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The runs test for randomness
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10.55. Determine the number of runs, V, for each of these sequences: (a) A B A B B A A A B B A B (b) H
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H T H H H T T T T H H T H H T H T
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10.56. Twenty-five individuals were sampled as to whether they liked or did not like a product (indicated by Y and N, respectively). The resulting sample is shown by the following sequence: Y Y N N N N Y Y Y N Y N N Y N N N N N Y Y Y Y N N (a) Determine the number of runs, V. (b) Test at the 0.05 significance level whether the responses are random. 10.57. Use the runs test on sequences (10) and (11) in this chapter, and state any conclusions about randomness. 10.58. (a) Form all possible sequences consisting of two a s and one b, and give the number of runs, V, corresponding to each sequence. (b) Obtain the sampling distribution of V. (c) Obtain the probability distribution of V. 10.59. In Problem 10.58, find the mean and variance of V (a) directly from the sampling distribution, (b) by formula. 10.60. Work Problems 10.58 and 10.59 for the cases in which there are (a) two a s and two b s, (b) one a and three b s, (c) one a and four b s.
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CHAPTER 10 Nonparametric Tests
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10.61. Work Problems 10.58 and 10.59 for the cases in which there are (a) two a s and four b s, (b) three a s and three b s.
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10.62. Assuming a significance level of 0.05, determine whether the sample of 40 grades in Table 10-5 is random. 10.63. The closing prices of a stock on 25 successive days are given in Table 10-42. Determine at the 0.05 significance level whether the prices are random.
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Table 10-42
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10.375 11.625 10.875 11.875 11.375 11.125 11.250 10.750 11.375 12.125 10.875 11.375 11.500 11.875 11.750 10.625 10.750 11.250 11.125 11.500 11.500 11.000 12.125 11.750 12.250
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. 10.64. The first digits of !2 are 1.41421 35623 73095 0488 c What conclusions can you draw concerning the randomness of the digits 10.65. What conclusions can you draw concerning the randomness of the following digits (a) !3 (b) p 1.73205 08075 68877 2935 c 3.14159 26535 89793 2643 c
10.66. Work Problem 10.30 by using the runs test for randomness. 10.67. Work Problem 10.32 by using the runs test for randomness. 10.68. Work Problem 10.34 by using the runs test for randomness.
Rank correlation
10.69. In a contest, two judges were asked to rank eight candidates (numbered 1 through 8) in order of preference. The judges submitted the choices shown in Table 10-43. (a) Find the coefficient of rank correlation. (b) Decide how closely the judges agreed in their choices.
Table 10-43
First judge Second judge 5 4 2 5 8 7 1 3 4 2 6 8 3 1 7 6
10.70. The rank correlation coefficient is derived by using the ranked data in the product-moment formula of 8. Illustrate this by using both methods to work a problem. 10.71. Can the rank correlation coefficient be found for grouped data Explain this, and illustrate your answer with an example.
CHAPTER 10 Nonparametric Tests
ANSWERS TO SUPPLEMENTARY PROBLEMS
10.26. There is a difference at the 0.05 level, but not at the 0.01 level. 10.28. The program is effective at the 0.05 level. 10.29. We can reject the hypothesis of increased sales at the 0.05 level. 10.31. (a) Reject. (b) Accept. (c) Accept. (d) Reject. 10.34. There is no difference at the 0.05 level. 10.36. (a) Yes. (b) Yes. 10.41. 3 10.42. 6 10.37. Yes. 10.35. No. 10.30. No. 10.27. Yes.
10.38. (a) Yes. (b) Yes.
10.49. There is no significant difference at either level.
10.50. The difference is significant at the 0.05 level, but not at the 0.01 level. 10.51. The difference is significant at the 0.05 level, but not at the 0.01 level. 10.52. There is a significant difference between the grades at both levels. 10.55. (a) 8. (b) 10. 10.56. (a) 10. (b) The responses are random at the 0.05 level.
10.62. The sample is not random at the 0.05 level. There are too many runs, indicating a cyclic pattern. 10.63. The sample is not random at the 0.05 level. There are too few runs, indicating a trend pattern. 10.64. The digits are random at the 0.05 level. 10.65. (a) The digits are random at the 0.05 level. (b) The digits are random at the 0.05 level. 10.69. (a) 0.67. (b) The judges did not agree too closely in their choices.