You might think that if you looked at the first digit in randomly selected numbe

Question

You 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)

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 169 checks to a supposed company are as follows:

a. State the appropriate null and alternative hypotheses for this test.



b. Explain why =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
P-Value =


d. What is your decision?

Reject the Null Hypothesis

Fail to reject the Null Hypothesis

e. 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.

Digit Observed
Frequency 1 42 2 21 3 29 4 10 5 17 6 17 7 8 8 12 9 13

Explanation / Answer

Ans:

Chi square test for Goodness of fit:

H0:data fits the Benford's distribution.

H1:data does not fit Benford's distribution.

Total Calculated chi square score=18.560

degree of freedom=9-1=8

p-value=CHIDIST(18.560,8)=0.0174

Significance level=0.01

As, p-value=0.0174>0.01,we fail to reject the Null hypothesis.

Observed(O) Expected€ (O-E)^2/E Pi pi*169 1 42 0.301 50.869 1.546 2 21 0.176 29.744 2.571 3 29 0.125 21.125 2.936 4 10 0.097 16.393 2.493 5 17 0.079 13.351 0.997 6 17 0.067 11.323 2.846 7 8 0.058 9.802 0.331 8 12 0.051 8.619 1.326 9 13 0.046 7.774 3.513 Total 169 1 169 18.560

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