Question 5.75

5.75 Benford’s law, continued

Benford’s law suggests that the proportion of legitimate invoices with a first digit of 1, 2, or 3 is much greater than if the digits were distributed as equally likely outcomes. As a fraud investigator, you would be suspicious of some potential wrongdoing if the count of invoices with a first digit of 1, 2, or 3 is too low. You decide if the count is in the lower 5% of counts expected by Benford’s law, then you will call for a detailed investigation for fraud.

  1. Assuming the expected proportion of invoices with a first digit of 1, 2, or 3 given by Benford’s law, use software on the binomial distribution to find the smallest number out of invoices such that is no larger than 0.05.
  2. Based on the cutoff count value found in part (a), how small does the sample proportion of invoices with first digit of 1, 2, or 3 need to be for you to be suspicious of fraud?
  3. What is the standard deviation of the sample proportion , assuming again Benford’s law on the first digits of 1, 2, and 3?
  4. Using the Normal approximation, find the value such that . Compare with the cutoff proportion found in part (b).

5.75

(a) . (b) 0.575 or less. (c) 0.0155. (d) 0.5765. The values are very close; the Normal approximation works well here.