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 157 checks to a supposed company are as follows: a. State the appropriate null and alternative hypotheses for this test. b. Explain why alpha = 0.01 is an appropriate choice for the level of significance in this c. What is the P-Value? Report answer to 4 decimal places P-Value = _____ d. What is your decision? Reject the Null Hypothesis

Explanation / Answer

a) Ho : the data follow given distribution

Ha: the data don't follow given distribution

b) alpha should be small as this is type ! error , we don't want to make more error

alpha is normally 0.01 or 0.05

c) TS = 2 = [ (Oi - Ei)2 / Ei ]

df = r -1 = 8

p-value = P(X^2 > 36.1425) = 0.00001653

d)

since p-value < 0.01

we reject the null

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Oi Ei (Oi-Ei)^2/Ei 0.301 29 47.257 7.053305309 0.176 30 27.632 0.202932252 0.125 19 19.625 0.019904459 0.097 14 15.229 0.09918189 0.079 3 12.403 7.128630896 0.067 18 10.519 5.320406978 0.058 17 9.106 6.843316055 0.051 13 8.007 3.113531785 0.046 14 7.222 6.361296594 1 157 157 36.14250622

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