Recall that Benford's Law claims that numbers chosen from verylarge data files tend to have \"1\" as the first nonzero digitdisproportionately often. In fact, research has shown that if yourandomly draw a number from a very large data file, the probabilityof getting a number with \"1\" as the leading digit is about 0.301.Now suppose you are the auditor for a very large corporation. Therevenue file contains millions of numbers in a large computer databank. You draw a random sample of n = 229 numbers fromthis file and r = 87 have a first nonzero digit of 1. Letp represent the population proportion of all numbers inthe computer file that have a leading digit of 1.

(i) Test the claim that p is more than 0.301. Useα = 0.10.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.301;H₁:  p ≠0.301H₀: p = 0.301;H₁:  p <0.301     H₀:p = 0.301; H₁: p >0.301H₀: p > 0.301;H₁:  p = 0.301

(b) What sampling distribution will you use?

The Student's t, since np > 5 andnq > 5.The standard normal, since np > 5and nq > 5.     The Student'st, since np < 5 and nq < 5.Thestandard normal, since np < 5 and nq <5.

What is the value of the sample test statistic? (Round your answerto two decimal places.)


(c) Find the P-value of the test statistic. (Round youranswer to four decimal places.)


Sketch the sampling distribution and show the area corresponding tothe P-value.

(d) Based on your answers in parts (a) to (c), will you reject orfail to reject the null hypothesis? Are the data statisticallysignificant at level α?

At the α = 0.10 level, we reject the null hypothesisand conclude the data are statistically significant.At theα = 0.10 level, we reject the null hypothesis and concludethe data are not statisticallysignificant.     At the α = 0.10level, we fail to reject the null hypothesis and conclude the dataare statistically significant.At the α = 0.10 level, wefail to reject the null hypothesis and conclude the data are notstatistically significant.

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

There is sufficient evidence at the 0.10 level to conclude thatthe true proportion of numbers with a leading 1 in the revenue fileis greater than 0.301.There is insufficient evidence at the 0.10level to conclude that the true proportion of numbers with aleading 1 in the revenue file is greater than0.301.     

(ii) If p is in fact larger than 0.301, it would seemthere are too many numbers in the file with leading 1's. Could thisindicate that the books have been \"cooked\" by artificially loweringnumbers in the file? Comment from the point of view of the InternalRevenue Service. Comment from the perspective of the Federal Bureauof Investigation as it looks for \"profit skimming\" by unscrupulousemployees.

Yes. There seems to be too many entries with a leading digit1.No. There seems to be too many entries with a leading digit1.     Yes. There does not seem to be toomany entries with a leading digit 1.No. There does not seem to betoo many entries with a leading digit 1.

(iii) Comment on the following statement: If we reject the nullhypothesis at level of significance α , we have not provedH₀ to be false. We can say that the probabilityis α that we made a mistake in rejectingH_o. Based on the outcome of the test, would yourecommend further investigation before accusing the company offraud?

We have proved H₀ to be false. Because ourdata lead us to reject the null hypothesis, more investigation isnot merited.We have not proved H₀ to be false.Because our data lead us to reject the null hypothesis, moreinvestigation is not merited.     We havenot proved H₀ to be false. Because our datalead us to reject the null hypothesis, more investigation ismerited.We have not proved H₀ to be false.Because our data lead us to accept the null hypothesis, moreinvestigation is not merited.