Question

You might think that if you looked at the first digit in randomly selected numbers that the distribution would be uniform. Actually, it is not! Simon Newcomb and later Frank Benford both discovered that the digits occur according to the following distribution: (digit, probability)

(1,0.301),(2,0.176),(3,0.125),(4,0.097),(5,0.079),(6,0.067),(7,0.058),(8,0.051),(9,0.046)(1,0.301),(2,0.176),(3,0.125),(4,0.097),(5,0.079),(6,0.067),(7,0.058),(8,0.051),(9,0.046)

The IRS currently uses Benford's Law to detect fraudulent tax data. Suppose you work for the IRS and are investigating an individual suspected of embezzling. The first digit of 145 checks to a supposed company are as follows:

Digit	Observed Frequency
1	31
2	18
3	25
4	18
5	12
6	15
7	7
8	7
9	12

a. State the appropriate null and alternative hypotheses for this test.



b. Explain why ?=0.01?=0.01 is an appropriate choice for the level of significance in this situation.



c. What is the P-Value? Report answer to 4 decimal places  
P-Value = .0300


d. What is your decision?

Reject the Null Hypothesis

OR

Fail to reject the Null Hypothesis (answer)

e. Write a statement to the law enforcement officials that will use it to decide whether to pursue the case further or not. Structure your essay as follows: Given a brief explanation of what a Goodness of Fit test is. Explain why a Goodness of Fit test should be applied in this situation. State the hypotheses for this situation Interpret the answer to part c Use the answer to part c to justify the decision in part d Use the decision in part d to make a conclusion about whether the individual is likely to have embezzled. Use this to then tell the law enforcement officials whether they should pursue the case or not.

Need help with A, B, and E please

Answer 1

a. The appropriate null and alternative hypotheses for this test is

H₀: Data comes from the specified distribution.

H₁ : Data does not come from a specified distribution.

b. Small the significance level, the change of rejection of null hypothesis is more.

test statistic is

Digit	Observed(O)	Probability	Expected(E)	(O-E)^2/E
1	31	0.301	43.645	3.663559
2	18	0.176	25.52	2.215925
3	25	0.125	18.125	2.607759
4	18	0.097	14.065	1.100905
5	12	0.079	11.455	0.02593
6	15	0.067	9.715	2.875062
7	7	0.058	8.41	0.236397
8	7	0.051	7.395	0.021099
9	12	0.046	6.67	4.259205
Total	145

c. The p-value is used as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected.

p-value = 0.03

d. The p-value is greater than 0.01, we fail to reject the null hypothesis .