Q.13 Benford鈥檚 law and fraud A not-... [FREE SOLUTION]

91影视

The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 6: Q.13 (page 354)

Benford鈥檚 law and fraud A not-so-clever employee decided to fake his monthly expense report. He believed that the first digits of his expense amounts should be equally likely to be any of the numbers from $1$ to $9$ . In that case, the first digit $Y$ of a randomly selected expense amount would have the probability distribution shown in the histogram.
(a). Explain why the mean of the random variable Y is located at the solid red line in the figure.
(b) The first digits of randomly selected expense amounts actually follow Benford鈥檚 law (Exercise 5). What鈥檚 the expected value of the first digit? Explain how this information could be used to detect a fake expense report.
(c) What鈥檚 $P (Y > 6)$ ? According to Benford鈥檚 law, what proportion of first digits in the employee鈥檚 expense amounts should be greater than $6$ ? How could this information be used to detect a fake expense report?

Short Answer

Expert verified

a). The mean lies at the center, which is $5$ .

b). On average the first digit is $3.441$ .

c). If the proportion is closer to $0.3333$ than it is $0.155$ to , then it seems that the expense report is fake.

Step by step solution

Part (a) Step 1: Given Information

Given in the question that, the first digits of his expense amounts should be equally likely to be any of the numbers from $1$ to $9$ .

Part (a) Step 2: Explanation

The supplied distribution is symmetric (the graph to the left of $5$ is a mirror image of the graph to the right of $5$ ) and the mean is located in the middle (which is $5$ ).

Part (b) Step 1: Given Information

Given in the question is a graph

Part (b) Step 2: Explanation

Table $5$ shows the probability distribution of Benford's law.

The expected value is calculated by multiplying each possibility by its probability:

$E (X) = 鈭� x P (x) = 1 脳 0.301 + 2 脳 0.176 + 3 脳 0.125 + 4 脳 0.097 + 5 脳 0.079 + 6 脳 0.067 + 7 脳 0.058 + 8 脳 0.051 + 9 脳 0.046 = 3.441$

Part (c) Step 1: Given Information

Given that the first digits of his spending amounts might be any number from $1$ to $9$ ,

Part (c) Step 2: Explanation

Using the given uniform distribution, add the associated probabilities:

$P (Y > 6) = P (Y = 7) + P (Y = 8) + P (Y = 9) = \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{3}{9} = \frac{1}{3} 鈮� 0.3333$

Using Benford's law, add the corresponding probabilities:

$P (Y > 6) = P (Y = 7) + P (Y = 8) + P (Y = 9) = 0.058 + 0.051 + 0.046 = 0.155$

Determine the percentage of cost reports with first digits greater than $6$ . If this proportion is closer to $0.3333$ than to $0.155$ , the cost report appears to be false.