
Concept explainers
One of the most surprising
Following are the populations of the 50 states in a recent census. The first digit of each population number is listed separately.
Here is a frequency distribution of the first digits of the state populations:
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.
Here is a frequency distribution of the first digits of the stock market averages:
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
Here is the probability distribution of digits as predicted by Benford’s law:
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.
ii
iii

Want to see the full answer?
Check out a sample textbook solution
Chapter 5 Solutions
Essential Statistics
- please find the answers for the yellows boxes using the information and the picture belowarrow_forwardA marketing agency wants to determine whether different advertising platforms generate significantly different levels of customer engagement. The agency measures the average number of daily clicks on ads for three platforms: Social Media, Search Engines, and Email Campaigns. The agency collects data on daily clicks for each platform over a 10-day period and wants to test whether there is a statistically significant difference in the mean number of daily clicks among these platforms. Conduct ANOVA test. You can provide your answer by inserting a text box and the answer must include: also please provide a step by on getting the answers in excel Null hypothesis, Alternative hypothesis, Show answer (output table/summary table), and Conclusion based on the P value.arrow_forwardA company found that the daily sales revenue of its flagship product follows a normal distribution with a mean of $4500 and a standard deviation of $450. The company defines a "high-sales day" that is, any day with sales exceeding $4800. please provide a step by step on how to get the answers Q: What percentage of days can the company expect to have "high-sales days" or sales greater than $4800? Q: What is the sales revenue threshold for the bottom 10% of days? (please note that 10% refers to the probability/area under bell curve towards the lower tail of bell curve) Provide answers in the yellow cellsarrow_forward
- Business Discussarrow_forwardThe following data represent total ventilation measured in liters of air per minute per square meter of body area for two independent (and randomly chosen) samples. Analyze these data using the appropriate non-parametric hypothesis testarrow_forwardeach column represents before & after measurements on the same individual. Analyze with the appropriate non-parametric hypothesis test for a paired design.arrow_forward
- Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL


