Q.11 - The Practice of Statistics for AP

Question

Benford’s law Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford’s law.^$3$ Call the first digit of a randomly chosen record X for short. Benford’s law gives this probability model for X (note that a first digit can’t be 0):

A forensic accountant who is familiar with Benford’s law inspects a random sample of invoices from a company that is accused of committing fraud. The table below displays the sample data.

(a)  Are these data inconsistent with Benford’s law? Carry out an appropriate test at the $\alpha =0.05$ level to support your answer. If you find a significant result, perform follow-up analysis. 

(b)  Describe a Type I error and a Type II error in this setting, and give a possible consequence of each. Which do you think is more serious?

Step-by-Step Solution

Answer

From the given information, 

a) Using the chi-square table, the p-value for $8$ degrees of freedom is $0.005792.$

The p-value is below the level of significance. The null hypothesis is proved to be wrong. As a result, there is enough evidence to reject his assertion of Benford.

b)When the null hypothesis is rejected, even if it is valid, a type l error occurs. If it is determined that Benford's law is not applicable to the sample, a type I error will occur.

Part(a) Step 1: Given Information

It is given in the question that, Benford’s law gives this probability model for X  

The table below displays the sample data.

Part (a) Step 2: Explanation

The hypotheses are:

$H0: p1=0.301, p2=0.176, p3=0.125... p9=0.046$

H_$\alpha$: At least one of the pi is different 

The calculation for the test statistic is done as:

The test statistic is :

${\chi }^2=\sum (O-E{)}^2/E$

      =$21.5633$

Part (a) Step 3: Explanation

The degree of freedom is calculated as:

$\text{Degree of freedom}=\text{Number of categories}-1$

                                   $=9-1$

                                   $=8$

Using the chi-square table, the p-value for $8$ degrees of freedom is $0.005792.$

The p-value is below the level of significance. The null hypothesis is proved to be wrong. As a result, there is enough evidence to reject his assertion of Benford.

Part(b) Step 4: Given Information

It is given in the question 

Part(b) Step 5: Explanation

When the null hypothesis is rejected, even if it is valid, a type l error occurs. If it is determined that Benford's law is not applicable to the sample, a type I error will occur.

Type ll error, on the other hand, would occur if it was determined that Benford's law is applicable to the sample when it is not.

The Type I error is the worse in this case since it accuses an innocent corporation of fraud.