The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9.
ID: 3326106 • Letter: T
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 207 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer. Complete parts (a) and (b) below.
Distribution of first digits (Benford's Law)
Digit
1
2
3
4
5
Probability
0.301
0.176
0.125
0.097
0.079
Digit
6
7
8
9
Probability
0.067
0.058
0.051
0.046
First digits in allegedly fraudulent checks
First digit
1
2
3
4
5
6
7
8
9
Frequency
42
25
45
20
23
17
9
17
9
What is the test statistic?
2 / 0=___
(Round to three decimal places as needed.)
What is the P-value= __ of the test?
Distribution of first digits (Benford's Law)
Digit
1
2
3
4
5
Probability
0.301
0.176
0.125
0.097
0.079
Digit
6
7
8
9
Probability
0.067
0.058
0.051
0.046
Explanation / Answer
Solution:
Here, we have to use Chi square test for goodness of fit.
Part a
The test statistic formula is given as below:
Chi square = [(O – E)^2/E]
Where O is observed frequency and E is expected frequency
E = total*exp. Prob.
Digit
Prob.
O
E
(O - E)^2/E
1
0.301
42
207*0.301 = 62.307
6.618425683
2
0.176
25
207*0.176 = 36.432
3.587248134
3
0.125
45
207*0.125 = 25.875
14.13586957
4
0.097
20
20.079
0.000310822
5
0.079
23
16.353
2.701804501
6
0.067
17
13.869
0.706839787
7
0.058
9
12.006
0.752626687
8
0.051
17
10.557
3.932201288
9
0.046
9
9.522
0.028616257
Total
1
207
207
32.46394272
Chi square = [(O – E)^2/E] = 32.46394272
Part b
Degrees of freedom = n – 1 = 9 – 1 = 8
P-value = 0.0000769
(By using chi square table or excel command =CHIDIST(32.46394272,8))
Digit
Prob.
O
E
(O - E)^2/E
1
0.301
42
207*0.301 = 62.307
6.618425683
2
0.176
25
207*0.176 = 36.432
3.587248134
3
0.125
45
207*0.125 = 25.875
14.13586957
4
0.097
20
20.079
0.000310822
5
0.079
23
16.353
2.701804501
6
0.067
17
13.869
0.706839787
7
0.058
9
12.006
0.752626687
8
0.051
17
10.557
3.932201288
9
0.046
9
9.522
0.028616257
Total
1
207
207
32.46394272
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.