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**Introduction to Nonparametric Statistics**

CD-ROM Chapter 15 Introduction to Nonparametric Statistics

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**Chapter 15 - Chapter Outcomes**

After studying the material in this chapter, you should be able to: Recognize when and how to use the runs test and testing for randomness. Know when and how to perform a Mann-Whitney U test. Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it in a decision-making context. Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA.

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**Nonparametric Statistics**

Nonparametric statistical procedures are those statistical methods that do not concern themselves with population distributions and/or parameters.

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The Runs Test The runs test is a statistical procedure used to determine whether the pattern of occurrences of two types of observations is determined by a random process.

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The Runs Test A run is a succession of occurrences of a certain type preceded and followed by occurrences of the alternate type or by no occurrences at all.

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The Runs Test (Table 15-1)

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**The Runs Test (Small Sample Example)**

H0: Computer-generated numbers are random between 0.0 and 1.0. HA: Computer-generated numbers are not random . Runs: There are r = 10 runs From runs table (Appendix K) with n1 = 9 and n2 = 11, the critical value of r is 6

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**The Runs Test (Small Sample Example)**

Test Statistic: r = 10 runs Critical Values from Runs Table: Possible Runs: Reject H0 Reject H0 Do not reject H0 Decision: Since r = 10, we do not reject the null hypothesis.

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**MEAN AND STANDARD DEVIATION FOR r**

Large Sample Runs Test MEAN AND STANDARD DEVIATION FOR r where: n1 = Number of occurrences of first type n2 = Number of occurrences of second type

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**TEST STATISTIC FOR LARGE SAMPLE RUNS TEST**

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**Large Sample Runs Test (Example 15-2)**

Table 15-2 OOOUOOUOUUOOUUOOOOUUOUUOOO UUUOOOOUUOOUUUOUUOOUUUUU OOOUOUUOOOUOOOOUUUOUUOOOU OOUUOUOOUUUOUUOOOOUUUOOO n1 = 53 “O’s” n2 = 47 “U’s” r = 45 runs

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**Large Sample Runs Test (Example 15-2)**

H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level. H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce level. = 0.05 Rejection Region /2 = 0.025 Rejection Region /2 = 0.025 Since z= > and < 1.96, we do not reject H0,

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**Mann-Whitney U Test • The two samples are independent and random.**

The Mann Whitney U test can be used to compare two samples from two populations if the following assumptions are satisfied: • The two samples are independent and random. • The value measured is a continuous variable. • The measurement scale used is at least ordinal. • If they differ, the distributions of the two populations will differ only with respect to the central location.

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**Mann-Whitney U Test U-STATISTICS where:**

n1 and n2 are the two sample sizes R1 and R2 = Sum of ranks for samples 1 and 2

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**Mann-Whitney U Test - Large Samples -**

MEAN AND STANDARD DEVIATION FOR THE U-STATISTIC where: n1 and n2 = Sample sizes from populations 1 and 2

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**Mann-Whitney U Test - Large Samples -**

MANN-WHITNEY U-TEST STATISTIC

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**Mann-Whitney U Test (Example 15-4)**

Rejection Region = 0.05 Since z= > , we do not reject H0,

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**Wilcoxon Matched-Pairs Test**

The Wilcoxon matched pairs signed rank test can be used in those cases where the following assumptions are satisfied: • The differences are measured on a continuous variable. • The measurement scale used is at least interval. • The distribution of the population differences is symmetric about their median.

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**Wilcoxon Matched-Pairs Test**

WILCOXON MEAN AND STANDARD DEVIATION where: n = Number of paired values

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**Wilcoxon Matched-Pairs Test**

WILCOXON TEST STATISTIC

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**Kruskal-Wallis One-Way Analysis of Variance**

Kruskal-Wallis one-way analysis of variance can be used in one-way analysis of variance if the variables satisfy the following: • They have a continuous distribution. • The data are at least ordinal. • The samples are independent. • The samples come from populations whose only possible difference is that at least one may have a different central location than the others.

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**Kruskal-Wallis One-Way Analysis of Variance**

H-STATISTIC where: N = Sum of sample sizes in all samples k = Number of samples Ri = Sum of ranks in the ith sample ni = Size of the ith sample

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**Kruskal-Wallis One-Way Analysis of Variance**

CORRECTION FOR TIED RANKINGS where: g = Number of different groups of ties ti = Number of tied observations in the ith tied group of scores N = Total number of observations

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**Kruskal-Wallis One-Way Analysis of Variance**

H-STATISTIC CORRECTED FOR TIED RANKINGS

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**Key Terms • Kruskal-Wallis One-Way Analysis of Variance • Run**

• Mann-Whitney U Test • Nonparametric Statistical Procedure • Run • Runs Test • Wilcoxon Test

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