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)


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)Chi-sqaure goodness of fit test:

H0: Frequanyc distribution is equal to the proportion

Ha: Frequancy distribution is different from proportion

b) Significant value= 0.01

One of the proporitonal value is 0.046. So it is appropriate to use for the significance level.

c)

Digit Observed Freq P(x) E(X)=n*p(x) (O-E) (O-E)^2 (O-E)^2/E
1 49 0.301 60.501 -11.501 132.273 2.186294
2 31 0.176 35.376 -4.376 19.14938 0.54131
3 24 0.125 25.125 -1.125 1.265625 0.050373
4 14 0.097 19.497 -5.497 30.21701 1.549829
5 15 0.079 15.879 -0.879 0.772641 0.048658
6 20 0.067 13.467 6.533 42.68009 3.169235
7 21 0.058 11.658 9.342 87.27296 7.486101
8 20 0.051 10.251 9.749 95.043 9.271583
9 7 0.046 9.246 -2.246 5.044516 0.545589
Sum(n) 201 Sum 24.84897

c) The degree of freedom: df= c-1= 8

P-value: 0.002

d) Reject the null hypothesis because P-value is lessthan significant value.


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