Question

An investigator analyzed the leading digits from 787 checks issued by seven suspect companies. The frequencies were found to be 4​, 11​, 2​, 72​, 371​, 281​, 7​, 16​, and 23​, and those digits correspond to the leading digits of​ 1, 2,​ 3, 4,​ 5, 6,​ 7, 8, and​ 9, respectively. If the observed frequencies are substantially different from the frequencies expected with​ Benford's law shown​ below, the check amounts appear to result from fraud. Use a 0.025 significance level to test for​ goodness-of-fit with​ Benford's law. Does it appear that the checks are the result of​ fraud?

Leading Digit:      1    2    3 4 5 6    7 8 9
Actual Frequency:      4 11    2    72    371 281 7    16    23

Benford's Law:      30.1% 17.6% 12.5%    9.7%    7.9% ​ 6.7%    ​5.8% ​ 5.1%    ​4.6%

Determine the null and alternative hypotheses.

Ho​: (1)_________________    H1​: (2)_________________

Calculate the test​ statistic, χ2.

χ2 = _______________​

(Round to three decimal places as​ needed.)

Calculate the​ P-value.

P-value = _______________​

(Round to four decimal places as​ needed.)

State the conclusion.

(3)_________________Ho. There (4)______________sufficient evidence to warrant rejection of the claim that the leading digits are from a population with a distribution that conforms to​ Benford's law. It (5)________________that the checks are the result of fraud.

Choose from the following options:
(1) a. At least two leading digits have frequencies that do not conform to Benford's    law.
b. The leading digits are from a population that conforms to Benford's law.
c. At most three leading digits have frequencies that do not conform to Benford's law.
d. At least one leading digit has a frequency that does not conform to Benford's law.

(2) a. The leading digits are from a population that conforms to Benford's law.
b. At most three leading digits have frequencies that do not conform to Benford's    law.
c. At least one leading digit has a frequency that does not conform to Benford's    law.
   d. At least two leading digits have frequencies that do not conform to Benford's law.

(3) Do not reject
   Reject

(4) is
   is not

(5) does appear
   does not appear

Answer 1

Solution:

Test Hypothesis:

Ho​: (1)The amounts follow Benford's Law. That is the checks are not faulty.

H1​: (2)The amounts do not follow Benford's Law. That is the checks are faulty.

Test​ statistic, χ2:

Chi -square test Stat =

Test Stat = 3015.279

Calculate the​ P-value.

P-value =P( > Test stat) ....................df = n-1

= P(₈ > 3015.279)

p-value = 0.000

Since p-value < 0.025

We reject the null hypothesis at 2.5%. There is sufficient evidence to conclude that the amounts do not follow Benford's Law. That is the checks are faulty.

Reject H0.