Question

In: Statistics and Probability

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

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

Solutions

Expert Solution

(a)

H0: The distribution of first digit of allegedly fraudlent checks obey Benford's law.

Ha: The distribution of first digit of allegedly fraudlent checks do not law obey Benford's law.

(b)

Since the result of study do not have critical negative consequences so reasecher used very small tyep I error.

(c)

Following table shows the calculations for test statistics:

O p E=p*201 (O-E)^2/E
49 0.301 60.501 2.18629446
31 0.176 35.376 0.54130981
24 0.125 25.125 0.05037313
14 0.097 19.497 1.54982864
15 0.079 15.879 0.04865804
20 0.067 13.467 3.16923509
21 0.058 11.658 7.48610087
20 0.051 10.251 9.27158336
7 0.046 9.246 0.54558901
Total 201 1 201 24.8489724

So, the test statistics is:

Degree of freedom: df=9-1=8

The p-value is: 0.0016

(d)

Since p-value is ess than so we reject the null hypothesis.


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