Our number system consists of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9 because we do not write numbers such as 12 as 012. Although we may think that each first digit appears with equal frequency so that each digit has a 1/9 probability of being the first significant digit, this is not true. In 1881, Simon Newcomb discovered that first digits do not occur with equal frequency. This same result was discovered again in 1938 by physicist Frank Benford. After studying much data, he was able to assign probabilities of occurrence to the first digit in a number as shown.

The probability distribution is now known as Benford's Law and plays a major role in identifying fraudulent data on tax returns and accounting books. For example, the following distribution represents the first digits in 200 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.

(a) Because these data are meant to prove that someone is guilty of fraud, what would be an appropriate level of significance when performing a goodness-of-fit test?
(b) Using the level of significance chosen in part (a), test whether the first digits in the allegedly fraudulent checks obey Benford's Law.
(c) Based on the results of part (b), do you think that the employee is guilty of embezzlement?

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Digit Probability Digit 0.301 0.097 0.176 0.125 0.079 0.051 Probability 0.067 0.058 0.046 First digit Frequency 36 32 28 26 23 17 15 16 2.