Benford\'s Law states that the first nonzero digits of numbers drawn at random f

Question

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.†

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.

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. (a) What is the level of significance?


State the null and alternate hypotheses.

H0: The distributions are the same.
H1: The distributions are the same. H0: The distributions are the same.
H1: The distributions are different.     H0: The distributions are different.
H1: The distributions are the same. H0: The distributions are different.
H1: The distributions are different.

(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.)


Are all the expected frequencies greater than 5?

Yes No    

What sampling distribution will you use?

normal binomial     chi-square uniform Student's t

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.100 0.050 < P-value < 0.100     0.025 < P-value < 0.050 0.010 < P-value < 0.025 0.005 < P-value < 0.010 P-value < 0.005

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

Explanation / Answer

Ans:

(a) level of significance=0.01

State the null and alternate hypotheses.

H0: The distributions are the same.
H1: The distributions are different.

(b)

Test statistic:

Calculated chi square score=2.959
Are all the expected frequencies greater than 5?

Yes

What sampling distribution will you use?

chi-square

What are the degrees of freedom?
df=n-1=9-1=8

(c) P-value=CHIDIST(2.959,8)=0.9369

P-value > 0.100,do not reject H0.

we can conclude the distributions are same.

First Nonzero Digit 1 2 3 4 5 6 7 8 9 Total Observed freq(O) 86 45 35 23 22 18 13 17 16 275 Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 1 Expected freq(E) 82.775 48.4 34.375 26.675 21.725 18.425 15.95 14.025 12.65 275 (O-E)^2/E 0.126 0.239 0.011 0.506 0.003 0.010 0.546 0.631 0.887 2.959

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