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)(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 201 checks to a supposed company are as follows:

Digit	Observed Frequency
1	49
2	31
3	24
4	14
5	15
6	20
7	21
8	20
9	7

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
P-Value =


d. What is your decision?

Fail to reject the Null Hypothesis

Reject the Null Hypothesis

Answer 1

The statistical software output for this problem is:

Chi-Square goodness-of-fit results:
Observed: Oi
Expected: Ei

N	DF	Chi-Square	P-value
201	8	24.848972	0.0016

Observed	Expected
49	60.501
31	35.376
24	25.125
14	19.497
15	15.879
20	13.467
21	11.658
20	10.251
7	9.246

Hence,

a) Hypotheses:

Ho: The given distribution follows Benford's law.

Ha: The given distribution does not follow Benford's law.

b) Level of significance as 0.01 is an appropriate choice because we need more preciseness for testing this data.

c) p - Value = 0.0016

d) Reject the Null Hypothesis