Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank. You draw a random sample ofn=228numbers from this file andr=86have a first nonzero digit of 1. Letprepresent the population proportion of all numbers in the computer file that have a leading digit of 1.

(i) Test the claim thatpis more than 0.301. Use=0.10.

(a) What is the level of significance?

State the null and alternate hypotheses.

(b) What sampling distribution will you use?

What is the value of the sample test statistic?

(c) Find theP-value of the test statistic.

Sketch the sampling distribution and show the area corresponding to theP-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level?

(e) Interpret your conclusion in the context of the application.

(ii) Ifpis in fact larger than 0.301, it would seem there are too many numbers in the file with leading 1's. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees.

(iii) Comment on the following statement: If we reject the null hypothesis at level of significance, we have not provedH₀to be false. We can say that the probability isthat we made a mistake in rejectingH_o. Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?