Question

The first significant digit in any number must be​ 1, 2,​ 3, 4,​5, 6,​ 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as​ Benford's Law. For​example, the following distribution represents the first digits in 232 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.

a.) Using the level of significance alpha 0.01​, test whether the first digits in the allegedly fraudulent checks obey​Benford's Law. What is the null​ hypothesis?

b.) What is the alternative​ hypothesis?

c.) What is the test statistic?

d.) What is the p-value?

e.) Using the​ P-value approach, compare the​ P-value with the given alpha = 0.10, level of significance. Based on the​results, do the first digits obey the​ Benford's Law?

f.) (b) Based on the results of part (a)​,could one think that the employee is guilty of​ embezzlement?

Digit   Probability   Frequency
1 0.301   42
2 0.176   25
3 0.125   45
4 0.097   26
5 0.079   24
6 0.067   36
7    0.058   9
8    0.051   16
9 0.046   9

Answer 1

(a) The data follow Benford's Law

(b) The data does not follows Benford's Law

(c) 58.48

(d) 0.0000

(e) Since the p-value (0.0000) is less than the significance level (0.10), we can reject the null hypothesis.

(f) Therefore, we can conclude that the employee is guilty of​ embezzlement.

observed	expected	O - E	(O - E)² / E
42	69.832	-27.832	11.093
25	40.832	-15.832	6.139
45	29.000	16.000	8.828
26	22.504	3.496	0.543
24	18.328	5.672	1.755
36	15.544	20.456	26.920
9	13.456	-4.456	1.476
16	11.832	4.168	1.468
9	10.672	-1.672	0.262
232	232.000	0.000	58.483
58.48	chi-square
8	df
9.24E-10	p-value