Benford’s Law, Part I Our number system consists of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9 because we do not write numbers such as 12 as 012. Although we may think that each first digit appears with equal frequency so that each digit has a 1/9 probability of being the first significant digit, this is not true. In 1881, Simon Newcomb discovered that first digits do not occur with equal frequency. This same result was discovered again in 1938 by physicist Frank Benford. After studying much data, he was able to assign probabilities of occurrence to the first digit in a number as shown.
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Source: T. P. Hill, “The First Digit Phenomenon,” American Scientist, July—August, 1998.
The probability distribution is now known as Benford’s Law and plays a major role in identifying fraudulent data on tax returns and accounting books. For example, the following distribution represents the first digits in 200 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
a. Because these data are meant to prove that someone is guilty of fraud, what would be an appropriate level of significance when performing a goodness-of-fit test?

"In Problems 7–10, determine (a) the chi-square test statistic.

H0: pA=pB=pC=pD=pE=1/5

H1: At least one of the proportions is different from the others.

"In Problems 7–10, determine (b) the degrees of freedom.

H0: pA=pB=pC=pD=pE=1/5

H1: At least one of the proportions is different from the others.

"In Problems 7–10, determine (c) the critical value using.

H0: pA=pB=pC=pD=pE=1/5

H1: At least one of the proportions is different from the others.

"In Problems 7–10, determine (d) test the hypothesis at the level of significance.

H0: pA=pB=pC=pD=pE=1/5

H1: At least one of the proportions is different from the others.

[DATA] Putting It Together: The V-2 Rocket in London In Thomas Pynchon’s book Gravity Rainbow, the characters discuss whether the Poisson probabilistic model can be used to describe the locations that Germany’s feared V-2 rocket would land in. They divided London into 0.25-km2 regions. They then counted the number of rockets that landed in each region, with the following results:

b. Explain why the requirements for conducting a goodness-of-fit test are not satisfied.

Explain why chi-square goodness-of-fit tests are always right tailed.

If the expected count of a category is less than 1, what can be done to the categories so that a goodness-of-fit test can still be performed?

Many municipalities are passing legislation that forbids smoking in restaurants and bars. Bar owners claim that these laws hurt their business. Are their concerns legitimate? The following data represent the smoking status and frequency of visits to bars from the General Social Survey. Do smokers tend to spend more time in bars? Use the α = 0.05 level of significance.