11-1 Goodness-of-Fit 589 16. Baseball Player Births In his book Outliers, author Malcolm Gladwell argues that more baseball players have birth dates in the months immediately following July 31, because that was the age cutoff date for non-school baseball leagues. Here is a sample of frequency counts of months of birth dates of American-born Major League Baseball players starting with January: 387, 329, 366, 344, 336, 313, 313, 503, 421, 434, 398, 371. Using a 0.05 significance level, is there sufficient evidence to warrant rejection of the claim that American-born Major League Baseball players are born in different months with the same frequency? Do the sample values appear to support Gladwell’s claim? Exercises 17–20 are based on data sets included in Appendix B. The complete data sets can be found at www.TriolaStats.com. 17. Admissions for Birth Data Set 6 “Births” includes the days of the weeks that prospective mothers were admitted to a hospital to give birth. A physician claims that because many births are induced or involve cesarean section, they are scheduled for days other than Saturday or Sunday, so births do not occur on the seven different days of the week with equal frequency. Use a 0.01 significance level to test that claim. 18. Discharges After Birth Data Set 6 “Births” includes the days of the weeks that newborn babies were discharged from the hospital. A hospital administrator claims that such discharges occur on the seven different days of the week with equal frequency. Use a 0.01 significance level to test that claim. 19. M&M Candies Mars, Inc. claims that its M&M plain candies are distributed with the following color percentages: 16% green, 20% orange, 14% yellow, 24% blue, 13% red, and 13% brown. Refer to Data Set 38 “Candies” in Appendix B and use the sample data to test the claim that the color distribution is as claimed by Mars, Inc. Use a 0.05 significance level. 20. Last Digits of Weights Data Set 1 “Body Data” in Appendix B includes weights (kg) of 300 subjects. Use a 0.05 significance level to test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency. Do the results suggest that the weights were reported? Benford’s Law. According to Benford’s law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21–24, test for goodness-of-fit with the distribution described by Benford’s law. Leading Digit 1 2 3 4 5 6 7 8 9 Benford’s Law: Distribution of Leading Digits 30.1% 17.6% 12.5% 9.7% 7.9% 6.7% 5.8% 5.1% 4.6% 21. Detecting Fraud When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 15, 0, 76, 479, 183, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford’s law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodnessof-fit with Benford’s law. Does it appear that the checks are the result of fraud? 22. Author’s Check Amounts Exercise 21 lists the observed frequencies of leading digits from amounts on checks from seven suspect companies. Here are the observed frequencies of the leading digits from the amounts on 300 of the most recent checks written by the author at the time this exercise was created: 102, 45, 30, 34, 20, 27, 12, 18, 12. (Those observed frequencies correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively.) Using a 0.01 significance level, test the claim that these leading digits are from a population of leading digits that conform to Benford’s law. 23. Tax Cheating? Frequencies of leading digits from IRS tax files are 152, 89, 63, 48, 39, 40, 28, 25, and 27 (corresponding to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively, based on data from Mark Nigrini, who provides software for Benford data analysis). Using a 0.05 significance level, test for goodness-of-fit with Benford’s law. Does it appear that the tax entries are legitimate?
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