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## Statistics for Business and Economics

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### Statistics for Business and Economics

Frequently Used Nonparametric TestsFrequently Used Nonparametric TestsFrequently Used Nonparametric TestsFrequently Used Nonparametric Tests

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

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

- 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

- 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 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: =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

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

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

- 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

© 1984-1994 T/Maker Co.

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

- 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 SolutionMach1 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

Raw Data

Ranks

Mach1 Mach2Mach3 2 13

Kruskal-Wallis H-Test SolutionMach1 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

Raw Data

Ranks

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

Kruskal-Wallis H-Test SolutionMach1 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

Raw Data

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

- 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

- 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

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