Book edition 4th

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

Pages 809 pages

ISBN 9781319113339

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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 $α = 0.05$ 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.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

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:

$H 0 : p 1 = 0.301, p 2 = 0.176, p 3 = 0.125 . . . p 9 = 0.046$

H_$α$: At least one of the pi is different

The calculation for the test statistic is done as:

The test statistic is :

$χ^{2} = \sum (O - E)^{2} / E$

= $21.5633$

Part (a) Step 3: Explanation

The degree of freedom is calculated as:

$Degree of freedom = Number of categories - 1$

$= 9 - 1$

$= 8$

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.

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The Practice of Statistics for AP

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