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 an auditor for a very large corporation. Therevenue report involves millions of numbers in a large computerfile. Let us say you took a random sample of n = 216numerical entries from the file and r = 52 of the entrieshad a first nonzero digit of 1. Let p represent thepopulation proportion of all numbers in the corporate file thathave a first nonzero digit of 1.
(i) Test the claim that p is less than 0.301. Use? = 0.10.
(a) What is the level of significance?
State the null and alternate hypotheses.
H0: p = 0.301;H1: p < 0.301
H0: p = 0.301;H1: p ?0.301
H0: p < 0.301;H1: p = 0.301
H0: 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 > 5 and nq> 5.
The Student's t, since np > 5 andnq > 5.
The standard 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 rejector fail to reject the null hypothesis? Are the data statisticallysignificant at level ??
At the ? = 0.10 level, we reject the null hypothesis andconclude the data are statistically significant.
At the ? = 0.10 level, we reject the null hypothesis andconclude the data are not statisticallysignificant.
At the ? = 0.10 level, we fail to reject the null hypothesis andconclude the data are statistically significant.
At the ? = 0.10 level, we fail to reject the null hypothesis andconclude the data are not statistically 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 less than 0.301.
There is insufficient evidence at the 0.10 level to concludethat the true proportion of numbers with a leading 1 in the revenuefile is less than 0.301.
(ii) If p is in fact less than 0.301, would it make yoususpect that there are not enough numbers in the data file withleading 1's? Could this indicate that the books have been "cooked"by "pumping up" or inflating the numbers? Comment from theviewpoint of a stockholder. Comment from the perspective of theFederal Bureau of Investigation as it looks for money laundering inthe form of false profits.
No. The revenue data file seems to include more numbers withhigher first nonzero digits than Benford's law predicts.
No. The revenue data file does not seem to include more numberswith higher first nonzero digits than Benford's lawpredicts.
Yes. The revenue data file seems to include more numbers withhigher first nonzero digits than Benford's law predicts.
Yes. The revenue data file does not seem to include more numberswith higher first nonzero digits than Benford's law predicts.
(iii) Comment on the following statement: If we reject the nullhypothesis at level of significance ?, we have not provedHo to be false. We can say that the probabilityis ? that we made a mistake in rejecting Ho.Based on the outcome of the test, would you recommend furtherinvestigation before accusing the company of fraud?
We have not proved H0 to be false. Becauseour data lead us to reject the null hypothesis, more investigationis merited.
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. Becauseour data lead us to reject the null hypothesis, more investigationis not merited.
We have not proved H0 to be false. Becauseour data lead us to accept the null hypothesis, more investigationis not merited.
