2 rather than with 3, and so on.Based on this observation and many others, he discovered that more numbers in the real world started with the digit 1 rather than with 2, and that more started with the digit 2 rather than with 3, and so on. He later published a formula which describes the proportion of times a number will begin with the digit 1, 2, 3, etc.He later published a formula which describes the proportion of times a number will begin with the digit 1, 2, 3, etc.

5. 14-5 14-4 Benford’s Law This formula is now called Benford’s Law.This formula is now called Benford’s Law. The Table on the next slide shows the distribution of the proportions, to three decimal places, for the leading digits of numbers based on Benford’s Law.The Table on the next slide shows the distribution of the proportions, to three decimal places, for the leading digits of numbers based on Benford’s Law.

6. 14-6 14-4 Benford’s Law The next slide shows a graphical Depiction of Benford’s Law.

7. 14-7 14-4 Benford’s Law

8. 14-8 14-4 Benford’s Law Example: Students who attend college and apply for student loans must submit a FAFSA (Free Application for Federal Student Aid) form. Part of the information that is required is the annual income of the parent or parents. A sample of 3,633 forms was sampled from a college records and the proportion, to three decimal places, of the leading digits for the total annual income for the parents were recorded. This information is presented on the next slide.Example: Students who attend college and apply for student loans must submit a FAFSA (Free Application for Federal Student Aid) form. Part of the information that is required is the annual income of the parent or parents. A sample of 3,633 forms was sampled from a college records and the proportion, to three decimal places, of the leading digits for the total annual income for the parents were recorded. This information is presented on the next slide.

9. 14-9 14-4 Benford’s Law Test at the 5 percent significance level whether the distribution of the first digits for the reported total salaries for the parents follow Benford’s Law.

10. 14-10 14-4 Benford’s Law Solution: Plots of the proportions of the leading digits for both Benford’s Law and the parents’ salaries are shown below.Solution: Plots of the proportions of the leading digits for both Benford’s Law and the parents’ salaries are shown below.

11. 14-11 14-4 Benford’s Law Solution (continued): The Table on the next slide shows the computations needed to compute the  2 test statistic.Solution (continued): The Table on the next slide shows the computations needed to compute the  2 test statistic. The value of the test statistic is equal to 507.527.The value of the test statistic is equal to 507.527. To obtain the expected frequencies based on Benford’s Law one should multiply the total of 3,633 by Benford’s proportions.To obtain the expected frequencies based on Benford’s Law one should multiply the total of 3,633 by Benford’s proportions. For example, from the table, the expected frequency value of 639.408 is obtained from 3,633×0.176 = 639.408, etc.For example, from the table, the expected frequency value of 639.408 is obtained from 3,633×0.176 = 639.408, etc.

12. 14-12 14-4 Benford’s Law

13. 14-13 14-4 Benford’s Law

14. 14-14 EXAMPLE (Continued) Solution (continued): Diagram showing the rejection region. Solution (continued): Diagram showing the rejection region.