Nonparametric Statistics Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Nonparametric Statistics Many of the nonparametric statistical tests answer the same sorts of questions as the parametric tests. With nonparametric tests the assumptions can be relaxed considerably. Consequently, nonparametric methods are used for situations that violate the assumptions of parametric procedures. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Runs Test (Small Samples) Sequence 1: HHHHH TTTTT Run 1Run 2 Sequence 3: T HHH T H T H TT Run 1Run 7 R = number of Runs Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Runs Test (Small Samples) H o : The sequence was generated in a random manner H a : The sequence was not generated in a random manner Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Runs Test (Large Samples) H o : The sequence was generated in a random manner H a : The sequence was not generated in a random manner Reject H o if |Z| > Z  /2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Nonparametric Test Central Tendency: Two Populations Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U (Independent Samples) H o : The two populations have identical probability distributions H a : The two populations differ in location T 1 = sum of the ranks of the observations from the first sample T 2 = sum of the ranks of the observations from the second sample Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Small Samples Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Small Samples Procedure: 1. Assume that n 1 < n 2 (reverse the samples if necessary) 2. Determine U 1 and U 2 3. Use the value from Table A.10 to test H o vs. H a Two-Sided Test H a : the two populations differ in location Reject H o if Table A.10 value for U is <  /2, where U is the minimum of U 1 and U 2. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Small Samples Procedure: 1. Assume that n 1 < n 2 (reverse the samples if necessary) 2. Determine U 1 and U 2 3. Use the value from Table A.10 to test H o vs. H a One-Sided Test H a : Population 1 is shifted toH a : Population 1 is shifted to the right of population2 the left of population 2 Reject H o if Table A.10 Reject H o if Table A.10 value for U is less than  value for U is less than  where U = U 1. Where U = U 2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Large Samples Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Large Samples H o : The two populations have identical probability distribution Determine U 2 Two-Sided Test H a : The two populations differ in location Reject H o if |Z| > Z  /2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Mann-Whitney U Large Samples H o : The two populations have identical probability distribution Determine U 2 One-Sided Test H a : Population 1 is shifted toH a : Population 1 is shifted to the right of population 2 the left of population 2 Reject H o if Z > Z  Reject H o if Z < - Z  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples When small samples from suspected nonnormal populations are used, a nonparametric technique is required. The Wilcoxon test is used for such situations Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples Determine the difference for each sample. Arrange the absolute value of these differences in order, assigning a rank to each. T + = sum of ranks having a positive value and T - = sum of ranks having a negative value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples (small n) H o : The population differences are centered at 0 Two-Sided Test H a : the population differences are not centered at 0 Using the two-sided value from Table A.11, Reject H o if T  table value, where T = the minimum of T + and T -. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples (small n) H o : The population differences are centered at 0 One-Sided Test H a :the population differences are centered at a value >0 are centered at a value < 0 Using the one-sided value from Table A.11, Table A.11, Reject H o if T -  table value. Reject H o if T -  table value. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples (Large n) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples (Large n) H o : The population differences are centered at 0 Two-Sided Test H a : the population differences are not centered at 0 Reject H o if |Z| > Z  /2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples (Large n) H o : The population differences are centered at 0 One-Sided Test H a :the population differences are centered at a value >0 are centered at a value < 0 Reject H o if Z > Z  Reject H o if Z < -Z  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Wilcoxon Signed Rank Test for Paired Samples Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Kruskal-Wallis Test The nonparametric counterpart to the one- way ANOVA test. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Kruskal-Wallis Test H o : the k populations have identical probability distributions H a : at least two of the populations differ in location Reject H o if KW >  2 .df Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

p-value for KW Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Kruskal-Wallis Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Friedman Test The nonparametric counterpart to the randomized block ANOVA test. Reject H o if FR >  2 .df Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Friedman Test H o : the k populations have identical probability distributions H a : at least two of the populations differ in location Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Spearman’s Rank Correlation Spearman’s is the nonparametric counterpart to the Pearson Correlation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Spearman’s Rank Correlation If there are no ties then a shortcut equation may be used.0 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Spearman’s Rank Correlation H o : No association between X and Y exists H a : An association between X and Y exists Two-Sided Test Determine r s Use Table A.12 for  /2 Reject H o if |r s | > table value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Spearman’s Rank Correlation H o : No association between X and Y exists One-Sided Test H a : A positive association H a : A negative association between X and Y exists between X and Y exists Determine r s Use Table A.12 for  Reject H o if r s > table value Reject H o if r s < table value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing