An investigator analyzed the leading digits from 787 checks issued by seven suspect companies. The frequencies were found to be 387​, 201​, 142​, 70, 71​, 28​, 7​, 16​, and 23​, 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 shown​ below, 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? Leading Digit: 1 2 3 4 5 6 7 8 9 Actual Frequency: 387 201 142 70 71 28 7 16 23 Benford's Law: 30.1% 17.6% 12.5% 9.7% 7.9% ​ 6.7% ​5.8% ​ 5.1% 4.6%

To assess whether the observed frequencies of leading digits from checks issued by the suspect companies match the expected frequencies according to Benford's Law, we can conduct a Chi-Square Goodness-of-Fit test. This test will help us determine if there are statistically significant differences between the observed and expected frequencies.

### Step-by-Step Solution

1. **State the Hypotheses:**
- **Null Hypothesis (H₀):** The leading digits of the check amounts conform to Benford's Law.
- **Alternative Hypothesis (Hₐ):** The leading digits of the check amounts do not conform to Benford's Law.

2. **Collect Data and Expected Frequencies According to Benford's Law:**
- Total number of checks (n): 787
- Benford's Law percentages for leading digits are:
1 (30.1%), 2 (17.6%), 3 (12.5%), 4 (9.7%), 5 (7.9%), 6 (6.7%), 7 (5.8%), 8 (5.1%), 9 (4.6%).
- Expected frequencies for each digit are calculated as:
\text{Expected Frequency} = n \times \text{Percentage}

3. **Calculate Expected Frequencies:**
- Expected frequency for digit 1 = 787 \times 0.301
- Expected frequency for digit 2 = 787 \times 0.176 , and so on for each digit.

4. **Compute the Chi-Square Statistic:**
- Formula: \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
- O_i is the observed frequency for each digit, and E_i is the expected frequency for each digit.

5. **Determine the Critical Value or p-value:**
- Use a chi-square distribution with 8 degrees of freedom ( df = number of categories - 1 = 9 - 1 ).

6. **Decision:**
- If the chi-square statistic exceeds the critical value from the chi-square distribution table at \alpha = 0.01 , or if the p-value is less than 0.01, reject the null hypothesis.

### Calculations

First, let's calculate the expected frequencies for each leading digit according to Benford's Law and then the chi-square statistic.

### Results of Chi-Square Goodness-of-Fit Test:

- **Chi-Square Statistic**: \chi^2 \approx 208.088
- **Critical Value at \alpha = 0.01 with 8 degrees of freedom**: \chi^2_{0.01, 8} \approx 20.090
- **p-value**: \approx 1.26 \times 10^{-40}

### Conclusion:
The chi-square statistic (208.088) is significantly higher than the critical value (20.090), and the p-value is exceedingly small (practically zero), far below the significance level of 0.01. This indicates that we reject the null hypothesis.

**Interpretation**:
The leading digits of the check amounts issued by the suspect companies do not conform to Benford's Law. This significant discrepancy suggests that the check amounts could indeed be the result of fraud. The data does not follow the natural occurrence of leading digits as expected under normal, unmanipulated conditions.