When looking at a collection of numbers, such as population sizes, the figures on tax returns or the charges on a credit card, one may think that the leading digit of the number is equally likely to be any of the numbers from 1 to 9. However, such data sets often show a distribution for the leading digit that is quite different from uniform and is described by Benford€™s Law. This law stipulates that the probability for the leading digit to be 1 is 30.1%, whereas the probability of the leading digit to be 9 is only 4.6% instead of a uniform 1/9 = 11.1% probability for each. The following table shows the distribution of the leading digit of the last 130 transactions on the credit card of one of the authors. The raw data file is available for download on the book€™s website.

a. Find the expected counts, assuming the leading digits for the transactions follow Benford€™s Law.
b. Is there evidence that the distribution of the leading digit does not follow Benford€™s Law? Write a conclusion based on an appropriate test. If available, use software to read in the raw data and compute the test statistic and P-value.

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Leading digit: 6 7 8 9 Total 2 3 5 4 Benford probabilities (%): 30.1 17.6 12.5 9.7 7.9 6.7 5.8 5.1 4.6 100 24 Observed counts: 7 130 39 6 10 11 20 7