91Ó°ÊÓ

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 \(\frac{1}{9}\) 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. $$ \begin{array}{lccccc} \text { Digit } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & 0.301 & 0.176 & 0.125 & 0.097 & 0.079 \\ \hline \text { Digit } & 6 & 7 & 8 & 9 & \\ \hline \text { Probability } & 0.067 & 0.058 & 0.051 & 0.046 & \\ \hline \end{array} $$ 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. $$ \begin{array}{lrrrrrrrrr} \hline \text { First digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text { Frequency } & 36 & 32 & 28 & 26 & 23 & 17 & 15 & 16 & 7 \\ \hline \end{array} $$ (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?

Step by step solution

01

- Choosing the Level of Significance

Since this test is meant to prove fraud, a common choice of significance level is 0.01 (1%). This stringent level minimizes the probability of incorrectly accusing someone of fraud (Type I error).

02

- State the Hypotheses

Formulate the null hypothesis (H_0) and the alternative hypothesis (H_A):H_0: The first digit distribution follows Benford's Law.H_A: The first digit distribution does not follow Benford's Law.

03

- Calculate Expected Frequencies

Using Benford's probabilities and the total sample size (200 checks), calculate the expected frequency for each digit:1: 0.301 x 200 = 60.22: 0.176 x 200 = 35.23: 0.125 x 200 = 254: 0.097 x 200 = 19.45: 0.079 x 200 = 15.86: 0.067 x 200 = 13.47: 0.058 x 200 = 11.68: 0.051 x 200 = 10.29: 0.046 x 200 = 9.2

04

- Chi-Squared Goodness-of-Fit Test

Calculate the chi-squared statistic using the formula: df = k - 1(observed - expected)^2/expected.Chi-squared sum: ((36-60.2)^2/60.2) + ((32-35.2)^2/35.2) + ((28-25)^2/25) + ((26-19.4)^2/19.4) + ((23-15.8)^2/15.8) + ((17-13.4)^2/13.4) + ((15-11.6)^2/11.6) + ((16-10.2)^2/10.2) + ((7-9.2)^2/9.2) = 26.42

05

- Compare to the Critical Value

Check the chi-squared distribution table for 8 degrees of freedom (df) at the 0.01 significance level. The critical value is 20.09.Since 26.42 > 20.09, we reject the null hypothesis.

06

- Conclusion on Fraud

Since the first digit distribution does not follow Benford's Law and the null hypothesis is rejected, there is significant evidence to suggest that fraud may have occurred.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution

A probability distribution represents how likely it is for different outcomes to occur in a random experiment. In the context of Benford's Law, the probability distribution specifies how frequently each digit (1 through 9) is expected to appear as the first significant digit in a large set of naturally occurring numbers. For example:

• The digit '1' appears as the first digit about 30.1% of the time.
• The digit '2' appears approximately 17.6% of the time.

This distribution is counterintuitive because we might expect each digit to appear with the same probability, around 1/9 or roughly 11.11%.
Benford's Law has been observed in a wide variety of datasets, including financial records, census data, and even the lengths of rivers. This makes it a valuable tool for detecting anomalies and potential fraud.

Goodness-of-Fit Test

A goodness-of-fit test helps determine if a sample matches a specified distribution. In this exercise, it's used to see if the distribution of first digits in fraudulent checks matches the distribution expected by Benford's Law.
Key steps to perform a goodness-of-fit test:

• **Null Hypothesis (H₀):** The observed data follows Benford's Law.
• **Alternative Hypothesis (H_A):** The observed data does not follow Benford's Law.
• Calculate expected frequencies by multiplying the total number of observations by the probabilities given by Benford's Law.
• Use a chi-squared test to compare the observed and expected frequencies.

This test is significant in contexts like auditing and forensic accounting, where identifying deviations from expected distributions can flag potential fraud or errors.

Chi-Squared Statistic

The chi-squared statistic is a measure used in statistics to compare observed data with data we would expect to obtain according to a specific hypothesis. In this exercise, it is used to test if the observed frequencies of first digits in the checks match those expected under Benford's Law.
Steps to compute the chi-squared statistic:

• Calculate the difference between observed and expected frequencies for each digit.
• Square these differences to avoid negative values.
• Divide each squared difference by the expected frequency.
• Sum all these values to get the chi-squared statistic.

Mathematically, this can be expressed as: \[ \text{Chi-squared} = \sum \frac{(O_i - E_i)^2}{E_i} \] where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency for each digit.
Once computed, the chi-squared statistic is compared to a critical value from the chi-squared distribution table to decide whether to reject the null hypothesis. If the calculated statistic is higher than the critical value, the null hypothesis is rejected, indicating that the observed distribution does not follow Benford's Law, as seen in our exercise.