APPENDIX D 809 3. The critical values are 8 and 19. Because G = 17 is not less than or equal to 8 nor is G = 17 greater than or equal to 19, fail to reject randomness. There is not sufficient evidence to conclude that the sequence of political parties is not random. The sequence appears to be random. 5. The median is 161.0. n1 = 15, n2 = 15, G = 16. From Table A-10, the critical values are 10 and 22. There is not sufficient evidence to reject randomness. There does not appear to be a trend of increasing or decreasing values. 7. n1 = 20, n2 = 10, G = 16, critical values: 9, 20. Fail to reject randomness. There is not sufficient evidence to reject the claim that the dates before and after July 1 are randomly selected. 9. n1 = 27, n2 = 26, G = 22, mG = 27.49057, sG = 3.603576. Test statistic: z = -1.52. P-value: 0.1276. Critical values: z = { 1.96. Fail to reject randomness. There is not sufficient evidence to reject randomness. The runs test does not test for disproportionately more occurrences of one of the two categories, so the runs test does not suggest that either conference is superior. 11. The median is 3291.0. n1 = 27, n2 = 27, G = 2, mG = 28, sG = 3.639407. Test statistic: z = -7.14. Tech: P@value = 0.0000. Critical values: z = {1.96. There is sufficient evidence to reject randomness. All of the values below the median occur at the beginning and all of the values above the median occur at the end, so there appears to be an upward trend, and the stock market appears to be a profitable investment for the long term. 13. b. The 84 sequences yield these results: 2 sequences have 2 runs, 7 sequences have 3 runs, 20 sequences have 4 runs, 25 sequences have 5 runs, 20 sequences have 6 runs, and 10 sequences have 7 runs. c. With P12 runs2 = 2>84, P13 runs2 = 7>84, P14 runs2 = 20>84, P15 runs2 = 25>84, P16 runs2 = 20>84, and P17 runs2 = 10>84, each of the G values of 3, 4, 5, 6, 7 can easily occur by chance, whereas G = 2 is unlikely because P12 runs2 is less than 0.025. The lower critical value of G is therefore 2, and there is no upper critical value that can be equaled or exceeded. d. Critical value of G = 2 agrees with Table A-10. The table lists 8 as the upper critical value, but it is impossible to get 8 runs using the given elements. Chapter 13: Quick Quiz 1. 3, 8.5, 8.5, 3, 6, 1, 5, 10, 7, 3 2. The efficiency rating of 0.91 indicates that with all other factors being the same, rank correlation requires 100 pairs of sample observations to achieve the same results as 91 pairs of observations with the parametric test for linear correlation, assuming that the stricter requirements for using linear correlation are met. 3. a. Distribution-free test b. The term “distribution-free test” suggests correctly that the test does not require that a population must have a particular distribution, such as a normal distribution. The term “nonparametric test” incorrectly suggests that the test is not based on a parameter, but some nonparametric tests are based on the median, which is a parameter; the term “distribution-free test” is better because it does not make that incorrect suggestion. Section 13-6 1. Jackpot 9 1 8 6 5 4 3 2 7 Tickets 9 1.5 8 7 5 3.5 3.5 1.5 6 3. r represents the linear correlation coefficient computed from sample paired data; r represents the parameter of the linear correlation coefficient computed from a population of paired data; rs denotes the rank correlation coefficient computed from sample paired data; rs represents the rank correlation coefficient computed from a population of paired data. The subscript s is used so that the rank correlation coefficient can be distinguished from the linear correlation coefficient r. The subscript does not represent the standard deviation s. It is used in recognition of Charles Spearman, who introduced the rank correlation method. 5. rs = -1. Critical values are -0.786 and 0.786. Reject the null hypothesis of rs = 0 (no correlation). There is sufficient evidence to support a claim of a correlation between altitude and temperature. 7. rs = -0.644. Critical values: -0.648 and 0.648. Fail to reject the null hypothesis of rs = 0. There is not sufficient evidence to support the claim of a correlation between the quality ranks and the costs. It does not appear that more expensive brands have better quality. 9. rs = 1. Critical values: -0.886, 0.886. Reject the null hypothesis of rs = 0. There is sufficient evidence to conclude that there is a correlation between overhead widths of seals from photographs and the weights of the seals. 11. rs = -0.434. Critical values: -0.700 and 0.700. Fail to reject the null hypothesis of rs = 0. There is not sufficient evidence to support the claim of a correlation between heights of winning presidential candidates and heights of their main opponents. Hopefully, we do not elect our presidents based on heights or any other physical appearance. 13. rs = 0.429. Critical values: -0.074 and 0.074. Reject the null hypothesis of rs = 0. There is sufficient evidence to support the claim of a correlation between distance of the ride and the amount of the tip. It does appear that riders base their tips on the distance of the ride. 15. rs = 0.024. Critical values: -0.207 and 0.207. Fail to reject the null hypothesis of rs = 0. There is not sufficient evidence to support the claim of a correlation between ages of Best Actresses and Best Actors at the times they won Oscars. There does not appear to be a correlation between ages of Best Actresses and Best Actors. 17. With n = 91, the number of degrees of freedom is 89. Use t = 1.987 to get critical values of -0.206 and 0.206, which are very close to the critical values of -0.207 and 0.207 found by using Formula 13-1 on page 678. Section 13-7 1. No. The runs test can be used to determine whether the sequence of political parties is not random, but the runs test does not show whether the proportion of Republicans is significantly greater than the proportion of Democrats.
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