Question

In: Statistics and Probability

You might think that if you looked at the first digit in randomly selected numbers that...

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 )

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 133 checks to a supposed company are as follows:

Digit 1 2 3 4 5 6 7 8 9
Observed
Frequency
37 22 18 9 9 12 7 10 9

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

b. Explain why ? = 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 =

d. What is your decision? Fail to reject the Null Hypothesis Reject the Null Hypothesis

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.

Solutions

Expert Solution

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: All the 9 digits are in accordance with Benford's Law.

Alternative hypothesis: At least one of the proportions in the null hypothesis is false.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a chi-square goodness of fit test of the null hypothesis.

Analyze sample data. Applying the chi-square goodness of fit test to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = k - 1 = 9 - 1
D.F = 8
(Ei) = n * pi


X2 = 5.844

where DF is the degrees of freedom, k is the number of levels of the categorical variable, n is the number of observations in the sample, Ei is the expected frequency count for level i, Oi is the observed frequency count for level i, and X2 is the chi-square test statistic.

The P-value is the probability that a chi-square statistic having 8 degrees of freedom is more extreme than 5.844.

c) We use the Chi-Square Distribution Calculator to find P(X2 > 5.844) = 0.665

Interpret results. Since the P-value (0.665) is greater than the significance level (0.01), we have to accept the null hypothesis.

d) Do not reject H0.

e) From the above test we have sufficient evidence in the favor of the claim that all the 9 digits are in accordance with Benford's Law.


Related Solutions

You might think that if you looked at the first digit in randomly selected numbers that...
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 177 checks to a supposed company...
You might think that if you looked at the first digit in randomly selected numbers that...
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) 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 201 checks to a supposed company...
You might think that if you looked at the first digit in randomly selected numbers that...
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 201 checks to a supposed company...
How many​ 7-digit telephone numbers are possible if the first digit cannot be eight and ​(a)...
How many​ 7-digit telephone numbers are possible if the first digit cannot be eight and ​(a) only even digits may be​ used? ​(b) the number must be a multiple of 10​ (that is, it must end in​ 0)? ​(c) the number must be a multiple of 1,000? ​(d) the first 2 digits are 92? ​(e) no repetitions are​ allowed?
1. What do you think the graphs might have looked like if you added strong acid...
1. What do you think the graphs might have looked like if you added strong acid (HCl) to the water and acetate buffer? Sketch a graph to illustrate your answer. 2. If you needed to prepare a buffer solution of pH 7.4, what conjugate acid/base system might you choose and why? 3. What would be the most optimal pH range for a buffer system made of TRIS-acetate buffer? Its pKa = 8.3.
a) How many 3-digit numbers are there? b) How many 3-digit numbers can you make with...
a) How many 3-digit numbers are there? b) How many 3-digit numbers can you make with all three digits different? c) How many of the numbers is part b) are odd?
Four numbers are selected without replacement from the set of 1,2,3,4,5,6,7 to form a 4 digit...
Four numbers are selected without replacement from the set of 1,2,3,4,5,6,7 to form a 4 digit number. What is the probability that the number is greater than 5432?
Write a computer program for a logic bomb that continually generates 8-digit numbers randomly and increases...
Write a computer program for a logic bomb that continually generates 8-digit numbers randomly and increases a counter by one each time. If the random number meets the current date in a format mmddyyyy, it will display 6 times on screen the following message: Today is [date]! The count is: [nnnn] Hint: Since everyday is a different date, don’t hard code the date in your program. And the [nnnn] should be the number from your counter.
Four numbers are selected without replacement from {1,2,3,4,5,6,7} to form a 4-digit number. What is the...
Four numbers are selected without replacement from {1,2,3,4,5,6,7} to form a 4-digit number. What is the probability that the number is greater than 6543?
Write a program in C++ that generates and displays the first N three digit odd numbers....
Write a program in C++ that generates and displays the first N three digit odd numbers. Whereas the number N is provided by the user.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT