Analyzed The Leading Digits Of Amounts From 200 Checks
Analyzed the Leading Digits Of The Amounts From 200 Checks and Other
An investigator analyzed the leading digits of the amounts from 200 checks issued by three suspect companies. The frequencies were found to be 68, 40, 18, 19, 8, 20, 6, 9, 12 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 expected frequencies under Benford's law, it suggests possible fraud.
Use a 0.05 significance level to perform a goodness-of-fit chi-square test comparing the observed frequencies to the expected frequencies generated by Benford's law. You need to:
- Calculate the chi-square test statistic.
- Calculate the chi-square critical value for the given degrees of freedom and significance level.
- Determine whether there is sufficient evidence to conclude that the checks are the result of fraud based on your calculations.
Paper For Above instruction
Introduction
Benford's law, also known as the first-digit law, predicts the distribution of the leading digits in many naturally occurring datasets. According to Benford's law, the probability of a leading digit d (1 through 9) is given by P(d) = log10(1 + 1/d). This law is often employed in forensic accounting and fraud detection to identify anomalies in financial data, such as check amounts. When the observed distribution significantly deviates from Benford's distribution, it raises suspicion of manipulation or fraudulent activity.
Methodology and Calculation of χ² Test Statistic
In this scenario, we analyze the leading digits from 200 checks issued by three suspect companies. The observed frequencies are recorded for each digit as follows: 68 (for 1), 40 (2), 18 (3), 19 (4), 8 (5), 20 (6), 6 (7), 9 (8), 12 (9). To evaluate whether these frequencies conform to Benford’s law, we first calculate the expected frequencies for each digit based on Benford's law probabilities and the total number of observations (N=200).
Expected probabilities based on Benford's law are:
- Digit 1: P(1) = log10(1 + 1/1) ≈ 0.301
- Digit 2: P(2) ≈ 0.176
- Digit 3: P(3) ≈ 0.125
- Digit 4: P(4) ≈ 0.097
- Digit 5: P(5) ≈ 0.079
- Digit 6: P(6) ≈ 0.067
- Digit 7: P(7) ≈ 0.058
- Digit 8: P(8) ≈ 0.051
- Digit 9: P(9) ≈ 0.046
Multiplying these probabilities by 200 gives the expected frequencies.
Expected frequencies:
- Digit 1: 200 × 0.301 ≈ 60.2
- Digit 2: 200 × 0.176 ≈ 35.2
- Digit 3: 200 × 0.125 ≈ 25.0
- Digit 4: 200 × 0.097 ≈ 19.4
- Digit 5: 200 × 0.079 ≈ 15.8
- Digit 6: 200 × 0.067 ≈ 13.4
- Digit 7: 200 × 0.058 ≈ 11.6
- Digit 8: 200 × 0.051 ≈ 10.2
- Digit 9: 200 × 0.046 ≈ 9.2
Next, the chi-square test statistic is calculated as:
χ² = Σ [(Observed - Expected)² / Expected]
Now, I will compute this step-by-step:
Calculations for Chi-Square Statistic
| Digit | Observed (O) | Expected (E) | (O - E) | (O - E)² / E |
|---|---|---|---|---|
| 1 | 68 | 60.2 | 7.8 | (7.8)² / 60.2 ≈ 1.01 |
| 2 | 40 | 35.2 | 4.8 | (4.8)² / 35.2 ≈ 0.65 |
| 3 | 18 | 25.0 | -7.0 | (-7.0)² / 25.0 ≈ 1.96 |
| 4 | 19 | 19.4 | -0.4 | (-0.4)² / 19.4 ≈ 0.008 |
| 5 | 8 | 15.8 | -7.8 | (-7.8)² / 15.8 ≈ 3.84 |
| 6 | 20 | 13.4 | 6.6 | 6.6² / 13.4 ≈ 3.24 |
| 7 | 6 | 11.6 | -5.6 | (-5.6)² / 11.6 ≈ 2.70 |
| 8 | 9 | 10.2 | -1.2 | (-1.2)² / 10.2 ≈ 0.14 |
| 9 | 12 | 9.2 | 2.8 | 2.8² / 9.2 ≈ 0.85 |
Sum of all values: 1.01 + 0.65 + 1.96 + 0.008 + 3.84 + 3.24 + 2.70 + 0.14 + 0.85 ≈ 14.59
Therefore, the chi-square statistic is approximately 14.59.
Degree of Freedom and Critical Value
The degrees of freedom for this test are (number of categories - 1) = 9 - 1 = 8.
Using chi-square tables or software, the critical value of chi-square at α = 0.05 and df = 8 is approximately 15.507.
Conclusion
Since the computed chi-square statistic (≈14.59) is less than the critical value (15.507), we fail to reject the null hypothesis at the 0.05 significance level. This indicates that there is not sufficient evidence to conclude the observed leading digit distribution significantly deviates from Benford's law.
Implication for Fraud Detection
Based on this analysis, the check amounts' leading digits are consistent with Benford's law, suggesting no strong evidence of fraudulent manipulation in the data. However, it is essential to recognize that statistical tests are only part of a comprehensive fraud investigation, and other evidence should be considered before making definitive conclusions.
References
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