n x Nonparametric Statistics

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1 Parametric Procedures 1. Involve Population Parameters Example: Population Mean 2. Require Interval Scale Ratio Scale Whole Numbers Fractions Example: Height in Inches (72, 60.5, 54.7) 3. Have Stringent Assumptions Example: Nmal Distribution 4. Examples: Z Test, t Test, F Test Nonparametric Procedures 1. Do Not Involve Population Parameters Example: Probability Distributions, Independence 2. Data Measured on Any Scale Ratio Interval Ordinal Example: Good-Better-Best Nominal Example: Male-Female Advantages of Nonparametric Procedures 1. Used With All Scales 2. Make Fewer Assumptions 3. Need Not Involve Population Parameters 4. Results May Be as Exact as Parametric Procedures Disadvantages of Nonparametric Procedures 1. May Waste Infmation If Data Permit Using Parametric Procedures Example: Converting Data From Ratio to Ordinal Scale 2. Difficult to Compute by Hand f Large Samples 3. Tables Not Widely Available Sign Test f a Population Median h ( Binomial Test) Binomial probability density mass function: n x P( X = x) = π ( 1 π) x n! x = π ( 1 π) x!( n x)! n x n x 1. Tests One Population Median, η (eta) 2. Cresponds to t-test f 1 Mean 3. Assumes Population Is Continuous 4. Small Sample Test Statistic: # Sample Values Above ( Below) Median 5. Can Use Nmal Approximation If n 10 A. Chang 1

2 Example: You re an analyst f Chef-Boy-R-Dee. You ve asked 7 people to rate a new ravioli on a 5-point Likert scale (1 = terrible to 5 = excellent. The ratings are: At the.05 level, is there evidence that the median rating is less than 3? H 0 : η = 3 H a : η < 3 α =.05 Test Statistic: S = 2 (Ratings 1 & 2 Are Less Than η 0 = 3) p-value = P(X 2) = 1 - P(X 1) =.937 (Use Binomial p.m.f. Binomial Table, n = 7, p = 0.50) Conclusion: There Is No Evidence that Median Is Less Than 3 Sign Test f a Population Median (Assumption: The sample is randomly selected from a continuous distribution) H 0 : η = η 0 H a : η < η 0 ( η > η 0, η η 0 ) Test Statistic: S = # of sample measurements less than η 0 if H a : η < η 0 S = # of sample measurements greater than η 0 if H a : η > η 0 S = Larger of S1 and S2, where S1 is the # of measurements less than η 0 and S2 is the # of measurements greater than η 0 Observed significant level: (Use Binomial p.m.f. Binomial Table, n, p = 0.50) p-value = P(X S), if H a : η < η 0 ( η > η 0 ) p-value = 2P(X S), if H a : η η 0 Reject H 0 if p-value < α. Large-Sample Sign Test f a Population Median (Assumption: The sample is randomly selected from a continuous distribution) H 0 : η = η 0 H a : η < η 0 ( η > η 0, η η 0 ) ( S -. 5) -. 5n Test Statistic: z = (( Standard deviation is npq = n(. 5 )(. 5) =. 5 n ). 5 n where S = # of sample measurements less than η 0 if H a : η < η 0 S = # of sample measurements greater than η 0 if H a : η > η 0 S = Larger of S1 and S2, where S1 is the # of measurements less than η 0 and S2 is the # of measurements greater than η 0 Observed significant level: (Use Binomial p.m.f. Binomial Table, n, p = 0.50) p-value = P(Z z), if H a : η < η 0 ( η > η 0 ) p-value = 2P(Z z), if H a : η η 0 Reject H 0 if p-value < α. A. Chang 2

3 Example : To determine the median life span of certain spices of animal is greater than 5 years, a random sample of 25 observations were made and life span in year is the following: At 0.05 level of significant, use sign test to test if the median life span is greater than 5 years. H 0 : η = 5 H a : η > 5 Test Statistic: S = 14 (# of + signs), p-value = P(Z > 0.4) =.484 >.05 ( 14. 5) z = = Conclusion: Fail to reject H 0. There is no sufficient evidence to suppt that the median life span of this animal is greater than 5 years. Wilcoxon Rank Sum Test 1. Tests Two Independent Population Probability Distributions 2. Cresponds to t-test f 2 Independent Means 3. Assumptions Independent, Random Samples Populations Are Continuous 4. Can Use Nmal Approximation If n i 10 Example : You re a production planner. You want to see if the operating rates f 2 facties are the same. F facty 1, the rates (% of capacity) are 71, 82, 77, 92, 88. F facty 2, the rates are 85, 82, 94 & 97. Do the facty rates have the same probability distributions at the.10 level? H a : D 1 and D 2 are Not Identical Distributions (D 1 is shifted either to the right to the left of D 2 ) Test Statistic: T 2 = = 25.5 (Rank Sum of the Smallest Sample) Facty 1 Facty 2 Rate Rank Rate Rank Rank Sum A. Chang 3

4 α =.10 n 1 = 5 n 2 = 4 Critical Value(s): T L = 13, T U = 27 Wilcoxon Rank Sum Table α =.05 one-tailed; α =.10 two-tailed n T L T U T L T U T L T U n Rejection Region Rejection Region T L = 13 T U = 27 T L = T U = 27 Conclusion: There Is No Evidence That Distributions Are Not Equal Wilcoxon Rank Sum Test Procedure (Assumption: two independent random samples from continuous distributions.) H a : D 1 is shifted to the right of D 2 D 1 is shifted to the left of D 2 D 1 is shifted either to the right to the left of D 2 Test Statistic: T 1 (Rank sum of sample 1), if n 1 < n 2 T 2 (Rank sum of sample 2), if n 2 < n 1, and either rank sum above (denoted by T) can be used if n 1 = n 2 Rejection region: If H a : D 1 is shifted to the right of D 2, the rejection region is T 1 T U T 2 T L If H a : D 1 is shifted to the left of D 2, the rejection region is T 1 T L T 2 T U If H a : D 1 is shifted either to the right to the left of D 2 the rejection region is T T L T T U (T L and T U can be found from the Wilcoxon Rank Sum Table with an α level and sample sizes.) Wilcoxon Rank Sum Test f Large Samples (n 1 10 and n 2 10) (Assumption: two independent random samples from continuous distributions.) H a : D 1 is shifted to the right of D 2 D 1 is shifted to the left of D 2 D 1 is shifted either to the right to the left of D 2 A. Chang 4

5 Test Statistic: n1( n1 + 1) T1 z = 2, µ T = n1n2 ( n1 + 1) 12 n ( n ), σ T = n n2( n ) Rejection region: If H a : D 1 is shifted to the right of D 2, the rejection region is z > z α If H a : D 1 is shifted to the left of D 2, the rejection region is z < z α If H a : D 1 is shifted either to the right to the left of D 2 the rejection region is z > z α/2 Example: The following are the weight gains (in pounds) of two random samples of young turkeys fed two different diets but otherwise kept under identical conditions: Diet 1: Diet 2: At the 0.01 level of significance to test the null hypothesis that the two populations sampled are identical against the alternative hypothesis that on the average the second diet produces a greater gain in weight. H a : D 1 is shifted to the left of D 2 ( H 0 : µ X = µ Y, Ha : µ X < µ Y ) Reject Ho if z < T = = µ T = (16)(33)/2 = 264, σ T 2 = (16)(16)(33)/12 = 704 z = ( )/ (704) 1/2 = 3.11 < 2.33 => reject the null hypothesis. We conclude that on the average the second diet produces a greater gain in weight. A. Chang 5