Faked numbers in tax returns, payment records, invoices, etc. often display patterns that aren't present in legitimate

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Question:

Faked numbers in tax returns, payment records, invoices, etc. often display patterns that aren't present in legitimate records. It is a striking fact that the first digits of numbers in legitimate records often have probabilities that follow the model (known as Benford's Law) partially shown in the following table.

first digit	probability
1	?
2	0.176
3	0.125
4	0.097
5	0.079
6	0.067
7	0.058
8	0.051
9	0.046

d) Excel: Suppose you receive 1,000 invoices from a company that is under investigation. You are asked to provide evidence as to the legitimacy of the records. We simulate the first digits of these invoices on Excel with the following steps: i. Click on cell A1 > Click on the function icon ???????? > Choose the “Math & Trig” category > Select “RANDBETWEEN” function > OK ii. Enter “1” for “Bottom” > Enter “9” for “Top” > OK iii. After getting the random number in the first cell, highlight that cell, scroll down 1,000 cells, hold the “Shift” button and then click the 1,000th cell. The column should now be highlighted. Underneath the “Home” tab click “Fill – Down” ( ). All entries should now be filled with random numbers. Obtain a histogram of the first digits for the 1,000 invoices. Use upper-class limits of 1.5, 2.5, …, 9.5. On your chart and histogram, rename the labels for each class to be 1, 2, …, 9. Print the chart and histogram.

e) Using the chart output from d), obtain empirical probabilities for P(A), P(B), and P(C) using relative frequency approximations.

f) Compare your empirical probabilities in e) to the theoretical probabilities in b). Is there any evidence to suggest the invoices are not legitimate? Explain.