Question

ou might think that if you looked at the first digit in randomly selected numbers that the distribution would be uniform. Actually, it is not! Simon Newcomb and later Frank Benford both discovered that the digits occur according to the following distribution: (digit, probability)

(1,0.301),(2,0.176),(3,0.125),(4,0.097),(5,0.079),(6,0.067),(7,0.058),(8,0.051),(9,0.046)(1,0.301),(2,0.176),(3,0.125),(4,0.097),(5,0.079),(6,0.067),(7,0.058),(8,0.051),(9,0.046)

The IRS currently uses Benford's Law to detect fraudulent tax data. Suppose you work for the IRS and are investigating an individual suspected of embezzling. The first digit of 166 checks to a supposed company are as follows:

Digit	Observed Frequency
1	40
2	20
3	14
4	17
5	19
6	14
7	8
8	17
9	17

a. State the appropriate null and alternative hypotheses for this test.
b. Explain why ?=0.01?=0.01 is an appropriate choice for the level of significance in this situation.
c. What is the P-Value? Report answer to 4 decimal places.
d. What is your decision?
Write a statement to the law enforcement officials that will use it to decide whether to pursue the case further or not. Structure your essay as follows: Given a brief explanation of what a Goodness of Fit test is. Explain why a Goodness of Fit test should be applied in this situation. State the hypotheses for this situation Interpret the answer to part c Use the answer to part c to justify the decision in part d Use the decision in part d to make a conclusion about whether the individual is likely to have embezzled. Use this to then tell the law enforcement officials whether they should pursue the case or not.

Answer 1

c)

	Oi	Ei	(Oi-Ei)^2/Ei
0.301	40	47.257	1.114417949
0.176	20	27.632	2.10796989
0.125	14	19.625	1.612261146
0.097	17	15.229	0.205951868
0.079	19	12.403	3.508861485
0.067	14	10.519	1.1519499
0.058	8	9.106	0.134332967
0.051	17	8.007	10.10041826
0.046	17	7.222	13.2386159
1	166	157	33.17477936

df= n-1= 9-1 = 8

p-value =

=1 - CHISQ.DIST(36.142506,8,1)

= 0.00001

d)

since p-value < 0.05

we reject the null hypothesis

we conclude that the individual is likely to have embezzled.

the law enforcement officials should pursue the case