chapter 14: nonparametric methods to accompany introduction to business statistics fourth edition,...

CHAPTER 14:Nonparametric Methods

to accompany

Introduction to Business Statisticsfourth edition, by Ronald M. Weiers

Presentation by Priscilla Chaffe-Stengel Donald N. Stengel

© 2002 The Wadsworth Group

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.


Chapter 14 Key TermsNonparametric 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


Nonparametric Tests• Advantages:

– 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


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 {, >, <} m0

• Critical Value/Decision Rule: W, Wilcoxon signed rank test


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.

8 9 8 10 9 7 7 9 7 7 9 8 11 910 7 8 11 8 12


An Example, continuedhrs. di|di| hrs. di|di|

8 –2 2 9 –1 1 9 –1 1 8 –2 2 8 –2 2 11 +1 110 0 0 9 –1 1 9 –1 1 10 0 0 7 –3 3 7 –3 3 7 –3 3 8 –2 2 9 –1 1 11 +1 1 7 –3 3 8 –2 2 7 –3 3 12 +2 2

There are:7 with rank 1

1, 2, 3, 4, 5, 6, 7average rank = 4

6 with rank 28, 9, 10, 11, 12, 13

average rank = 10.5

5 with rank 314, 15, 16, 17, 18

average rank = 16


An Example, continuedhrs. di|di| Rank R+ R– hrs. di|di| Rank R+ R– 8 –2 2 10.5 - 10.5 9–1 1 4 - 4 9 –1 1 4 - 4 8–2 2 10.5 - 10.58 –2 2 10.5 - 10.5 11+1 1 4 4 -

10 0 0 - - - 9–1 1 4 - 49 –1 1 4 - 4 10 0 0 - - -7 –3 3 16 - 16 7–3 3 16 - 167 –3 3 16 - 16 8–2 2 10.5 - 10.59 –1 1 4 - 4 11+1 1 4 4 -7 –3 3 16 - 16 8–2 2 10.5 - 10.57 –3 3 16 - 16 12+2 2 10.5 10.5 -

So, R+ = 18.5, R– = 152.5 © 2002 The Wadsworth Group

An Example, continued• I. H0: m = 10 hours H1: m 10 hours

• II. Rejection Region: = 0.05,n = 18 data values not equal to the hypothesized median of 10If R+ < 41 or R+ > 130, reject H0.

• III. Test Statistics:R+ = 18.5 R– = 152.5


An Example, concluded• IV. Conclusion: Since the test

statistic of R+ = 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.


Wilcoxon Signed Rank Test forComparing 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 {, >, <} 0

• Critical Value/Decision Rule: W, Wilcoxon rank sum test


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


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


The Sign Test• Requirements:

– Scale - Ordinal scale of measurement

• The Research Question (H1):– One sample: The population median, m

{, >, <} a single value.– Two sample: The difference between two

populations medians {, >, <} 0.

• Critical Value/Decision Rule: p-value , the binomial distribution


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


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?


An Example, continued• nA = 12, nN = 19, T = 9, n = 31

• Compute the appropriate z statistic:

z

T –2n

1n2

n 1

2n1n2

(2n1n2

– n)

n2(n – 1)

9– 21219

31 1

21219(21219 – 31)312(31 – 1)

– 6.7097

6.7222 – 2.59


An Example, continued• I. H0: The sequence is random.

H1: The sequence is not random.

• II. Rejection Region: = 0.05If z > 1.96 orz < –1.96, reject H0.

• III. Test Statistic:z = – 2.59

z=-1.96 z=1.96

Do NotReject H 0

00Reject HReject H


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.


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

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