2. The physicist Frank Benford noticed in the 1930s that, for many sets of numbers in the real world, the leading digit* of the number is more likely to be a small number than a larger one (i.e., 1’s and 2’s are more common than 8’s and 9’s). In fact, many wide-ranging number sets obey “Benford’s Law,” which says that 1’s are about 30.1% of the leading digits, 2’s are 17.6%, 3’s are 12.5%, 4’s are 9.7%, 5’s are 7.9%, 6’s are 6.7%, 7’s are 5.8%, 8’s are 5.1%, and 9’s are 4.6%. Benford’s Law has been used to identify accounting fraud and made-up scientific data. *leading digit is the first significant digit; e.g., “1” in the number 132, “2” in the number .0026.S uppose I suspect my research collaborator of faking a dataset that represents the price that people say is the average amount they pay each month in health care deductibles and copayments, rounded to the nearest dollar.10 50 15 11 412 80 48 13 4250 10 68 41 790 15 139 147 1433 61 145 7 406 99 50 27 7513 40 3 21 1475 203 148 27 2423 13 10 6 13626 39 80 30 12 Check if my colleague is possibly faking data by testing whether his numbers conform to Benford’s Law or not. a) What is the null and alternative hypothesis, in words?b) Generate the expected frequencies. Do the data violate the assumptions of thetest? Explain why or why not.c) If you find the data don’t violate the assumptions, perform the test, generating the 2 statistic. If you do find the data violate the assumptions, lump the data in a way that permits you to perform the test. (Lumping = combining into different categories; if you do this, you should obey common sense about what categories get combined together, and do the least amount of lumping necessary to meet the test assumptions.)d) What are the degrees of freedom for the test? What is the critical value of 2 ? e) Report your complete finding, with statistical details in parentheses, in a plain English sentence.
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
2. The physicist Frank Benford noticed in the 1930s that, for many sets of numbers in the real world, the leading digit* of the number is more likely to be a small number than a larger one (i.e., 1’s and 2’s are more common than 8’s and 9’s). In fact, many wide-
*leading digit is the first significant digit; e.g., “1” in the number 132, “2” in the number .0026.
S
uppose I suspect my research collaborator of faking a dataset that represents the price that people say is the average amount they pay each month in health care deductibles and copayments, rounded to the nearest dollar.
10 50 15 11 41
2 80 48 13 42
50 10 68 41 7
90 15 139 147 14
33 61 145 7 40
6 99 50 27 75
13 40 3 21 14
75 203 148 27 24
23 13 10 6 136
26 39 80 30 12
Check if my colleague is possibly faking data by testing whether his numbers conform to Benford’s Law or not.
a) What is the null and alternative hypothesis, in words?
b) Generate the expected frequencies. Do the data violate the assumptions of the
test? Explain why or why not.
c) If you find the data don’t violate the assumptions, perform the test, generating the
2 statistic. If you do find the data violate the assumptions, lump the data in a way that permits you to perform the test. (Lumping = combining into different categories; if you do this, you should obey common sense about what categories get combined together, and do the least amount of lumping necessary to meet the test assumptions.)
d) What are the degrees of freedom for the test? What is the critical value of 2 ?
e) Report your complete finding, with statistical details in parentheses, in a plain English sentence.
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