1. CHAPTER 14: Nonparametric Methods
to accompany
Introduction to Business Statistics
fourth edition, by Ronald M. Weiers
Presentation by Priscilla Chaffe-Stengel
Donald N. Stengel
© 2002 The Wadsworth Group

2. Chapter 14 - Learning Objectives
Differentiate between nonparametric and parametric hypothesis tests.
Determine when a nonparametric test should be used instead of its parametric counterpart.
Appropriately apply each of the nonparametric methods introduced.
© 2002 The Wadsworth Group

3. Chapter 14 Key Terms Nonparametric tests Wilcoxon signed rank test:
One sample
Paired samples
Wilcoxon rank sum test, two independent samples
Kruskal-Wallis Test, three or more independent samples
Friedman test, randomized block design
Sign Test, paired samples
Runs test for randomness
Lilliefors test for normality
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4. Nonparametric Tests Advantages: Disadvantages:
Fewer assumptions about the population
Shape
Variance
Valid for small samples
Defined over a range of variables, nominal and ordinal scales included
Calculations simple
Disadvantages:
Sample data used less efficiently
Power of nonparametric analysis lower
Places greater reliance on statistical tables if computer statistical package or spreadsheet not being used
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5. Wilcoxon Signed Rank Test, One Sample
Requirements:
Variable - Continuous data
Scale - Interval or ratio scale of measurement
The Research Question (H1): Test the value of a single population median, m {¹, >, <} m0
Critical Value/Decision Rule: W, Wilcoxon signed rank test
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6. An Example
Problem 14.8: According to the director of a county tourist bureau, there is a median of 10 hours of sunshine per day during the summer months. For a random sample of 20 days during the past three summers, the number of hours of sunshine has been recorded below. Use the 0.05 level in evaluating the director’s claim.
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7. An Example, continued There are: hrs. di|di| hrs. di|di| 8 –2 2 9 –1 1
8 –2 2 9 –1 1
9 –1 1 8 –2 2
8 –
–1 1
9 –
7 –3 3 7 –3 3
7 –3 3 8 –2 2
9 –
7 –
There are:
7 with rank 1
1, 2, 3, 4, 5, 6, 7
average rank = 4
6 with rank 2
8, 9, 10, 11, 12, 13
average rank = 10.5
5 with rank 3
14, 15, 16, 17, 18
average rank = 16
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8. An Example, continued hrs. di|di| Rank R+ R– hrs. di|di| Rank R+ R–
8 – –
9 – –
8 – –
9 –
7 – –
7 – –
9 –
7 –
So, SR+ = 18.5, SR– = 152.5
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9. An Example, continued I. H0: m = 10 hours H1: m ¹ 10 hours
II. Rejection Region: a = 0.05,
n = 18 data values not equal to the
hypothesized median of 10
If SR+ < 41 or SR+ > 130, reject H0.
III. Test Statistics:
SR+ = SR– = 152.5
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10. An Example, concluded
IV. Conclusion: Since the test statistic of SR+ = 18.5 falls below the critical value of W = 41, we reject H0 with at least 95% confidence.
V. Implications: There is enough evidence to dispute the director’s claim that this county has a median of 10 days of sunshine per day during the summer months.
© 2002 The Wadsworth Group

11. Wilcoxon Signed Rank Test for Comparing Paired Samples
Requirements:
Variable - Continuous data
Scale - Interval or ratio scale of measurement
The Research Question (H1): Test the difference in two population medians, paired samples, md {¹, >, <} 0
Critical Value/Decision Rule: W, Wilcoxon rank sum test
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12. Kruskal-Wallis Test, Comparing Two Independent Samples
Requirements:
Scale - Ordinal, interval or ratio scale
Independent samples from populations with identical distributions
The Research Question (H1): At least one of the medians differs from the others.
Critical Value/Decision Rule: H, approximated by the chi-square distribution
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13. Friedman Test for the Randomized Block Design
Requirements:
Scale - Ordinal, interval or ratio scale
The Research Question (H1): At least one of the treatment medians differs from others, where block effect has been taken into account.
Critical Value/Decision Rule: Fr, approximated by the chi-square distribution
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14. The Sign Test Requirements: The Research Question (H1):
Scale - Ordinal scale of measurement
The Research Question (H1):
One sample: The population median, m {¹, >, <} a single value.
Two sample: The difference between two populations medians {¹, >, <} 0.
Critical Value/Decision Rule: p-value , the binomial distribution
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15. The Runs Test for Randomness
Requirements:
Scale - Nominal scale of measurement
Two categories
The Research Question (H1):
The sequence in which observations from the two categories appear is not random.
Critical Value/Decision Rule: z, the standard normal distribution
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16. An Example
Problem: For the first 31 Super Bowls, the winner is listed below according to “A” (American Conference) or “N” (National Conference). At the 0.05 level of significance, can this sequence be considered as other than random?
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17. An Example, continued nA = 12, nN = 19, T = 9, n = 31
Compute the appropriate z statistic:
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18. An Example, continued I. H0: The sequence is random.
H1: The sequence is not random.
II. Rejection Region: a = 0.05
If z > 1.96 or
z < –1.96, reject H0.
III. Test Statistic:
z = – 2.59
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19. An Example, concluded
IV. Conclusion: Since the test statistic of z = –2.59 falls below the critical bound of z = –1.96, we reject H0 with at least 95% confidence.
V. Implications: There is enough evidence to show that the sequence is not random.
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20. Lilliefors Test for Normality
Requirements:
Scale - Interval or ratio scale
Hypothesized distribution must be completely specified.
The Research Question (H1): The sample was not drawn from a normal distribution.
Critical Value/Decision Rule:
D = max|Fi – Ei|
© 2002 The Wadsworth Group