Benford’s Law, introduced in Chapter 3 Exercise 19, postulates that the lead digits (1, 2, …, 9) in a large data set should follow the rule 

(So, for example, the proportion of numbers with a leading 1 is predicted to be log₁₀(2) ≈ .3, and the probabilities decrease as d increases.) The author of the article “Benford’s Law Applies to Online Social Networks” (PLoS One 2015) used Twitter’s API to randomly sample 78,226 Twitter users and record the number of followers each person has. The lead digits of those counts are summarized below.

Does the data indicate that the lead digits of the variable “number of Twitter followers” indeed conforms to Benford’s Law? Test at the .05 significance level.

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Suppose that you read through this year's issues of the New York Times and record each number that appears in a news article the income of a CEO, the number of cases of wine produced by a winery, the total char- itable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be 1, 2, ..., 8, or 9. Your first thought might be that the leading digit X of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical argu- ments suggest an alternative probability distribution called Benford's law: