692 CHAPTER 13 Nonparametric Tests 2. Notation Identify the values of n1, n2, and G that would be used in the runs test for randomness. 3. Runs Test If we use a 0.05 significance level to test for randomness, what are the critical values from Table A-10? Based on those values and the number of runs from Exercise 2, what should be concluded about randomness? 4. Good Sample? Given the sequence of data, if we fail to reject randomness, does it follow that the sampling method is suitable for statistical methods? Explain. Using the Runs Test for Randomness. In Exercises 5–8, use the runs test with a significance level of A = 0.05. (All data are listed in order by row.) 5. Law Enforcement Fatalities Listed below are numbers of law enforcement fatalities for recent and consecutive years. First find the median, and then test for randomness above and below the median. Is there a trend? 196 195 162 149 163 158 179 184 140 174 172 146 162 242 159 151 167 166 160 203 159 139 171 185 141 120 148 160 159 129 6. Odd and Even Digits in Pi A New York Times article about the calculation of decimal places of p noted that “mathematicians are pretty sure that the digits of p are indistinguishable from any random sequence.” Given below are the first 25 decimal places of p. Test for randomness in the way that odd (O) and even (E) digits occur in the sequence. Based on the result, does the statement from the New York Times appear to be accurate? 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 7. Draft Lottery In 1970, a lottery was used to determine who would be drafted into the U.S. Army. The 366 dates in the year were placed in individual capsules, they were mixed, and then capsules were selected to identify birth dates of men to be drafted first. The first 30 results are listed below. Test for randomness before and after the middle of the year, which is July 1. Sept. 14 Apr. 24 Dec. 30 Feb. 14 Oct. 18 Sept. 6 Oct. 26 Sept. 7 Nov. 22 Dec. 6 Aug 31 Dec. 7 July 8 Apr. 11 July 12 Dec. 29 Jan. 15 Sept. 26 Nov. 1 June 4 Aug. 10 June 26 July 24 Oct. 5 Feb. 19 Dec. 14 July 21 June 5 Mar. 2 Mar. 31 8. CD Sales Listed below are amounts of CD sales in billions of dollars (based on data from the Recording Industry Association of America). The amounts are listed in order by year beginning with the year 1984. First find the mean, identify each value as being above the mean (A) or below the mean (B), then test for randomness above and below the mean. Is there a trend? 0.1 0.4 0.9 1.6 2.1 2.6 3.5 4.3 5.3 6.5 8.5 9.4 9.9 9.9 11.4 12.8 13.2 12.9 12.0 11.2 11.4 10.5 9.4 7.5 5.5 4.3 3.4 3.1 2.3 2.1 1.8 1.4 1.1 1.1 0.9 Runs Test with Large Samples. In Exercises 9–12, use the runs test with a significance level of A = 0.05. (All data are listed in order by row.) 9. Testing for Randomness of Super Bowl Victories Listed below are the conference designations of teams that won the Super Bowl, where N denotes a team from the NFC and A denotes a team from the AFC. Do the results suggest that either conference is superior? NNAAANAAAAANAAANNANNNNNNNNN NNNNAANAANAAAANANNNANAAANA
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