582 CHAPTER 11 Goodness-of-Fit and Contingency Tables Reject p0 5p 1 5• • • 5p 9 Fail to reject p0 5p 1 5• • • 5p 9 0 Critical Value: x2 5 16.919 Critical Region Test Statistic: x2 5 4490.174 FIGURE 11-2 Test of p0 = p1 = p2 = p3 = p4 = p5 = p6 = p7 = p8 = p9 INTERPRETATION This goodness-of-fit test suggests that the last digits do not provide a good fit with the claimed uniform distribution of equally likely frequencies. Instead of actually weighing the subjects, it appears that the subjects reported their weights. Visual examination of the frequencies in Table 11-2 reveals that the two highest frequencies correspond to the last digits of 0 and 5, and that also strongly suggests that the weights were reported instead of being measured. Because the weights are reported, the reliability of the data is questionable. Step7: If we use the P-value method of testing hypotheses, we see that the P-value is small (less than 0.0001), so we reject the null hypothesis. If we use the critical value method of testing hypotheses, Figure 11-2 shows that the test statistic falls in the critical region, so there is sufficient evidence to reject the null hypothesis. Step8: There is sufficient evidence to support the claim that the last digits do not occur with the same relative frequency. YOUR TURN. Do Exercise 5 “Heights: Measured or Reported?” Example 1 involves a situation in which the expected frequencies E for the different categories are all equal. The methods of this section can also be used when the expected frequencies are different, as in Example 2. Benford’s Law According to Benford’s law, many data sets have the property that the leading (leftmost) digits of numbers have a distribution described by the top two rows of Table 11-4 in the following example. Data sets with values having leading digits that conform to Benford’s law include numbers of Twitter followers, stock market prices, population sizes, amounts on tax returns, lengths of rivers, and check amounts. The real-world applications of Benford’s law are widespread. In the New York Times article “Following Benford’s Law, or Looking Out for No. 1,” Malcolm Browne writes that “the income tax agencies of several nations and several states, including California, are using detection software to identify fraud based on Benford’s Law. Many accounting firms and other large businesses use Benford’s law to identify corporate fraud. Which Car Seats Are Safest? Many people believe that the back seat of a car is the safest place to sit, but is it? University of Buffalo researchers analyzed more than 60,000 fatal car crashes and found that the middle back seat is the safest place to sit in a car. They found that sitting in that seat makes a passenger 86% more likely to survive than those who sit in the front seats, and they are 25% more likely to survive than those sitting in either of the back seats nearest the windows. An analysis of seat belt use showed that when not wearing a seat belt in the back seat, passengers are three times more likely to die in a crash than those wearing seat belts in that same seat. Passengers concerned with safety should sit in the middle back seat and wear a seat belt. M b b c p is s
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