In: Statistics and Probability
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 = 215
numerical entries from the file and r = 47 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
α = 0.05.
(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 and nq > 5.The Student's t, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5.The standard normal, since np < 5 and nq < 5.
What is the value of the sample test statistic? (Round your answer
to two decimal places.)
(c) Find the P-value of the test statistic. (Round your
answer to four 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 α?
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
(e) Interpret your conclusion in the context of the
application.
There is sufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.There is insufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.
(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.
No. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.Yes. 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 does not seem 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 α, we have not proved
Ho to be false. We can say that the probability
is α 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 H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is merited.We have not proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited. We have not proved H0 to be false. Because our data lead us to accept the null hypothesis, more investigation is not merited.We have proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.
a)
level of significance =0.05
H0: p = 0.301; H1: p < 0.301
b)
The standard normal, since np > 5 and nq > 5.
sample success x = | 47 | |
sample size n = | 215 | |
std error σp =√(p*(1-p)/n) = | 0.0313 | |
sample proportion p̂ = x/n= | 0.2186 | |
z =(p̂-p)/σp=(0.219-0.301)/0.219= | -2.63 |
c)
p value = | 0.0043 | (from excel:1*normsdist(-2.63) |
d)
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
e)
There is sufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.
f)
Yes. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts
We have not proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is merited.