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Statistics for Business and Economics. Nonparametric Statistics Chapter 14. Learning Objectives. 1. Distinguish Parametric & Nonparametric Test Procedures 2. Explain a Variety of Nonparametric Test Procedures 3. Solve Hypothesis Testing Problems Using Nonparametric Tests

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statistics for business and economics

Statistics for Business and Economics

Nonparametric StatisticsChapter 14

learning objectives
Learning Objectives
  • 1. Distinguish Parametric & Nonparametric Test Procedures
  • 2. Explain a Variety of Nonparametric Test Procedures
  • 3. Solve Hypothesis Testing Problems Using Nonparametric Tests
  • 4. Compute Spearman’s Rank Correlation
hypothesis testing procedures
Hypothesis Testing Procedures

Many More Tests Exist!

parametric test procedures
Parametric Test Procedures
  • 1. Involve Population Parameters
    • Example: Population Mean
  • 2. Require Interval Scale or Ratio Scale
    • Whole Numbers or Fractions
    • Example: Height in Inches (72, 60.5, 54.7)
  • 3. Have Stringent Assumptions
    • Example: Normal Distribution
  • 4. Examples: Z Test, t Test, 2 Test
nonparametric test procedures
Nonparametric Test Procedures
  • 1. Do Not Involve Population Parameters
    • Example: Probability Distributions, Independence
  • 2. Data Measured on Any Scale
    • Ratio or Interval
    • Ordinal
      • Example: Good-Better-Best
    • Nominal
      • Example: Male-Female
  • 3. Example: Wilcoxon Rank Sum Test
advantages of nonparametric tests
Advantages of Nonparametric Tests
  • 1. Used With All Scales
  • 2. Easier to Compute
    • Developed Originally Before Wide Computer Use
  • 3. Make Fewer Assumptions
  • 4. Need Not Involve Population Parameters
  • 5. Results May Be as Exact as Parametric Procedures

© 1984-1994 T/Maker Co.

disadvantages of nonparametric tests
Disadvantages of Nonparametric Tests
  • 1. May Waste Information
    • If Data Permit Using Parametric Procedures
    • Example: Converting Data From Ratio to Ordinal Scale
  • 2. Difficult to Compute by Hand for Large Samples
  • 3. Tables Not Widely Available

© 1984-1994 T/Maker Co.

frequently used nonparametric tests
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
frequently used nonparametric tests1
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
sign test1
Sign Test
  • 1. Tests One Population Median,  (eta)
  • 2. Corresponds to t-Test for 1 Mean
  • 3. Assumes Population Is Continuous
  • 4. Small Sample Test Statistic: # Sample Values Above (or Below) Median
    • Alternative Hypothesis Determines
  • 5. Can Use Normal Approximation If n 10
sign test uses p value to make decision
Sign Test Uses P-Value to Make Decision

Binomial: n = 8 p = 0.5

P-Value Is the Probability of Getting an Observation At Least as Extreme as We Got. If 7 of 8 Observations ‘Favor’ Ha, Then P-Value = P(x 7) = .031 + .004 = .035. If  = .05, Then Reject H0 Since P-Value .

sign test example
Sign Test Example
  • You’re an analyst for 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: 2 5 3 4 1 4 5. At the .05 level, is there evidence that the median rating is less than 3?
sign test solution
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:
  • Ha:
  •  =
  • Test Statistic:
sign test solution1
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  =
  • Test Statistic:
sign test solution2
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  = .05
  • Test Statistic:
sign test solution3
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  = .05
  • Test Statistic:

S = 2 (Ratings 1 & 2 Are Less Than  =3:2, 5, 3, 4, 1, 4, 5)

sign test solution4
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  = .05
  • Test Statistic:

P(x 2) = 1 - P(x 1) = .937(Binomial Table, n = 7, p = 0.50)

S = 2 (Ratings 1 & 2 Are Less Than  =3:2, 5, 3, 4, 1, 4, 5)

sign test solution5
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  = .05
  • Test Statistic:

P(x 2) = 1 - P(x 1) = .937(Binomial Table, n = 7, p = 0.50)

S = 2 (Ratings 1 & 2 Are Less Than  =3:2, 5, 3, 4, 1, 4, 5)

Do Not Reject at  = .05

sign test solution6
Sign Test Solution

P-Value:

Decision:

Conclusion:

  • H0:  =3
  • Ha:  < 3
  •  = .05
  • Test Statistic:

P(x 2) = 1 - P(x 1) = .937(Binomial Table, n = 7, p = 0.50)

S = 2 (Ratings 1 & 2 Are Less Than  =3:2, 5, 3, 4, 1, 4, 5)

Do Not Reject at  = .05

There Is No Evidence Median Is Less Than 3

frequently used nonparametric tests2
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
wilcoxon rank sum test1
Wilcoxon Rank Sum Test
  • 1. Tests Two Independent Population Probability Distributions
  • 2. Corresponds to t-Test for 2 Independent Means
  • 3. Assumptions
    • Independent, Random Samples
    • Populations Are Continuous
  • 4. Can Use Normal Approximation If ni 10
wilcoxon rank sum test procedure
Wilcoxon Rank Sum Test Procedure
  • 1. Assign Ranks, Ri, to the n1 + n2 Sample Observations
    • If Unequal Sample Sizes, Let n1 Refer to Smaller-Sized Sample
    • Smallest Value = 1
    • Average Ties
  • 2. Sum the Ranks, Ti, for Each Sample
  • 3. Test Statistic Is TA (Smallest Sample)
wilcoxon rank sum test example
Wilcoxon Rank Sum Test Example
  • You’re a production planner. You want to see if the operating rates for 2 factories is the same. For factory 1, the rates (% of capacity) are 71, 82, 77, 92, 88. For factory 2, the rates are 85, 82, 94& 97. Do the factory rates have the same probability distributions at the .10 level?
wilcoxon rank sum test solution
Wilcoxon Rank Sum Test Solution
  • H0:
  • Ha:
  •  =
  • n1 = n2 =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 Ranks

wilcoxon rank sum test solution1
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  =
  • n1 = n2 =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 Ranks

wilcoxon rank sum test solution2
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  = .10
  • n1 = 4 n2 = 5
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 Ranks

wilcoxon rank sum table portion
Wilcoxon Rank Sum Table (Portion)

 = .05 one-tailed;  = .10 two-tailed

wilcoxon rank sum test solution3
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  = .10
  • n1 = 4 n2 = 5
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

Reject

13

27

 Ranks

wilcoxon rank sum test computation table
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

Rank Sum

wilcoxon rank sum test computation table1
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

85

82

82

77

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table2
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

82

82

77

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table3
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

82

82

77

2

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table4
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

82

3

82

4

77

2

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table5
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

82

3

3.5

82

4

3.5

77

2

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table6
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

92

97

88

...

...

Rank Sum

wilcoxon rank sum test computation table7
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

92

97

88

6

...

...

Rank Sum

wilcoxon rank sum test computation table8
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

92

7

97

88

6

...

...

Rank Sum

wilcoxon rank sum test computation table9
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

8

92

7

97

88

6

...

...

Rank Sum

wilcoxon rank sum test computation table10
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

8

92

7

97

9

88

6

...

...

Rank Sum

wilcoxon rank sum test computation table11
Wilcoxon Rank Sum Test Computation Table

Factory 1

Factory 2

Rate

Rank

Rate

Rank

71

1

85

5

82

3

3.5

82

4

3.5

77

2

94

8

92

7

97

9

88

6

...

...

Rank Sum

19.5

25.5

wilcoxon rank sum test solution4
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  = .10
  • n1 = 4 n2 = 5
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

Do Not Reject

Reject

Reject

13

27

 Ranks

wilcoxon rank sum test solution5
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  = .10
  • n1 = 4 n2 = 5
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

Do Not Reject at  = .10

Do Not Reject

Reject

Reject

13

27

 Ranks

wilcoxon rank sum test solution6
Wilcoxon Rank Sum Test Solution
  • H0: Identical Distrib.
  • Ha: Shifted Left or Right
  •  = .10
  • n1 = 4 n2 = 5
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

Do Not Reject at  = .10

Do Not Reject

Reject

Reject

There Is No Evidence Distrib. Are Not Equal

13

27

 Ranks

frequently used nonparametric tests3
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
wilcoxon signed rank test1
Wilcoxon Signed Rank Test
  • 1. Tests Probability Distributions of 2 Related Populations
  • 2. Corresponds to t-test for Dependent (Paired) Means
  • 3. Assumptions
    • Random Samples
    • Both Populations Are Continuous
  • 4. Can Use Normal Approximation If n 25
signed rank test procedure
Signed Rank Test Procedure
  • 1. Obtain Difference Scores, Di= X1i- X2i
  • 2. Take Absolute Value of Differences, Di
  • 3. Delete Differences With 0 Value
  • 4. Assign Ranks, Ri, Where Smallest = 1
  • 5. Assign Ranks Same Signs as Di
  • 6. Sum ‘+’ Ranks (T+) & ‘-’ Ranks (T-)
    • Test Statistic Is T- (One-Tailed Test)
    • Test Statistic Is Smaller of T- or T+ (2-Tail)
signed rank test example
Signed Rank TestExample
  • You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times:
  • UserCurrentNew
  • Donna 9.98 9.88
  • Santosha 9.88 9.86
  • Sam 9.90 9.83
  • Tamika 9.99 9.80
  • Brian 9.94 9.87
  • Jorge 9.84 9.84

© 1984-1994 T/Maker Co.

signed rank test solution
Signed Rank Test Solution
  • H0:
  • Ha:
  •  =
  • n’ =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

T0

signed rank test solution1
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  =
  • n’ =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

T0

signed rank test solution2
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  = .05
  • n’ = 5 (not 6; 1 elim.)
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

T0

signed rank test solution3
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  = .05
  • n’ = 5 (not 6; 1 elim.)
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

1

T0

signed rank test solution4
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  = .05
  • n’ = 5 (not 6; 1 elim.)
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Since One-Tailed Test & Current Shifted Right, Use T-: T- = 0

Do Not Reject

Reject

1

T0

signed rank test solution5
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  = .05
  • n’ = 5 (not 6; 1 elim.)
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Since One-Tailed Test & Current Shifted Right, Use T-: T- = 0

Reject at  = .05

Do Not Reject

Reject

1

T0

signed rank test solution6
Signed Rank Test Solution
  • H0: Identical Distrib.
  • Ha: Current Shifted Right
  •  = .05
  • n’ = 5 (not 6; 1 elim.)
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Since One-Tailed Test & Current Shifted Right, Use T-: T- = 0

Reject at  = .05

Do Not Reject

Reject

There Is Evidence New Package Is Faster

1

T0

frequently used nonparametric tests4
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
kruskal wallis h test1
Kruskal-Wallis H-Test
  • 1. Tests the Equality of More Than 2 (p) Population Probability Distributions
  • 2. Corresponds to ANOVA for More Than 2 Means
  • 3. Used to Analyze Completely Randomized Experimental Designs
  • 4. Uses 2 Distribution with p - 1 df
    • If At Least 1 Sample Size nj > 5
kruskal wallis h test assumptions
Kruskal-Wallis H-Test Assumptions
  • 1. Independent, Random Samples
  • 2. At Least 5 Observations Per Sample
  • 3. Continuous Population Probability Distributions
kruskal wallis h test procedure
Kruskal-Wallis H-Test Procedure
  • 1. Assign Ranks, Ri , to the n Combined Observations
    • Smallest Value = 1; Largest Value = n
    • Average Ties
  • 2. Sum Ranks for Each Group
kruskal wallis h test procedure1
Kruskal-Wallis H-Test Procedure
  • 1. Assign Ranks, Ri , to the n Combined Observations
    • Smallest Value = 1; Largest Value = n
    • Average Ties
  • 2. Sum Ranks for Each Group
  • 3. Compute Test Statistic

Squared total of each group

kruskal wallis h test example
Kruskal-Wallis H-Test Example
  • As production manager, you want to see if 3 filling machines have different filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the .05 level, is there a difference in the distributionof filling times?

Mach1 Mach2Mach3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

kruskal wallis h test solution
Kruskal-Wallis H-Test Solution
  • H0:
  • Ha:
  •  =
  • df =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

kruskal wallis h test solution1
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  =
  • df =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

kruskal wallis h test solution2
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

kruskal wallis h test solution3
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 = .05

2

0

5.991

kruskal wallis h test solution4
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach3

kruskal wallis h test solution5
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach31

kruskal wallis h test solution6
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach32 1

kruskal wallis h test solution7
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach3 2 13

kruskal wallis h test solution8
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach314 9 2 15 6 7 12 10 1 11 8 4 13 5 3

kruskal wallis h test solution9
Kruskal-Wallis H-Test Solution

Raw Data

Mach1 Mach2Mach325.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

Ranks

Mach1 Mach2Mach314 9 2 15 6 7 12 10 1 11 8 413 5 365 38 17

Total

kruskal wallis h test solution11
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

H = 11.58

 = .05

2

0

5.991

kruskal wallis h test solution12
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

H = 11.58

Reject at  = .05

 = .05

2

0

5.991

kruskal wallis h test solution13
Kruskal-Wallis H-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

H = 11.58

Reject at  = .05

 = .05

There Is Evidence Pop. Distrib. Are Different

2

0

5.991

frequently used nonparametric tests5
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
friedman f r test
Friedman Fr-Test
  • 1. Tests the Equality of More Than 2 (p) Population Probability Distributions
  • 2. Corresponds to ANOVA for More Than 2 Means
  • 3. Used to Analyze Randomized Block Experimental Designs
  • 4. Uses 2 Distribution with p - 1 df
    • If either p, the number of treatments, or b, the number of blocks, exceeds 5
friedman f r test assumptions
Friedman Fr-Test Assumptions
  • The p treatments are randomly assigned to experimental units within the b blocks Samples
  • The measurements can be ranked within the blocks
  • 3. Continuous population probability distributions
friedman f r test procedure
Friedman Fr-Test Procedure
  • 1. Assign Ranks, Ri = 1 – p, to the p treatments in each of the b blocks
    • Smallest Value = 1; Largest Value = p
    • Average Ties
  • 2. Sum Ranks for Each Treatment
friedman f r test procedure1
Friedman Fr-Test Procedure
  • 1. Assign Ranks, Ri = 1 – p, to the p treatments in each of the b blocks
    • Smallest Value = 1; Largest Value = p
    • Average Ties
  • Sum Ranks for Each Treatment
  • Compute Test Statistic

Squared total of each treatment

friedman f r test example
Friedman Fr-Test Example
  • Three new traps were tested to compare their ability to trap mosquitoes. Each of the traps, A, B, and C were placed side-by-side at each five different locations. The number of mosquitoes in each trap was recorded. At the .05 level, is there a difference in the distributionof number of mosquitoes caught by the three traps?

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

friedman f r test solution
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  =
  • df =
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

friedman f r test solution1
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

friedman f r test solution2
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 = .05

2

0

5.991

friedman f r test solution3
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC

friedman f r test solution4
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC1

friedman f r test solution5
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC 2 3 1 1

friedman f r test solution6
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC2 3 1 3 2 1

friedman f r test solution7
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC2 3 1 3 2 1 3 1 2.

.

friedman f r test solution8
Friedman Fr-Test Solution

Raw Data

TrapATrapBTrapC 3 5 0 23 17 15 11 5 7 8 4 2 19 11 5

Ranks

TrapATrapBTrapC2 3 1 3 2 1 3 1 2 3 2 1

3 2 1

14 10 6

Total

friedman f r test solution10
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Fr = 6.64

 = .05

2

0

5.991

friedman f r test solution11
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Fr = 6.64

Reject at  = .05

 = .05

2

0

5.991

friedman f r test solution12
Friedman Fr-Test Solution
  • H0: Identical Distrib.
  • Ha: At Least 2 Differ
  •  = .05
  • df = p - 1 = 3 - 1 = 2
  • Critical Value(s):

Test Statistic:

Decision:

Conclusion:

Fr = 6.64

Reject at  = .05

 = .05

There Is Evidence Pop. Distrib. Are Different

2

0

5.991

frequently used nonparametric tests6
Frequently Used Nonparametric Tests
  • 1. Sign Test
  • 2. Wilcoxon Rank Sum Test
  • 3. Wilcoxon Signed Rank Test
  • 4. Kruskal Wallis H-Test
  • Friedman’s Fr-Test
  • Spearman’s Rank Correlation Coefficient
spearman s rank correlation coefficient1
Spearman’s Rank Correlation Coefficient
  • 1. Measures Correlation Between Ranks
  • 2. Corresponds to Pearson Product Moment Correlation Coefficient
  • 3. Values Range from -1 to +1
spearman s rank correlation coefficient2
Spearman’s Rank Correlation Coefficient
  • 1. Measures Correlation Between Ranks
  • 2. Corresponds to Pearson Product Moment Correlation Coefficient
  • 3. Values Range from -1 to +1
  • 4. Equation (Shortcut)
spearman s rank correlation procedure
Spearman’s Rank Correlation Procedure
  • 1. Assign Ranks, Ri , to the Observations of Each Variable Separately
  • 2. Calculate Differences, di , Between Each Pair of Ranks
  • 3. Square Differences, di 2, Between Ranks
  • 4. Sum Squared Differences for Each Variable
  • 5. Use Shortcut Approximation Formula
spearman s rank correlation example
Spearman’s Rank Correlation Example
  • You’re a research assistant for the FBI. You’re investigating the relationship between a person’s attempts at deception & % changes in their pupil size. You ask subjects a series of questions, some of which they must answer dishonestly. At the .05 level, what is the correlation coefficient?

Subj.Deception Pupil 1 87 10 2 63 6 3 95 11 4 50 7 5 43 0

conclusion
Conclusion
  • 1. Distinguished Parametric & Nonparametric Test Procedures
  • 2. Explained a Variety of Nonparametric Test Procedures
  • 3. Solved Hypothesis Testing Problems Using Nonparametric Tests
  • 4. Computed Spearman’s Rank Correlation
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End of Chapter

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