Benford's Law

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.
First digit 1 2 3 4 5 6 7 8 9
Proportion 0.289 0.156 0.135 0.089 0.052 0.064 0.043 0.038 0.134
Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1223 randomly chosen invoices there are no more than 676 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.)

Answer 1

This is the normal approximation to the binomial where p=.289+.156+.135=.580 so q=1-p=.420

z=(X-np)/sqrt(npq)

p(X<=676)=

 

z=(676-1223*.58)/sqrt(1223*.58*.42)

z=-1.9

so probability =.0287