You may think that in lists of real-life numbers, such as a list of the areas of the countries of the

world, that the first digit "1" has a 1/9th chance of occurence, the same as any other digit 2-9. But

in 1938, physicist Frank Benford discovered that the following distribution is an accurate model.

Table 1: Benford's Law.

Digit Probability

1 0.301

2 0.176

3 0.125

4 0.097

5 0.079

6 0.067

7 0.058

8 0.051

9 0.046

Benford's Law, and the Goodness-of-fit Test, have been used as

evidence against check fraud, accounting fraud, etc. In the 2009

Iranian elections, Benford's Law was used as evidence of election

fraud (the counts of the votes for various states didn't follow the

distribution).

In the case State of Arizona vs. Wayne James Nelson, an employee wrote 200 fraudulent checks. The frequency of first digits is summarized below.

First digit 1 2 3 4 5 6 7 8 9

Frequency 36 32 28 26 23 17 15 16 7

1. Write the null and alternative hypothesis.

2. Select an -level of significance (Wayne Nelson may go to prison

for life if the null is rejected).

3. Fill in the expected frequencies according to Benford's Law.

First digit 1 2 3 4 5 6 7 8 9

Frequency

4. Calculate the test statistic (Goodness off fit test) 2 = (OiEi)²

Ei

.

5. We use 8 degrees of freedom because there are 9 categories of probability (9 digits). Either, (1) for your , calculate the critical value, or (2) estimate the P-value of your 2 with an inequality.

6.Will youreject the null hypothesis or not reject the null hypothesis?

7.Is Wayne James Nelson guilty orWayne James Nelson is innocent?