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# Statistics for Business and Economics - PowerPoint PPT Presentation

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

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### Statistics for Business and Economics

Nonparametric StatisticsChapter 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
• 4. Compute Spearman’s Rank Correlation
Hypothesis Testing Procedures

Many More Tests Exist!

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
• 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
• 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

• 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

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 Test

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 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

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
• 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

P-Value:

Decision:

Conclusion:

• H0:
• Ha:
•  =
• Test Statistic:
Sign Test Solution

P-Value:

Decision:

Conclusion:

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

P-Value:

Decision:

Conclusion:

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

### Wilcoxon Rank Sum Test

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 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
• 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
• 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
• H0:
• Ha:
•  =
• n1 = n2 =
• Critical Value(s):

Test Statistic:

Decision:

Conclusion:

 Ranks

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 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)

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

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

Factory 1

Factory 2

Rate

Rank

Rate

Rank

Rank Sum

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 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 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 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 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 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 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 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 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 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 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 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 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 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

### Wilcoxon Signed Rank Test

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 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
• 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 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

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

Test Statistic:

Decision:

Conclusion:

Do Not Reject

Reject

T0

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 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 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 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 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 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

### Kruskal-Wallis H-Test

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-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
• 1. Independent, Random Samples
• 2. At Least 5 Observations Per Sample
• 3. Continuous Population Probability Distributions
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 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
• 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
• H0:
• Ha:
•  =
• df =
• Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

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 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 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 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 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 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 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 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 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 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 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 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

### Friedman Fr-Test for a Randomized Block Design

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 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 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 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 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 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 Fr-Test Solution
• H0: Identical Distrib.
• Ha: At Least 2 Differ
•  =
• df =
• Critical Value(s):

Test Statistic:

Decision:

Conclusion:

2

0

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 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 Fr-Test Solution

Raw Data

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

Ranks

TrapATrapBTrapC

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 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 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 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 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 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 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 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

### Spearman’s Rank Correlation Coefficient

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 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 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
• 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
• 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
• 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

End of Chapter

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