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. First digit 1 2 3 4 5 6 7 8 9 Proportion 0.288 0.169 0.134 0.081 0.062 0.05 0.037 0.041 0.138 Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1231 randomly chosen invoices there are no more than 672 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.)

Explanation / Answer

here probability of having 1,2,or 3 =p=0.288+0.169+0.134=0.591

therefore expected 1,2 or 3 in 1231 invoices =np=1231*0.591=727.521

and std deviation =(np(1-p))1/2 =17.25

hence  probability of  no more than 672 in amounts with first digit 1, 2, or 3 =P(X<=672)

=P(Z<(672.5-727.521)/17.25)=P(Z<-3.1897)=0.0007

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