Detecting FraudThe Brooklyn District Attorney’s office analyzed the leading (leftmost) digits of check amounts in order to identify fraud. The leading digit of 1 is expected to occur 30.1% of the time, according to “Benford’s law,” which applies in this case. Among 784 checks issued by a suspect company, there were none with amounts that had a leading digit of 1.

a. If there is a 30.1% chance that the leading digit of the check amount is 1, what is the expected number of checks among 784 that should have a leading digit of 1?

b. Assume that groups of 784 checks are randomly selected. Find the mean and standard deviation for the numbers of checks with amounts having a leading digit of 1.

c. Use the results from part (b) and the range rule of thumb to identify the values that are significantly low.

d. Given that the 784 actual check amounts had no leading digits of 1, is there very strong evidence that the suspect checks are very different from the expected results? Why or why not?

The number of checks issued by a suspect company is$n=784$.

The probability of a check having a lead digit of ‘1’is $p=0.301$.

a.

The expectednumber of checks among the 784 that should have a leading digit of 1 is computed as follows.

$$\begin{gathered} \mu =np \\ =784\times 0.301 \\ =235.984 \\ \approx 236.0 \end{gathered}$$

Therefore, theexpectednumber of checks among the 784 that should have a leading digit of 1 is236.0.

b.

The mean number of checks among the 784 that should have a leading digit of 1 is computed as follows.

$$\begin{gathered} \mu =np \\ =784\times 0.301 \\ =235.984 \\ \approx 236 \end{gathered}$$

Thus, the mean is 236 checks.

The standard deviation of thenumber of checks among the 784 that should have a leading digit of 1 is computed as follows.

$$\begin{gathered} \sigma =\sqrt{np\left(1-p\right)} \\ =\sqrt{784\times 0.301\times \left(1-0.301\right)} \\ =12.84 \\ \approx 12.8 \end{gathered}$$

.

Therefore, the standard deviation of the number of checks among the 784 that should have a leading digit of 1 is 12.8 checks.

c.

Using the range rule of thumb, we get that the significant low values are the values less than $\mu -2\sigma$.

The calculation is as follows.

$$\begin{gathered} \mu -2\sigma =236.0-\left(2\times 12.8\right) \\ =210.4 \end{gathered}$$

Therefore, the significant low values are the values that are less than 210.4 checks.

d.

It can be observed that 0 is less than or equal to 210.4.

Therefore, there isstrong evidence that the suspect checks are very different from the expected results.