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Q.11

Expert-verifiedFound in: Page 693

Book edition
4th

Author(s)
David Moore,Daren Starnes,Dan Yates

Pages
809 pages

ISBN
9781319113339

**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?

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.

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

The table below displays the sample data.

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 :

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

=$21.5633$

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.

It is given in the question

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.

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