Essential Statistics
Essential Statistics
2nd Edition
ISBN: 9781259570643
Author: Navidi
Publisher: MCG
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Chapter 5, Problem 1CS

One of the most surprising probability distributions found in practice is given by a rule known as Benford’s law. This probability distribution concerns the first digits of numbers. The first digit of a number may be any of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9. It is reasonable to believe that, for most sets of numbers encountered in practice, these digits would occur equally often. In fact, it has been observed that for many naturally occurring data sets, smaller numbers occur more frequently as the first digit than larger numbers do. Benford’s law is named for Frank Benford, an engineer at General Electric, who stated it in 1938.

Following are the populations of the 50 states in a recent census. The first digit of each population number is listed separately.

Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  1

Here is a frequency distribution of the first digits of the state populations:

Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  2

For the state populations, the most frequent first digit is 1, with 7, 8, and 9 being the least frequent.

Now here is a table of the closing value of the Dow Jones Industrial Average for each of the years 1974–2008.

Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  3

Here is a frequency distribution of the first digits of the stock market averages:

Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  4

For the stock market averages, the most frequent first digit by far is 1.

The stock market averages give a partial justification for Benford’s law. Assume the stock market starts at 1000 and goes up 10% each year. It will take 8 years for the average to exceed 2000. Thus, the first eight averages will begin with the digit 1. Now imagine that the average starts at 5000. If it goes up 10% each year, it would take only 2 years to exceed 6000, so there would be only 2 years starting with the digit 5. In general, Benford’s law applies well to data where increments occur as a result of multiplication rather than addition, and where there is a wide range of values. It does not apply to data sets where the range of values is small.

Here is the probability distribution of digits as predicted by Benford’s law:

Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  5

The surprising nature of Benford’s law makes it a useful tool to detect fraud. When people make up numbers, they tend to make the first digits approximately uniformly distributed; in other words, they have approximately equal numbers of 1s, 2s, and so on. Many tax agencies, including the Internal Revenue Service, use software to detect deviations from Benford’s law in tax returns.

Following are results from three hypothetical corporate tax returns. Each purports to be a list of expenditures, in dollars, that the corporation is claiming as deductions. Two of the three are genuine, and one is a fraud. Which one is the fraud?

i.    Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  6

ii    Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  7

iii    Chapter 5, Problem 1CS, One of the most surprising probability distributions found in practice is given by a rule known as , example  8

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Chapter 5 Solutions

Essential Statistics

Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 19ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 21ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 29ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 31ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 33ECh. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - Prob. 35ECh. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - Prob. 39ECh. 5.1 - 40. Fill in the missing value so that the...Ch. 5.1 - 41. Put some air in your tires: Let X represent...Ch. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - 48. Pain: The General Social Survey asked 827...Ch. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - 51. Lottery: In the New York State Numbers...Ch. 5.1 - 52. Lottery: In the New York State Numbers...Ch. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.1 - Prob. 59ECh. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.2 - 1. Determine whether X is a binomial random...Ch. 5.2 - Prob. 2CYUCh. 5.2 - Prob. 3CYUCh. 5.2 - Prob. 4CYUCh. 5.2 - Prob. 5ECh. 5.2 - In Exercises 5–7, fill in each blank with the...Ch. 5.2 - Prob. 7ECh. 5.2 - In Exercises 8–10, determine whether the statement...Ch. 5.2 - Prob. 9ECh. 5.2 - In Exercises 8–10, determine whether the statement...Ch. 5.2 - Prob. 11ECh. 5.2 - Prob. 12ECh. 5.2 - Prob. 13ECh. 5.2 - In Exercises 11–16, determine whether the random...Ch. 5.2 - Prob. 15ECh. 5.2 - In Exercises 11–16, determine whether the random...Ch. 5.2 - Prob. 17ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 19ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 22ECh. 5.2 - Prob. 23ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 25ECh. 5.2 - Prob. 26ECh. 5.2 - Prob. 27ECh. 5.2 - 28. Take another guess: A student takes a...Ch. 5.2 - Prob. 29ECh. 5.2 - Prob. 30ECh. 5.2 - Prob. 31ECh. 5.2 - 32. What should I buy? A study conducted by the...Ch. 5.2 - Prob. 33ECh. 5.2 - Prob. 34ECh. 5.2 - Prob. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Prob. 37ECh. 5.2 - 38. Stress at work: In a poll conducted by the...Ch. 5.2 - Prob. 39ECh. 5.2 - Prob. 40ECh. 5.2 - Prob. 41ECh. 5 - Prob. 1CQCh. 5 - Prob. 2CQCh. 5 - Prob. 3CQCh. 5 - Prob. 4CQCh. 5 - Prob. 5CQCh. 5 - Prob. 6CQCh. 5 - Prob. 7CQCh. 5 - Prob. 8CQCh. 5 - 9. At a cell phone battery plant, 5% of cell phone...Ch. 5 - Prob. 10CQCh. 5 - Prob. 11CQCh. 5 - Prob. 12CQCh. 5 - Prob. 13CQCh. 5 - Prob. 14CQCh. 5 - Prob. 15CQCh. 5 - Prob. 1RECh. 5 - Prob. 2RECh. 5 - Prob. 3RECh. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - 6. Lottery tickets: Refer to Exercise 5. What is...Ch. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - Prob. 11RECh. 5 - Prob. 12RECh. 5 - Prob. 13RECh. 5 - Prob. 14RECh. 5 - Prob. 15RECh. 5 - Prob. 1WAICh. 5 - Prob. 2WAICh. 5 - Prob. 3WAICh. 5 - Prob. 4WAICh. 5 - Prob. 5WAICh. 5 - Prob. 6WAICh. 5 - One of the most surprising probability...
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