Frequencies of leading digits from IRS tax files are...

Frequencies of leading digits from IRS tax files are $152,89,63,48,39,40$, 28,25, and 27 (corresponding to the leading digits of $1,2,3,4,5,6,7,8$, and 9, respectively, based on data from Mark Nigrini, who provides software for Benford data analysis). Using a $0.05$ significance level, test for goodness-of-fit with Benford's law. Does it appear that the tax entries are legitimate?

Key Concepts

Benford's Law

Benford's Law is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. According to the law, smaller digits occur as the leading digit more frequently than larger ones. This concept is used in forensic accounting and fraud detection to determine whether data, such as tax returns, follow the expected distribution or if anomalies may suggest manipulation.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical method used to compare observed frequencies with expected frequencies based on a specified theoretical distribution. In the context of Benford's Law, it assesses whether the observed digit frequencies diverge significantly from the distribution predicted by Benford's Law, providing a quantitative basis for testing legitimacy.

Null Hypothesis

In a goodness-of-fit test, the null hypothesis is the assumption that the observed data follow the expected distribution—in this case, that the tax file leading digits follow Benford's Law. Rejecting this hypothesis indicates a statistically significant deviation from the expected pattern, which might suggest irregularities or fraud.

Significance Level

The significance level, often denoted as alpha (?), is the threshold for determining whether the results of a statistical test are statistically significant. In this scenario, a 0.05 significance level means that if the probability of observing the data (or more extreme alike) under the null hypothesis is less than 5%, the null hypothesis is rejected, suggesting the data may not follow Benford's Law.

Degrees of Freedom

Degrees of freedom in a chi-square test refer to the number of independent values that can vary when estimating statistical parameters. For a goodness-of-fit test, it is typically calculated as the number of categories minus one. This concept is crucial for determining the appropriate chi-square distribution to assess the test statistic and to interpret the p-value correctly.

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