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16- 2 Chapter Sixteen Nonparametric Methods: Analysis of Ranked Data GOALS When you have completed this chapter, you will be able to: ONE Conduct the sign test for dependent samples using the binomial and standard normal distributions as the test statistics. TWO Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test. THREE Conduct and interpret the Wilcoxon rank-sum test for independent samples. FOUR Conduct and interpret the Kruskal-Wallis test for several independent samples.

16- 3 Chapter Sixteen GOALS When you have completed this chapter, you will be able to: FIVE Compute and interpret Spearman’s coefficient of rank correlation. SIX Conduct a test of hypothesis to determine whether the correlation among the ranks in the population is different from zero. continued Nonparametric Methods: Analysis of Ranked Data Goals

16- 4 The test requires dependent (related) samples. The Sign Test No assumption is necessary regarding the shape of the population of differences. Based on the sign of a difference between two related observations. The binomial distribution is the test statistic for small samples and the standard normal (z) for large samples.

16- 5 The Sign Test continued n is the number of usable pairs (without ties), X is the number of pluses or minuses, and the binomial probability p=.5. Determine the sign of the difference between related pairs. The Sign Test Determine the number of usable pairs. Compare the number of positive (or negative) differences to the critical value

16- 6 Normal Approximation If both n  and n(1-  ) are greater than 5, the z distribution is appropriate. Normal Approximation If the number of pluses or minuses is more than n/2, then If the number of pluses or minuses is less than n/2, then

16- 7 Example 1 The 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 2001. At the.05 significance level has the R&D expense declined? Use the sign test.

16- 8 Example 1 Company 20002001 DifferenceSign Savoth Glass20164+ Ruisi Glass14131+ Rubin Inc.23203+ Vaught24177+ Lambert Glass31229+ Pimental22202+ Olson Glass1420-6- Flynn Glass18117+

16- 9 Example 1 Step 2: H 0 : is rejected if the number of negative signs is 0 or 1. Step 1 H 0 : p >.5 H 1 : p <.5 Step 3: There is one negative difference. That is there was an increase in the percent for one company Step 4: H 0 is rejected. We conclude that R&D expense as a percent of income declined from 2000 to 2001.

16- 10 Testing a Hypothesis About a Median When testing the value of the median, we use the normal approximation to the binomial distribution. The z distribution is used as the test statistic

16- 11 Example 2 The Gordon Travel Agency claims that their median airfare for all their clients to all destinations is \$450. This claim is being challenged by a competing agency, who believe the median is different from \$450. A random sample of 300 tickets revealed 170 tickets were below \$450. Use the 0.05 level of significance.

16- 12 Example 2 Continued H 0 is rejected. We conclude that the median is not \$450. H 0 is rejected if z is less than –1.96 or greater than 1.96. The value of z is 2.252.

16- 13 Wilcoxon Signed-Rank Test The observations must be related or dependent. Use if assumption of normality is violated for the paired-t test Wilcoxon Signed-Rank Test Requires the ordinal scale of measurement

16- 14 Wilcoxon Signed-Rank Test Compare the smaller of the two rank sums with the T value, obtained from Appendix H. Wilcoxon Signed-Rank Test Compute the differences between related observations. Rank the absolute differences from low to high. Return the signs to the ranks and sum positive and negative ranks.

16- 15 Example 3 Step 2 H 0 is rejected if the smaller of the rank sums is less than or equal to 5. See Appendix H. From Example 1 Have R&D expenses declined as a percent of income? Use.05 significance level. Step 1: H 0 : The percent stayed the same. H 1 : The percent declined.

16- 16 Example 3 Continued The smaller rank sum is 5, which is equal to the critical value of T. H0 is rejected. The percent has declined from one year to the next. Company 2000 2001 Difference ABS-Diff RankR+R- Savoth Glass20164444* Ruisi Glass14131111* Rubin Inc.23203333* Vaught24177777* Lambert Glass31229988* Pimental22202222* Olson Glass1420-665*5 Flynn Glass18117766*

16- 17 Wilcoxon Rank-Sum Test Each sample must contain at least eight observations Wilcoxon Rank-Sum Test No assumption about the shape of the population is required The data must be at least ordinal scale Used to determine if two independent samples came from the same or equal populations

16- 18 Wilcoxon Rank-Sum Test Rank all data values from low to high as if they were from a single population Determine the sum of ranks for each of the two samples Use the smaller of the two sums W to compute the test statistic Wilcoxon Rank-Sum Test

16- 19 Example 4 Hills Community College purchased two vehicles, a Ford and a Chevy, for the administration’s use when traveling. The repair costs for the two cars over the last three years is shown on the next slide. At the.05 significance level is there a difference in the two distributions?

16- 20 Example 4 Ford (\$)Rank Chevy(\$)Rank 25.31 3.0 14.89 1.0 33.68 5.5 20.31 2.0 46.89 7.0 25.97 4.0 51.83 8.0 33.68 5.5 87.6513.0 68.98 9.0 87.9014.0 78.2310.0 90.8915.0 80.3111.0 120.6716.0 81.7512.0 157.9017.0 81.50 71.5

16- 21 Example 4 Step 2: H 0 is rejected if z >1.96 or z is less than –1.96 Step 1: H 0 : The populations are the same. H 1 : The populations are not the same. Step 3: The value of the test statistic is 0.914. Step 4: 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.

16- 22 Kruskal-Wallis Test: Analysis of Variance by Ranks Kruskal-Wallis Test Analysis of Variance by Ranks Used to compare three or more samples to determine if they came from equal populations The ordinal scale of measurement is required It is an alternative to the one-way ANOVA The chi-square distribution is the test statistic Each sample should have at least five observations The sample data is ranked from low to high as if it were a single group

16- 23 Kruskal-Wallis Test: Analysis of Variance by Ranks continued Test Statistic Kruskal-Wallis Test Analysis of Variance by Ranks

16- 24 Example 5 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?

16- 25 EXAMPLE 5 continued Ranked Increases in Managers’ Salaries

16- 26 There is no difference in the percent increases in manager salaries in the four plants. Step 1: H 0 : The populations are the same. H 1 : The populations are not the same. Step 2: H 0 is rejected if  is greater than 7.185. There are 3 degrees of freedom at the.05 significance level. The null hypothesis is not rejected.

16- 27 Rank-Order Correlation Spearman’s Coefficient of Rank Correlation Reports the association between two sets of ranked observations Similar to Pearson’s coefficient of correlation, but is based on ranked data. Ranges from –1.00 up to 1.00 d is the difference in the ranks and n is the number of observations

16- 28 Testing the Significance of r s H 1 : Rank correlation in population is not 0. H o : Rank correlation in population is 0

16- 29 Example 6 Continued School Coaches Writers Maryland23 NC State34 NC66 Virginia55 Clemson42 Wake Forest78 Duke87 Florida State11 Preseason Football Rankings for the Atlantic Coast Conference by the coaches and sports writers

16- 30 Example 6 continued School Coaches Writersdd 2 Maryland23-11 NC State34-11 NC66 00 Virginia55 00 Clemson42 24 Wake Forest78-11 Duke87 11 Florida State11 00 Total 8  d 2

16- 31 Example 6 Continued There is a strong correlation between the ranks of the coaches and the sports writers. Coefficient of Rank Correlation