Recall that Benford\'s Law claims that numbers chosen from very large data files
ID: 3221184 • Letter: R
Question
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 an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 223 numerical entries from the file and r = 45 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1.
(i) Test the claim that p is less than 0.301. Use = 0.05.
(a) What is the level of significance?
State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?
Ho: p = 0.301; H1: p > 0.301; right-tailed
Ho: p < 0.301; H1: p = 0.301; left-tailed
Ho: p = 0.301; H1: p 0.301; two-tailed
Ho: p = 0.301; H1: p < 0.301; left-tailed
(b) What sampling distribution will you use? Do you think the sample size is sufficiently large?
The normal distribution, since the sample size is large
The t distribution, since the sample size is large.
What is the value of the sample test statistic? (Use 2 decimal places.)
(c) Find the P-value of the test statistic. (Use 4 decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-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 ?
At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
(e) State your conclusion in the context of the application.
Fail to reject the null hypothesis, there is insufficient evidence that the true proportion of numbers with a leading 1 in the revenue file is less than .301.
Fail to reject the null hypothesis, there is sufficient evidence that the true proportion of numbers with a leading 1 in the revenue file is less than .301.
Reject the null hypothesis, there is insufficient evidence that the true proportion of numbers with a leading 1 in the revenue file is less than .301.
Reject the null hypothesis, there is sufficient evidence that the true proportion of numbers with a leading 1 in the revenue file is less than .301.
(ii) If p is in fact less than 0.301, would it make you suspect that there are not enough numbers in the data file with leading 1's? Could this indicate that the books have been "cooked" by "pumping up" or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits.
Yes. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.
No. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.
Yes. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.
No. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.
(iii) Comment on the following statement: If we reject the null hypothesis at level of significance , we have not proved Ho to be false. We can say that the probability is that we made a mistake in rejecting Ho. Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?
We have not proved Ho to be false. Because our data lead us to reject the null hypothesis, more investigation is merited.
We have not proved Ho to be false. Because our data lead us to accept the null hypothesis, more investigation is not merited.
We have not proved Ho to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.
We have proved Ho to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.
Explanation / Answer
Here we have to test
H0 : p = 0.301 Vs H1 : p < 0.301
This is a left tailed test since in alternative hypothesis contains < sign
assume alpha = significance level = 0.05
(b) What sampling distribution will you use? Do you think the sample size is sufficiently large?
The normal distribution, since the sample size is large
What is the value of the sample test statistic?
Here test statistic is,
Z = (p^ - p) / sqrt((p*q)/n)
where p^ = x/n = sample proportion
q = 1 - p
n = 223
THis test we can done in TI_83 calculator.
steps :
STAT --> TESTS --> 5:1-PropZTest --> ENTER --> Input all the values --> Alternative : select "<P0" --> ENTER --> Calculate --> ENTER
Test statistic (Z) = -3.23
P-value = 6.1954E-4 = 0.000
P-value < alpha
Reject H0 at 5% level of significance.
Conclusion : There is sufficient evidence to say that the true proportion of numbers with a leading 1 in the revenue file is less than .301.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.