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 226 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.

Using the table below and a significance level of a=0.01, complete part​ (a) below.

Digit	Probability	Frequency
1	0.301	36
2	0.176	32
3	0.125	45
4	0.097	20
5	0.079	24
6	0.067	17
7	0.058	9
8	0.051	16
9	0.046	7

(a) What is the test​ statistic? (round to three decimal places as needed)

Answer 1

According to the question, the given distribution represents the first digits in 226 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.The probabilities of occurrence of each of the first significant numbers has been provided in the question.

Here, we apply Pearson's chi-squared goodness of fit test for the given frequency table. Pearson's chi-squared test statistic is given by:

= Pearson's cumulative test statistic, which asymptotically approaches a distribution.

= the number of observations of type i.

= total number of observations

= the expected (theoretical) count of type i, asserted by the null hypothesis that the fraction of type i in the population is

= the number of cells in the table

Significant Numbers(n=9)	Frequency(Oi)	Probability(pi)	Expected(Ei)	(Oi-Ei)	(Oi-Ei)2	(Oi-Ei)2/Ei
1	36	0.301	62.006	-26.006	676.312	10.907
2	32	0.176	36.256	-4.256	18.1135	0.499
3	45	0.125	25.75	19.25	370.56	14.39
4	20	0.097	19.982	0.018	0.000324	0.0000162
5	24	0.079	17.854	6.146	37.7733	2.1156
6	17	0.067	13.802	3.198	10.2272	0.74
7	9	0.058	11.948	-2.948	8.69	0.7273
8	16	0.051	10.506	5.494	30.184	2.873
9	7	0.046	9.476	-2.476	6.13	0.6468
Total	206	1	206(approx.)		1157.99	32.899

From the table the test statistics .

Further, according to the table for Chi-square distribution , chi-squared value for n=9 is 2.088 < 32.899(observed.)