It is a striking fact that the first digits of numbers in legitimate records oft

Question

It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford's Law, shown below.

Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1152 randomly chosen invoices there are no more than 677 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.)

First digit 1 2 3 4 5 6 7 8 9 Proportion 0.295 0.167 0.132 0.075 0.07 0.069 0.03 0.031 0.131

Explanation / Answer

Solution:

This is the normal approximation to the binomial where
p=0.295+0.167+0.132= 0.594 so q=1-p = 0.406
z=(X-np)/sqrt(npq)
p(X<=677)=

z=(677-1152*0.594)/sqrt(1158*0.594*.406)
z = -7.288/16.711
z= -0.4361
so probability =0.3314

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