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
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
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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 9
10 7 8 11 8 12
? 2002 The Wadsworth Group
An Example,continued
hrs,di|di| hrs,di|di|
8 –2 2 9 –1 1
9 –1 1 8 –2 2
8 –2 2 11 +1 1
10 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
1,2,3,4,5,6,7
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
? 2002 The Wadsworth Group
An Example,continued
hrs,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 - 10.5
8 –2 2 10.5 - 10.5 11 +1 1 4 4 -
10 0 0 - - - 9 –1 1 4 - 4
9 –1 1 4 - 4 10 0 0 - - -
7 –3 3 16 - 16 7 –3 3 16 - 16
7 –3 3 16 - 16 8 –2 2 10.5 - 10.5
9 –1 1 4 - 4 11 +1 1 4 4 -
7 –3 3 16 - 16 8 –2 2 10.5 - 10.5
7 –3 3 16 - 16 12 +2 2 10.5 10.5 -
So,SR+ = 18.5,SR– = 152.5 ? 2002 The Wadsworth Group
An Example,continued
? I,H0,m = 10 hours H1,m ? 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+ = 18.5 SR– = 152.5
? 2002 The Wadsworth Group
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
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 {?,>,<} 0
? Critical Value/Decision Rule,W,Wilcoxon
rank sum test
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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?
? 2002 The Wadsworth Group
An Example,continued
? nA = 12,nN = 19,T = 9,n = 31
? Compute the appropriate z statistic:
z ?
T –
2 n
1
n
2
n
? 1
??
??
??
??
??
??
??
??
??
??
??
??
??
??
2 n
1
n
2
( 2 n
1
n
2
n )
n 2 ( n 1 )
?
9 –
2 ?12 ?19
31
? 1
??
??
??
??
??
??
??
??
??
??
2 ?12 ?19 ( 2 ?12 ?19 31 )
31 2 ( 31 1 )
?
– 6, 70 97
6, 72 22
? 2, 59
? 2002 The Wadsworth Group
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
??? ?? ??? ????? ?
z=-1,96 z=1.9 6
Do Not
Rejec t H
0
00
Rejec t HRejec t H
? 2002 The Wadsworth Group
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.
? 2002 The Wadsworth Group
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
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
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
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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 9
10 7 8 11 8 12
? 2002 The Wadsworth Group
An Example,continued
hrs,di|di| hrs,di|di|
8 –2 2 9 –1 1
9 –1 1 8 –2 2
8 –2 2 11 +1 1
10 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
1,2,3,4,5,6,7
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
? 2002 The Wadsworth Group
An Example,continued
hrs,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 - 10.5
8 –2 2 10.5 - 10.5 11 +1 1 4 4 -
10 0 0 - - - 9 –1 1 4 - 4
9 –1 1 4 - 4 10 0 0 - - -
7 –3 3 16 - 16 7 –3 3 16 - 16
7 –3 3 16 - 16 8 –2 2 10.5 - 10.5
9 –1 1 4 - 4 11 +1 1 4 4 -
7 –3 3 16 - 16 8 –2 2 10.5 - 10.5
7 –3 3 16 - 16 12 +2 2 10.5 10.5 -
So,SR+ = 18.5,SR– = 152.5 ? 2002 The Wadsworth Group
An Example,continued
? I,H0,m = 10 hours H1,m ? 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+ = 18.5 SR– = 152.5
? 2002 The Wadsworth Group
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
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 {?,>,<} 0
? Critical Value/Decision Rule,W,Wilcoxon
rank sum test
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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
? 2002 The Wadsworth Group
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?
? 2002 The Wadsworth Group
An Example,continued
? nA = 12,nN = 19,T = 9,n = 31
? Compute the appropriate z statistic:
z ?
T –
2 n
1
n
2
n
? 1
??
??
??
??
??
??
??
??
??
??
??
??
??
??
2 n
1
n
2
( 2 n
1
n
2
n )
n 2 ( n 1 )
?
9 –
2 ?12 ?19
31
? 1
??
??
??
??
??
??
??
??
??
??
2 ?12 ?19 ( 2 ?12 ?19 31 )
31 2 ( 31 1 )
?
– 6, 70 97
6, 72 22
? 2, 59
? 2002 The Wadsworth Group
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
??? ?? ??? ????? ?
z=-1,96 z=1.9 6
Do Not
Rejec t H
0
00
Rejec t HRejec t H
? 2002 The Wadsworth Group
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.
? 2002 The Wadsworth Group
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
