Refer to the discussion of Benford€™s law in Exercise 7.25. While this may seem like a curious oddity, researchers have developed some important applications for these proportions. One involves auditing company records to look for instances of fraud or other financial malfeasance. In many cases accounting records tend to follow Benford€™s law and significant departures can help auditors discover patterns that should be examined more closely. For example, if a company€™s policy requires cosignatures for expenses over $10,000 and auditors find an unusually high number of claims starting with the digit €˜€˜9,€ they might be suspicious and examine those claims more closely.
Two of Professor Cleary€™s students obtained data for 7273 invoices at a company. The observed counts for the leading digits of the invoice amounts are shown in Table 7.17 and stored in the Invoices variable of the Benford data file. Test if these counts are inconsistent with the probabilities given by Benford€™s law.

Table 7.17

Exercise 7.25

Frank Benford, a physicist working in the 1930s, discovered an interesting fact about some sets of numbers. While you might expect the first digits of numbers such as street addresses or checkbook entries to be randomly distributed, Benford showed that in many cases the distribution of leading digits is not random, but rather tends to have more ones, with decreasing frequencies as the digits get larger. Table 7.15 shows the proportions of leading digits for data that satisfy Benford€™s law.

Table 7.15

Professor Rick Cleary of Bentley University has given several public lectures about Benford€™s law. As part of his presentation, he rips out pages of a telephone book and asks audience members to select entries at random and record the first digit of the street address. Counts for the leading digits of 1188 such addresses are shown in Table 7.16 and stored in a variable called Address in the dataset Benford. Test if these counts are inconsistent with the probabilities given by Benford€™s law.

Table 7.16

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Leading digit Observed count 5 6 7 8 9 3 4 2 4 2225 1214 881 639 655 532 433 362 332 Leading digit Proportion 3 0.125 4 0.097 5 0.079 6. 0.067 0.176 0.051 0.301 0.058 0.046