Benford’s Law
According to Benford’s law, a variety of d...

Question

Benford’s Law

According to Benford’s law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21–24, test for goodness-of-fit with the distribution described by Benford’s law.

Detecting Fraud When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 15, 0, 76, 479, 183, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford’s law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodness-of-fit with Benford’s law. Does it appear that the checks are the result of fraud?

Accepted Answer

All right. Hello everyone. So this question says, a forensic accountant is examining 650 sales invoices from a company suspected of financial irregularities. She records the leading digits of each invoice amount and compares the observed distribution to Benford's law, which describes the expected frequency of leading digits in naturally occurring data sets. The observed frequencies and Bedford's expected percentages are shown below. At the 0.01 significance level, test whether the observed digit distribution follows Benford's law. All right, so the first step, if you recall, is to state our hypotheses. First, we have the no hypothesis. Which states that the distribution of leading digits does indeed follow Benford's law. So I'm just writing yes here to describe that. Because of that, though, the alternative hypothesis. It would simply be that the distribution of leading digits does not follow Benford's law. And so with the data provided, let's go ahead and calculate the expected frequencies for each number for each leading digit, according to Benford's law. And so you can see here on the screen that I went ahead and pre-wrote. The expected frequencies here according to Benford's law for each digit, but as a decimal. And the reason why I did this is because here, to find the expected frequency, we're going to multiply each decimal, each proportion, by 650, which is the sample size. So each of these values gets multiplied by 650. And so what this tells you. Is how many Or rather the frequency of each digit. According to Benford Law. And so when you calculate all the values. They equal as follows. For number one, that's 195.65. For 2, that's 114. 40 For 3, that's 81.25. For 4 that's 63.05. For 5, that would be 51.35. 6 would be 43.55. For 7, it would be 37.7. For 8, that would be 33.15. And for 9, that would be 29.9. So here, You're going to have to use these values to find the chi squared statistic. Now, the chi square statistic is equal to the sum of. The difference between The observed value and the expected value squared. Divided by the expected frequency. And so what this means to say is that for each of. The leading digits. Given, you're going to subtract the observed frequency from the expected frequency, square the difference, and then divide by the expected frequency. And so after you repeat that calculation for all data points, you then add them together and then plug that value into. Or rather sol for Kai Square. So here, after evaluating this expression, chi square comes out to 2,908. 0.86. So now it's this value that we then compare to the critical value. For the critical value, the degrees of freedom would be equal to the 9 digits given, subtracted by 1, which equals 8. And so at 8 degrees of freedom. With alpha equal to 0.01, the critical value for chi square. Is approximately 20.09. So now, because our calculated value for chi square. Is greater than the critical value, we reject the null hypothesis. So now this leads us to our final answer, which reads as follows. There is sufficient evidence that the distribution of leading digits does not follow Benford's law. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Benford's Law

Goodness-of-Fit Test

Significance Level

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