Bài giảng Statistical Techniques in Business and Economics - Chapter 16 Analysis of Ranked Data

Chapter 16Analysis of Ranked Data1.Conduct the sign test for single and dependent samples using the binomial and standard normal distributions as the test statistics2.Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank testChapter GoalsWhen you have completed this chapter, you will be able to:and...3.Conduct and interpret the Wilcoxon rank-sum test for independent samplesChapter Goals4.5.6.Conduct and interpret the Kruskal-Wallis test for several independent samples Compute and interpret Spearman’s coefficient of rank correlationConduct a test of hypothesis to determine whether the correlation among the ranks in the population is different from zero TerminologyRangeis the difference between the largest and the smallest value Only two values are used in its calculation It is influenced by an extreme value It is easy to compute and understandThe Sign Test The Sign Test is based on the sign of a difference between two related observations... no assumption is necessary regarding the shape of the population of differences the binomial distribution is the test statistic for small samples and the standard normal (z) for large samples the test requires dependent (related) samples Determine the sign of the difference between related pairs Determine the number of usable pairs Compare the number of positive (or negative) differences to the critical value n is the number of usable pairs (without ties), x is the number of pluses or minuses, and the binomial probability p=.5 The Sign TestcontinuedProcedure to conduct the test:if the number of pluses or minuses is more than n/2, thenif the number of pluses or minuses is less than n/2, then Normal Approximationzxnn=--(.)..555zxnn=+-(.)..555if both and are greater than 5, the z distribution is appropriaten pn()1-pExampleThe Gagliano Research Institute for Business Studies is comparing the Research and Development expense (R&D) as a percent of income for a sample of glass manufacturing firms for 2000 and 2001At the .05 significance level has the R&D expense declined?Use the sign test Normal ApproximationExampleCompany 2000 2001 Difference SignSavoth Glass 20 16 4 +Ruisi Glass 14 13 1 +Rubin Inc.. 23 20 3 +Vaught 24 17 7 +Lambert Glass 31 22 9 +Pimental 22 20 2 +Olson Glass 14 20 - 6 -Flynn Glass 18 11 7 + Normal ApproximationExampleContinuedStep 1: H0: p=.5 H1: p than –1.96 or 1.96 or z is less than –1.96Wilcoxon Rank-Sum Test Step 3: The value of the test statistic is 0.914.12)198)(9(82)198(85.81++++-=12)1(2)1(2121211++++-=nnnnnnnWz= 0.914ContinuedWilcoxon Rank-Sum Test Step 4: The value of the test statistic is 0.914.We do not reject the null hypothesis! We cannot conclude that there is a difference in the distributions of the repair costs of the two vehicles!Kruskal-Wallis Test: Analysis of Variance by Ranksused to compare three or more samples to determine if they came from equal populationsthe ordinal scale of measurement is requiredit is an alternative to the one-way ANOVAthe chi-square distribution is the test statisticeach sample should have at least five observationsthe sample data is ranked from low to high as if it were a single groupNoteContinuedThe test statistic is given by:Kruskal-Wallis Test: Analysis of Variance by Rankscontinued)HNNRnRnRnNkk=++++ỡớợỹýỵ-+121311212222()()()...()(SSSExample... Keely Ambrose, director of Human Resources for Miller Industries, wishes to study the percent increase in salary for middle managers at the four manufacturing plants. She gathers a sample of managers and determines the percent increase in salary from last year to this year. At the 5% significance level, can Keely conclude that there is a difference in the percent increases for the various plants?Kruskal-Wallis Test: Analysis of Variance by RanksContinuedKruskal-Wallis Test: Analysis of Variance by RanksMillville Rank Camden Rank Eaton Rank Cliff Rank 2.2 2.0 1.9 1 3.7 6.0 5.7 9.0 3.6 5.0 2.7 3 4.5 7.0 6.8 10.5 4.9 8.0 3.1 4 7.1 13.5 8.9 16.0 6.8 10.5 6.9 12 9.3 17.0 11.6 18.5 7.1 13.5 8.3 15 11.6 18.5 13.9 20.0 39.0 35 62.0 74.0PlantsContinued Step 1: H0: The populations are the same H1: The populations are not the same Step 2: H0 is rejected if x2 is greater than 7.185. There are 3 degrees of freedom at the .05 significance level. Kruskal-Wallis Test: Analysis of Variance by RanksContinued)120(3574562535539)120(20122222+-ùỵùýỹùợùớỡ++++= )1(3)()()()()1(1224223222121+-ùỵùýỹùợùớỡS+S+S+S+=NnRnRnRnRNNHk= 5.949Kruskal-Wallis Test: Analysis of Variance by RanksMillville Rank Camden Rank Eaton Rank Cliff Rank 2.2 2.0 1.9 1 3.7 6.0 5.7 9.0 3.6 5.0 2.7 3 4.5 7.0 6.8 10.5 4.9 8.0 3.1 4 7.1 13.5 8.9 16.0 6.8 10.5 6.9 12 9.3 17.0 11.6 18.5 7.1 13.5 8.3 15 11.6 18.5 13.9 20.0 39.0 35 62.0 74.0ContinuedThe null hypothesis is not rejectedKruskal-Wallis Test: Analysis of Variance by RanksThere is no difference in the percent increases in the four plantsRank-Order Correlation Spearman’s coefficient of rank correlation reports the association between two sets of ranked observationsFeatures it can range from –1.00 up to 1.00 it is similar to Pearson’s coefficient of correlation, but is based on ranked dataContinuedSpearman’s Rank-Order CorrelationFormula (to find the coefficient of rank correlation)d is the difference in the ranks and n is the number of observations. rdnns=--16122S()ContinuedSpearman’s Rank-Order CorrelationTesting the significance of rs State the null hypothesis: Rank correlation in population is 0State the alternate hypothesis: Rank correlation in population is not 0.The value of the test statistic is computed from trsnrss=--21Example...Spearman’s Rank-Order Correlation The preseason football rankings for the Atlantic Coast Conference by the coaches and sports writers are shown below. What is the coefficient of rank correlation between the two groups?School Coaches WritersMaryland 2 3NC State 3 4NC 6 6Virginia 5 5Clemson 4 2Wake Forest 7 8Duke 8 7Florida State 1 1ContinuedSchool Coaches Writers d d2Maryland 2 3 -1 1 NC State 3 4 -1 1NC 6 6 0 0Virginia 5 5 0 0Clemson 4 2 2 4Wake Forest 7 8 -1 1Duke 8 7 1 1Florida State 1 1 0 0Spearman’s Rank-Order CorrelationTotal 8ContinuedSpearman’s Rank-Order CorrelationSchool Coaches Writers d d2Maryland 2 3 -1 1 NC State 3 4 -1 1NC 6 6 0 0Virginia 5 5 0 0Clemson 4 2 2 4Wake Forest 7 8 -1 1Duke 8 7 1 1Florida State 1 1 0 0Total 8= 0.905)18(8)8(612--=)1(6122-S-nnd=rsThere is a strong correlation between the ranks of the coaches and the sports writers!Test your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!This completes Chapter 16