Benford\'s Law is a probability rule frequently used by accounting auditors to d

Question

Benford's Law is a probability rule frequently used by accounting auditors to detect systematic fraud. It states that approximately 14.8% of numbers will begin with the number 2.

An auditor decides to investigate a particular firm further if a sample proportion of their invoices, with the invoiced number beginning with a 2, is in the highest 0.5% of all possible sample proportions.

What sample proportion will cause further investigation of the firm if a random sample of 611 invoices is selected? Express your answer as a percentage (2 decimal places).

Explanation / Answer

You want to test if the probability of observing a sample percentage, phat, is in the highest .005.

This is a bernoulli distribution since you are taking trials with a 14.8% chance of success (first number is 2) in each trial.

One way to go about doing this is using a normal approximation since your sample size is so large. A bernoulli variable is approximately Normal with mean n*p and standard deviation square root of np*(1-p)

Hence it will have a mean of 611*.148 = 90.428 and SD 9.4863

You now need to find the z critical value that leaves .005 area in the tail, so if you observe a z in this area, it is in the highest .5%. On a Z-table, the value of 2.5758 leaves this much area in the tail.

To get this value so it is distributed the same as our successes variable we multiply by the SD and add the mean:

2.5758*9.4863 + 90.428 = 114.8628

Thus if we observe a sample proportion greater than 114.8628/611 = .1879 or 18.79% then we believe it in the highest .5% of records.

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