1. The first significant digit in any number must be I, 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 215 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer. Complete parts (a) and (b) below.
(a) Using the level of significance a = 0.10, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. What is the null hypothesis?
- H0: The distribution of the first digits in the allegedly fraudulent checks does not obey Benford's Law.
- 110: The distribution of the first digits in the allegedly fraudulent checks obeys Benford's Law.
- None of these.
What is the alternative hypothesis?
- HI: The distribution of the first digits in the allegedly fraudulent checks does not obey Benford's Law.
- Ht: The distribution of the first digits in the allegedly fraudulent checks obeys Benford's Law.
- None of these.
What is the test statistic?
What is the P-value of the test?
Using the P-value approach, compare the P-value with the given ? = 0.10 level of significance. Based on the result, do the first digits obey the Benford's Law?
- Do not reject the Ho because the calculated P-value is less than the given ? level of significance.
- Do not the reject the Ho because the calculated P-value is greater than the given ? level of significance.
- Reject the Ho because the calculated P-value is less than the given ? level of significance.
- Reject the Ho because the calculated P-value is greater than the given ? level of significance.
b. Based on the result of part (a), could one think that the employee is guilty of embezzlement?
- No, the first digits obey the Benford's Law.
- Yes, the first digits obey the Benford's Law.
- Yes, the first digits do not obey the Benford's Law.
- No, the first digits do not obey the Benford's Law.
Distribution of First Digits
|
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
|
36
|
32
|
28
|
20
|
24
|
36
|
15
|
17
|
7
|
2. A traffic safety company publishes reports about motorcycle fatalities and helmet use. In the first accompanying data table, the distribution shows the proportion of fatalities by location of injury for motorcycle accidents. The second data table shows the location of injury and fatalities for 2062 riders not wearing a helmet. Complete parts (a) and (b) below.
(a) Does the distribution of fatal injuries for riders not wearing a helmet follow the distribution for all riders? Use a = 0.01 level of significance. What are the null and alternative hypotheses?
- Ho: The distribution of fatal injuries for riders not wearing a helmet follows the same distribution for all other riders.
- HI: The distribution of fatal injuries for riders not wearing a helmet does not follow the same distribution for all other riders.
- Ho: The distribution of fatal injuries for riders not wearing a helmet does not follow the same distribution for all other riders.
- HI: The distribution of fatal injuries for riders not wearing a helmet does follow the same distribution for all other riders.
- None of these.
Compute the expected counts for each fatal injury.
|
Location of injury
|
Observed Count
|
Expected Count
|
|
Multiple Locations
|
1047
|
|
|
Head
|
852
|
|
|
Neck
|
34
|
|
|
Thorax
|
83
|
|
|
Abdomen/Lumbar/Spine
|
46
|
|
What is the P-value of the test?
Based on the results, does the distribution of fatal injuries for riders not wearing a helmet follow the distribution for all other riders at a significance level of a = 0.01?
Reject Ho. There is sufficient evidence that the distribution of fatal injuries for riders not wearing a helmet does not follow the distribution for all riders.
Do not reject Ho. There is not sufficient evidence that the distribution of fatal injuries for riders not wearing a helmet does not follow the distribution for all riders.
Do not reject Ho. There is sufficient evidence that the distribution of fatal injuries for riders not wearing a helmet follows the distribution for all riders.
Reject Ho. There is not sufficient evidence that the distribution of fatal injuries for riders not wearing a helmet follows the distribution for all riders.
(b) Compare the observed and expected counts for each category. What does this information tell you?
- Motorcycle fatalities from head injuries occur less frequently for riders not wearing a helmet.
- Motorcycle fatalities from head injuries occur more frequently for riders not wearing a helmet.
- Motorcycle fatalities from thorax injuries occur more frequently for riders not wearing a helmet.
Distribution of fatalities by location of injury
|
Location of injuries and fatalities for 2062 riders not wearing a helmet
|
|
Location of injuries
|
Multiple Locations
|
Head
|
Neck
|
Thorax
|
Abdomen Lumar/Spine
|
|
Number
|
1047
|
852
|
34
|
83
|
46
|
|
Proportion of fatalities by location of injuries for motorcycle accidents
|
|
Location of injuries
|
Multiple Locations
|
Head
|
Neck
|
Thorax
|
Abdomen Lumar/Spine
|
|
Proportion
|
0.570
|
0.310
|
0.030
|
0.060
|
0.030
|
Attachment:- 8-2 problem 8.pdf
Attachment:- 8-2 problem 9.pdf