Ch18 Nonparametric Methods Analysis of Ranked Data.docx

Chapter 18 Nonparametric Methods: Analysis of Ranked Data True/False 1. Three tests—rank correlation, Kruskal-Wallis and Wilcoxon signed rank test—require that the data be at least ordinal (ranked) level of measurement. Answer: True 2. We can apply non-parametric tests to problems involving interval or ratio data. Answer: True 3. We can apply parametric tests, such as the t test, to ordinal or ranked level of measurement. Answer: False 4. If we have interval or ratio level of measurement and if certain specified assumptions are met, such as normality of the population, a parametric test should be performed. Answer: True 5. If we have interval or ratio level of measurement and if the assumptions of normality and other assumptions cannot be met, a distribution free test should be used. Answer: True 6. When testing the null hypothesis, Ho: ? = 100 with a sample of 150 observations, the central limit theorem supports a parametric test based on a z-statistic. Answer: True 7. When testing the null hypothesis, Ho: ? = 100 with a sample of 15 observations, and the knowledge that the population is normally distributed, a parametric test based on a t-statistic should be used. Answer: True 8. When testing the null hypothesis, with a sample of 15 observations, and no valid reason to believe that the population distribution is normally distributed, then nonparametric methods should be applied. Answer: True 9. The Kruskal-Wallis one-way analysis of variance by ranks is especially appropriate to test whether three or more population means are equal if the data is measured with an ordinal scale and/or the populations are not normal. Answer: True 10. To apply the Kruskal-Wallis test, the samples selected from the populations must be dependent. Answer: False 11. For the Kruskal-Wallis test, all the sample values are combined, combined values are ordered from low to high and ordered values are replaced by ranks starting with 1 for the smallest value. Answer: True 12. Samples selected from the populations must be independent to apply the Kruskal-Wallis test. Answer: True 13. The test statistic for the Kruskal-Wallis test is designated H and the distribution of the sample H statistic is very close to the chi-square distribution with k – 1 degrees of freedom if every sample size is at least five. Answer: True 14. The Kruskal-Wallis test is a distribution-free test. Answer: True AACSB: REF 15. If we cannot assume equal population standard deviations, the Kruskal-Wallis test is substituted for ANOVA. Answer: True 16. The Wilcoxon signed rank test can replace the paired t test when the assumptions for t cannot be met. Answer: True Difficulty: Easy 17. The Wilcoxon signed rank test of differences requires that the data be at least ordinal scaled and that the two samples are related. Answer: True Difficulty: Easy 18. The paired t test is used if we assume that the differences between two sets of paired observations approximate a normal distribution. Answer: True 19. The sign test is an appropriate nonparametric test for dependent samples. Answer: True 20. For a sign test, if the alternate hypothesis is , then the null hypothesis is Answer: True 21. For small samples, the test statistic for the sign test is the normal distribution. Answer: False 22. For small samples, the test statistic for the sign test is the binomial distribution. Answer: True 23. To apply the sign test, at least an interval level of measurement is required. Answer: False 24. The sign test is more efficient than the Wilcoxon signed-rank test. Answer: False 25. The sign test ignores the numerical difference between two numbers. Answer: True 26. The sign test is based on the sign of the difference between two numbers. Answer: True 27. The Wilcoxon signed rank test requires that the two samples be at least at the ordinal level of measurement and that the data be paired. Answer: True 28. If we want to find out if the median income of senior citizens is $17,680, the null hypothesis is: Ho: Median = $17,680. Answer: True 29. The rank correlation test should be applied to test a hypothesis about a median. Answer: False 30. Nonparametric tests require that the sampled populations are normally distributed and that the observations be at least interval scale. Answer: False 31. Spearman's rank-order correlation coefficient may assume a value from –1 to +1. Answer: True 32. A Spearman's rank-order correlation coefficient of 0.91 indicates a very weak relationship. Answer: False Difficulty: Easy 33. The alternate hypothesis states that the correlation between two populations is greater than zero. The test of significance is one-tailed. Answer: True Goal: 6 34. A Spearman's rank-order correlation is used when we can assume that two populations are normally distributed. Answer: False Difficulty: Easy 35. A Spearman's rank-order correlation coefficient is inversely related to the sum of square differences in the ranks of the paired observations. Answer: True Difficulty: Easy Multiple Choice 36. To determine whether four population means are equal, a sample from each population was selected at random and using the Kruskal-Wallis test, H was computed to be 2.09. What is our decision at the 0.05 level of risk? A) Fail to reject the null hypothesis because 0.05 < 2.09 B) Fail to reject the null hypothesis because 2.09 < 7.815 C) Reject the null hypothesis because 7.815 is > 2.09 D) Reject the null hypothesis because 2.09 > critical value of 1.96 Answer: B 37. A one-tailed sign test is applied to a set of paired data. There were 14 pairs and the 0.10 level of significance was selected. There were 9 pluses and 5 minuses. What is the decision regarding the null hypothesis? A) Fail to reject it B) Reject it C) Neither reject it nor fail to reject it based on information given D) None of the above Answer: A 38. If a sign test is applied, what is the null hypothesis for a two-tailed test? A) B) is not equal to 0.50 C) D) Answer: A 39. What is the test statistic for the sign test if the sample size is large? A) Chi-square B) Standard normal z C) Binomial D) Wilcoxon signed-rank Answer: B 40. For the sign test large-sample procedure, the sample size is at least A) 2 B) 8 C) 100 D) 10 Answer: D 41. To use the sign test with paired data, successive trials must be: A) Independent B) Dependent C) Both dependent and independent at the same time D) Equal to an a of 0.05 Answer: A 42. What is a null hypothesis for a one-tail sign test? A) B) C) D) Answer: D 43. What is the first step in applying the sign test? A) Rank the paired data from high to low or vice versa B) Apply a correction factor to each observation C) Determine sign of difference between each observed pair of data D) Determine the number of degrees of freedom Answer: C 44. What is the decision rule for small sample sign test? A) Fail to reject H0 if computed B) Reject Ho if computed does not equal 0.50 C) Fail to reject H0 if the computed value of chi-square > 11.070 D) Fail to reject H0 if z = 0.50 E) None of the above Answer: E Scrambling: Locked 45. Which statement is true when using the sign test and a large sample? A) Chi-square value is applied to arrive at the degrees of freedom B) A normal distribution should be used to approximate a binomial distribution by including a continuity correction factor C) Wilcoxon Z is used as the test statistic D) All of the above are correct Answer: B Scrambling: Locked 46. If a test is to be used to test H0: Median equals $12.40 and the alternate hypothesis: Median does not equal $12.40, what is the critical value of z at the 10% level of significance? A) ± 1.96 B) ± 1.65 C) ± 2.33 D) ±0.10 Answer: B 47. A two-tailed test of hypothesis was conducted and 60 values were above the median. The computed z = 1.61. At the 0.05 level, what is the decision regarding the null hypothesis? A) Reject it with a 0.05 Type I error rate B) Fail to reject it C) Reject it with a 0.025 Type I error rate D) None of the above Answer: B Scrambling: Locked 48. If there are three or more populations with ordinal data and we wish to test whether the distributions are equal, what is the appropriate test? A) z B) t C) ANOVA D) Kruskal-Wallis Answer: D 49. What test is an appropriate substitute for the ANOVA procedure if the assumptions for ANOVA cannot be met? A) z B) t C) Rank correlation D) Kruskal-Wallis Answer: D 50. If the assumptions for the paired t test cannot be met, what is the nonparametric alternative test? A) Rank correlation B) Kruskal-Wallis C) Wilcoxon signed rank D) Median Answer: C 51. Which of the following tests is not a distribution-free test? A) Rank correlation B) Student's t C) Kruskal-Wallis D) Wilcoxon Answer: B 52. This test is used when the assumptions for the parametric analysis of variance (ANOVA) cannot be met. Its purpose is to test whether three or more populations are equal. The data must be at least ordinal scaled. What nonparametric test is this? A) Students' t B) Kruskal-Wallis C) Mann-Whitney D) ANOVA Answer: B 53. This nonparametric test requires at least ordinal level data. Its purpose is to find out if there is any difference between two sets of related observations and it is used if the assumptions required for the paired t test cannot be met. Which test is it? A) Students' t B) ANOVA C) Rank correlation D) Kruskal-Wallis E) None of the above Answer: E Goal: 3 Scrambling: Locked 54. Which of the following statements is true regarding the sign test? A) It is based on the sign of the difference B) It is used in "before" and "after" experiments C) The binomial distribution is used as the test statistic. D) All of the above E) None of the above Answer: D 55. What is the distribution for Spearman's rank correlation? A) t B) z C) F D) Chi-square Answer: A 56. What is a requirement that must be met before the Kruskal-Wallis one-way analysis of variance by ranks test can be applied? A) Populations must be normal or near normal B) Samples must be independent C) Population standard deviations must be equal D) Data must be at least interval level E) None of the above Answer: B 57. The distribution of the sample statistic (H) for the Kruskal-Wallis test is approximately equal to A) F. B) t. C) chi-square. D) standard normal. Answer: C 58. Under what conditions would the Kruskal-Wallis test be chosen over ANOVA when the sample data is at least interval level? A) Assumption concerning the population being normal cannot be made B) ANOVA is too difficult to calculate C) ANOVA formulas are overly precise D) ANOVA is not helpful for unequal sample sizes Answer: A 59. The scale of measurement for the coefficient of rank correlation is the A) nominal. B) ordinal. C) interval. D) ratio. Answer: B 60. Which of the following values of Spearman's (rho) indicates the strongest relationship between two variables? A) –0.91 B) –0.05 C) +0.64 D) +0.89 E) 0 Answer: A 61. Suppose ranks are assigned to a set of data from low to high with $10 being ranked 1, $12 being ranked 2 and $21 being ranked 3. What ranks would be assigned to $26, $26 and $26? A) 4, 5, 6 B) 4, 4, 4 C) 5, 5, 5 D) 5.5, 5.5, 5.5 Answer: C 62. For a given set of twelve ranked data values, the sum of the squared differences is 63.18. What is Spearman's coefficient of rank correlation for the data? A) +.7791 B) –.7791 C) +.2209 D) –.2209 Answer: A 63. A hypothesis test that rho is zero is conducted at the 5% level of significance. If Spearman's (rho) is 0.86 for a sample of 15 observations, what is the computed value of the t statistic? A) 0.456 B) 6.08 C) 0.425 D) 2.16 Answer: B Fill-in-the-Blank 64. What kind of sign test should be applied if and the level of significance is 0.05?___________________ Answer: one-tailed 65. Which test requires that the observations be paired? _______________ Answer: Wilcoxon sign rank or sign test 66. Which test do we apply to test whether three or more population means are equal and the data nonparametric?_____________________ Answer: Kruskal-Wallis 67. In which test must the samples be dependent? ___________________ Answer: Wilcoxon signed-rank or sign test 68. If two movie critics ranked the current week's releases in opposite orders, what would Spearman's rank correlation be? _____________ Answer: -1 69. Which test statistic follows the chi-square distribution? __________ Answer: Kruskal-Wallis H statistic 70. Which test is used for before and after situations? _______________ Answer: sign test 71. Which test pairs the data? ______________ Answer: Wilcoxon or sign test 72. In testing the significance of Spearman's rank-order coefficient, what is the null hypothesis? _______________ Answer: 73. In testing the significance of Spearman's rank-order coefficient, what is the alternate hypothesis? ______________________ Answer: 74. What is the value of the Spearman rank-order correlation coefficient that indicates no association between the two variables? _____ Answer: zero Multiple Choice Use the following to answer questions 75-77: A study was conducted on the percent of total advertising dollars spent by ten local firms for advertising in the press and on cable television. Results were ranked with a resulting sum of squared differences equal to 128. 75. What is Spearman's coefficient of rank correlation? A) –0.871 B) +0.224 C) +0.234 D) –0.234 Answer: B 76. If = 0.345, then the computed value of t is A) 0.655. B) 0.968. C) 1.206. D) 1.0396. Answer: D 77. What is the sum of the differences in ranks? A) 128 B) 100 C) 0 D) 1 Answer: C Fill-in-the-Blank Use the following to answer questions 78-85: A soap manufacturer is experimenting with several formulas of soap powder and three of the formulas were selected for further testing by a panel of homemakers. The ratings for the three formulas are as follows: 78. The soap manufacturer decided to use the Kruskal-Wallis test. What is the null hypothesis? ___________________________________ Answer: No difference in the ratings 79. What is the number of degrees of freedom? _______ Answer: 2 80. What is the value of chi-square at the 5% level of significance? ________ Answer: 5.991 81. What is the sum of the ranks for Formula A? _______ Answer: 42 82. What is the sum of the ranks for Formula B? _______ Answer: 52 83. What is the sum of the ranks for Formula C? _______ Answer: 77 84. What is the value of H? _______ Answer: 3.801 85. If H is less than the critical value, what is our decision rule? __________________________________ Answer: Do not reject H0; no difference in mean rating scores Use the following to answer questions 86-88: In a market research study involving eleven people, seven preferred Product X, three preferred Product Y and one expressed no preference. A sign test is selected using the 0.05 level of significance. 86. What is the null hypothesis? ___________________ Answer: 87. What is the alternate hypothesis? _________________ Answer: 88. What is the decision rule if the test statistic is the number of people preferring Product X? _______________________________ Answer: Do not reject H0 if 2-8 people prefer X; else, reject H0 Use the following to answer questions 89-94: Two movie reviewers gave their ratings (0 to 4 stars) to seven movies released this past month as follows: The Wilcoxon signed-rank test of difference is to be used. 89. What is the null hypothesis? ___________________________________ Answer: No difference in the movie ratings 90. What is the alternate hypothesis? ______________________________ Answer: There is a difference in the movie ratings 91. What is the sum of the ranks? _______ Answer: 23 92. What is the critical value at the 5% level of significance? ______ Answer: 3 93. What is the test statistic? ______ Answer: 5 94. What is your decision at the 5% level of significance? __________ Answer: Do not reject H0 Use the following to answer questions 95-98: Two movie reviewers gave their ratings (0 to 4 stars) to seven movies released this past month as follows: The Wilcoxon rank-sum test is to be used. Assume 4 is ranked first. 95. What is the sum of the ranks for S? _____________________ Answer: 46 96. What is the sum of the ranks for T? _____________________ Answer: 59 97. What is W if S is identified as population #1? ________________ Answer: 46 98. What is the value of calculated z? __________________ Answer: -0.83 Use the following to answer questions 99-107: Two movie reviewers gave their ratings (0 to 4 stars) to ten movies released this past month as follows: The rank order correlation is to be used. 99. What is the value of d? __________ Answer: 0 100. What is the value of ? __________ Answer: 48 101. What is the rank order correlation? __________ Answer: 0.7091 102. What is the null hypothesis for rank order correlation? __________ Answer: 103. What is the alternate hypothesis for rank order correlation? ________ Answer: 104. What are the degrees of freedom? ________ Answer: 8 105. What is the critical value at a = 0.05? ________ Answer: 2.306 106. What is the computed value of t? ________ Answer: 2.844 107. What is your decision? ________ Answer: Reject H0 Multiple Choice Questions Use the following to answer questions 108-112: The following are the ratings (0 to 4) given by 12 individuals for two possible new flavors of soft drinks. Wilcoxon rank-sum is to be used. Assume that the "0" is ranked first. 108. What is the sum of the ranks for flavor #1? A) 144 B) 139 C) 156 D) 153 Answer: A 109. What is the sum of the ranks for flavor #2? A) 153 B) 139 C) 144 D) 156 Answer: D 110. What is W if flavor #1 is identified as population 1? A) 153 B) 156 C) 144 D) 139 Answer: C 111. What is the absolute value of calculated z? A) 0.3464 B) 0.1732 C) 0.0165 D) 0.2807 Answer: A 112. At the 0.05 level of significance, what is the decision? A) Fail to reject null hypothesis; critical value is ?1.65 B) Fail to reject null hypothesis; critical value is ?1.96 C) Reject null hypothesis; critical value is 0.1732 D) Reject null hypothesis; critical value is 0.3464 Answer: B Use the following to answer questions 113-116: 20 economists were sampled and asked to predict if the national economy would improve during the next twelve months. Eleven of the economists predicted an increase, two economists predicted no change, and seven economists predicted a decrease in the economy. Conduct a hypothesis test at the 0.10 significance level to determine if the majority of economists predict an increase. 113. The null hypothesis is: A) B) C) D) Answer: C 114. The correct analysis would be: A) A sign test based on the binomial distribution B) A sign test based on the standard normal distribution C) A chi square test D) A Wilcoxon signed rank test Answer: B 115. The test statistic is: A) 0.707 B) 1.179 C) 1.707 D) 0.179 Answer: A 116. Based on the analysis, we would conclude that: A) The economists favor an increase in the economy B) The economists favor a decrease in the economy C) The economists favor no change in the economy D) No conclusion can be reached. Answer: C Essay 117. What is the difference between chi-square nonparametric tests and nonparametric tests based on ranks? Answer: Chi-square tests are appropriate for nominal and ordinal data that can be classified. Nonparametric tests based on ranks can be applied to interval and ratio data. Goal: any 118. When are nonparametric methods best applied? Answer: When we can not make valid assumptions about the distribution of the data. Goal: any 119. Why is the nonparametric, sign test appropriate for testing a hypothesis about the median? Answer: We cannot make a valid assumption that the distribution of the population median is normally distributed.