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1、Nonparametric MethodsChapter 19Nonparametric MethodsLearning Objectives1.Learn the difference between parametric and nonparametric methods.2.Know the particular advantages of nonparametric methods and when they are and when they are not applicable.3.Learn how to use the sign test for the analysis of
2、 paired comparisons.4.Be able to use the sign test to conduct hypothesis tests about a median.5.Be able to use the Wilcoxon signed-rank test and the Mann-Whitney-Wilcoxon test to determine whether or not two populations have the same distribution.6.Be able to use the Kruskal-Wallis tests for the com
3、parison of k populations.7.Be able to compute the Spearman rank correlation coefficient and test for a significant correlation between two sets of rankings.Solutions:1.Binomial Probabilities for n = 10, p = .50.xProbabilityxProbability0.00106.20511.00987.11722.04398.04393.11729.00984.205110.00105.24
4、61P(0) + P(1) = .0108; adding P(2), exceeds .025 required in the tail. Therefore, reject H0 if the number of plus signs is less than 2 or greater than 8.Number of plus signs is 7.Do not reject H0 ; conclude that there is no indication that a difference exists.2.There are n = 27 cases in which a valu
5、e different from 150 is obtained.Use the normal approximation with = np = .5(27) = 13.5 and Use x = 22 as the number of plus signs and obtain the following test statistic:With a = .01, we reject if z 2.33; since z = 3.27 2.33 we reject H0 .Conclusion: the median is greater than 150.3.a.Let p = proba
6、bility the shares held will be worth more after the splitH0 : p .50 Ha : p .50If H0 cannot be rejected, there is no evidence to conclude stock splits continue to add value to stock holdings.b.Let x be the number of plus signs (increases in value).Use the binomial probability tables with n = 18 (ther
7、e were 2 ties in the 20 cases)P(x 12) = .0482Reject H0 if the number of + signs is greater than 12.c.With x = 14, we reject H0. The results support the conclusion that stock splits are beneficial for shareholders.4.We need to determine the number who said better and the number who said worse. The su
8、m of the two is the sample size used for the study.n = .34(1253) + .29(1253) = 789.4Use the large sample test using the normal distribution. This means the value of n (n = 789.4 above) need not be integer. Hence, = .5 n = .5(789.4) = 394.7Let p = proportion of adults who feel children will have a be
9、tter future.H0: p .50Ha: p .50With x = .34(1253) = 426With a = .05, we reject H0 if z 1.645Since z = 2.23 1.645, we reject H0Conclusion: more than half of the adults feel their children will have a better future.5.n = 185 + 165 = 350 = 0.5 n = 0.5(350) = 175Reject H0 if z 1.96Do not reject H0; canno
10、t conclude there is a difference in preference for the two shows.6.n = 202 + 158 = 360 = 0.5 n = 0.5(360) = 180Reject H0 if z 1.96Reject H0; conclude Packard Bell and Compaq have different market shares.7. = 0.5 n = 0.5(300) = 150p-value = 2(.5000 - .4582) = .0836Do not reject H0; we are unable to c
11、onclude that the median annual income differs.8. = .5 n = .5(150) = 75One tailed test: reject H0 if z 1.645For 98 + signsReject H0; conclude that a home team advantage exists.9.H0: Median 15Ha: Median 15Use binomial probabilities with n = 8 and p = .50:One tail test with a = .05,P(8 +s) = .0039 P(7
12、+s) = .0312 .0351 P(6 +s) = .1094Reject H0 if 7 or 8 +s. With 7 +s in the sample, reject H0 . Data does enable us to conclude that there has been an increase in the median number of part-time employees.10.H0: Median = 152Ha: Median 152 = .5n = .5(225) = 112.5Reject H0 if z 1.96For 122 casesDo not re
13、ject H0; we are unable to conclude that the median annual income needed differs from that reported in the survey.11.n = 50 = 0.5n = 0.5(50) = 25Reject H0 if z 1.64533 had wages greater than $585Reject H0; conclude that the median weekly wage is greater than $585.12.H0: The populations are identicalHa: The populations are not identicalAdditive 1Additive 2DifferenceAbsolute ValueRankSigned Rank20.1218.052.072.079+923.5621.771.791.797+722.0322.57-.54.543-319.1517.062.092.0910+1021.2321.22.01.011+124.7723.80.97.974+416.1617.20-1.041.045-518.55