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In: Statistics and Probability

Benford's Law states that the first nonzero digits of numbers drawn at random from a large...

Benford's Law states that the first nonzero digits of numbers drawn at random from a large complex data file have the following probability distribution.†

First Nonzero Digit 1 2 3 4 5 6 7 8 9
Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046

Suppose that n = 275 numerical entries were drawn at random from a large accounting file of a major corporation. The first nonzero digits were recorded for the sample.

First Nonzero Digit 1 2 3 4 5 6 7 8 9
Sample Frequency 85 50 34 22 20 18 13 17 16

Use a 1% level of significance to test the claim that the distribution of first nonzero digits in this accounting file follows Benford's Law.

b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)

Solutions

Expert Solution

Given information :

Digit Frequency(Oi) probability(Pi)
1 85 0.301
2 50 0.176
3 34 0.125
4 22 0.097
5 20 0.079
6 18 0.067
7 13 0.058
8 17 0.051
9 16 0.046
Total 275
N 275

We have to compute Expected frequencies

Ei = N * Pi

Digit Frequency(Oi) probability(Pi) Ei=N*Pi
1 85 0.301 82.775
2 50 0.176 48.400
3 34 0.125 34.375
4 22 0.097 26.675
5 20 0.079 21.725
6 18 0.067 18.425
7 13 0.058 15.950
8 17 0.051 14.025
9 16 0.046 12.650
Total 275 275
N 275

Hypothesis :

Ho :  The distribution of first nonzero digits in this accounting file follows Benford's Law.

V/s

H1 :  The distribution of first nonzero digits in this accounting file doesn't follows Benford's Law.

Under Ho,

Test statistic is,

Oi Ei (Oi-Ei)^2/Ei
85 82.775 0.060
50 48.4 0.053
34 34.375 0.004
22 26.675 0.819
20 21.725 0.137
18 18.425 0.010
13 15.95 0.546
17 14.025 0.631
16 12.65 0.887
Total 275 3.147

DF = k-1 = 8

We find critical value using chi square table.

Here <    at 0.01 level of significance and 8 df then we fail to reject Ho.

At 0.01 level of significance there is sufficient evidance to support the claim that the distribution of first nonzero digits in this accounting file follows Benford's Law.

orchestra answered 1 year ago
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