Newbold_chap15.ppt

Chap 15-1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 15
Nonparametric Statistics
Statistics for
Business and Economics
6th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-2
Chapter Goals
After completing this chapter, you should be able to:
 Use the sign test for paired or matched samples
 Use a sign test for a single population median
 Recognize when and how to use the Wilcoxon signed
rank test for a population median
 Apply a normal approximation for the Wilcoxon signed
rank test
 Know when and how to perform a Mann-Whitney U-test
 Explain Spearman rank correlation and perform a test
for association

Nonparametric Statistics
 Nonparametric Statistics
 Fewer restrictive assumptions about data
levels and underlying probability distributions
 Population distributions may be skewed
 The level of data measurement may only be
ordinal or nominal

Sign Test and
Confidence Interval
 A sign test for paired or matched samples:
 Calculate the differences of the paired observations
 Discard the differences equal to 0, leaving n observations
 Record the sign of the difference as + or –
 For a symmetric distribution, the signs are random and
+ and – are equally likely

Sign Test
 Define + to be a “success” and let P = the true
proportion of +’s in the population
 The sign test is used for the hypothesis test
 The test-statistic S for the sign test is
S = the number of pairs with a positive difference
 S has a binomial distribution with P = 0.5 and
n = the number of nonzero differences
(continued)
0.5
P
:
H0 

Determining the p-value
 The p-value for a Sign Test is found using the binomial
distribution with n = number of nonzero differences, S =
number of positive differences, and P = 0.5
 For an upper-tail test, H1: P > 0.5, p-value = P(x  S)
 For a lower-tail test, H1: P < 0.5, p-value = P(x  S)
 For a two-tail test, H1: P  0.5, 2(p-value)

Sign Test Example
 Ten consumers in a focus group have rated the
attractiveness of two package designs for a new product
Consumer Rating Difference Sign of Difference
Package 1 Package 2 Rating 1 – 2
1
2
3
4
5
6
7
8
9
10
5
4
4
6
3
5
7
5
6
7
8
8
4
5
9
9
6
9
3
9
-3
-4
0
+1
-6
-4
-1
-4
+3
-2
–
–
0
+
–
–
–
–
+
–

Sign Test Example
(continued)
 Test the hypothesis that there is no overall package preference
using  = 0.10
The proportion of consumers who prefer
package 1 is the same as the proportion
preferring package 2
A majority prefer package 2
 The test-statistic S for the sign test is
S = the number of pairs with a positive difference
= 2
 S has a binomial distribution with P = 0.5 and n = 9 (there was
one zero difference)
0.5
P
:
H0 
0.5
P
:
H1 

 The p-value for this sign test is found using the binomial
distribution with n = 9, S = 2, and P = 0.5:
 For a lower-tail test,
p-value = P(x  2|n=9, P=0.5)
= 0.090
Since 0.090 <  = 0.10 we reject the null hypothesis
and conclude that consumers prefer package 2
Sign Test Example
(continued)

Sign Test: Normal Approximation
 If the number n of nonzero sample observations is large,
then the sign test is based on the normal approximation to
the binomial with mean and standard deviation
 The test statistic is
 Where S* is the test-statistic corrected for continuity:
 For a two-tail test, S* = S + 0.5, if S < μ or S* = S – 0.5, if S > μ
 For upper-tail test, S* = S – 0.5
 For lower-tail test, S* = S + 0.5
n
0.5
0.25n
P)
nP(1
σ
0.5n
nP
μ






n
0.5
0.5n
*
S
σ
μ
*
S
Z





Sign Test for
Single Population Median
 The sign test can be used to test that a single
population median is equal to a specified value
 For small samples, use the binomial distribution
 For large samples, use the normal approximation

Wilcoxon Signed Rank Test for
Paired Samples
 Uses matched pairs of random observations
 Still based on ranks
 Incorporates information about the magnitude
of the differences
 Tests the hypothesis that the distribution of
differences is centered at zero
 The population of paired differences is
assumed to be symmetric

Conducting the test:
 Discard pairs for which the difference is 0
 Rank the remaining n absolute differences in ascending order
(ties are assigned the average of their ranks)
 Find the sums of the positive ranks and the negative ranks
 The smaller of these sums is the Wilcoxon Signed Rank Statistic T:
T = min(T+ , T- )
Where T+ = the sum of the positive ranks
T- = the sum of the negative ranks
n = the number of nonzero differences
 The null hypothesis is rejected if T is less than or equal to the value in
Appendix Table 10
Wilcoxon Signed Rank Test for
Paired Samples
(continued)

Signed Rank Test Example
T+ = 3 T– = 42
Consumer Rating Difference
Package 1 Package 2 Diff (rank) Rank (+) Rank (–)
1
2
3
4
5
6
7
8
9
10
5
4
4
6
3
5
7
5
6
7
8
8
4
5
9
9
6
9
3
9
-3 (5)
-4 (7 tie)
0 (-)
+1 (2)
-6 (9)
-4 (7 tie)
-1 (3)
-4 (7 tie)
+3 (1)
-2 (4)
2
1
5
7
9
7
3
7
4
 Ten consumers in a focus group have
rated the attractiveness of two package
designs for a new product

Test the hypothesis that the distribution of paired
differences is centered at zero, using  = 0.10
Conducting the test:
 The smaller of T+ and T- is the Wilcoxon Signed Rank Statistic T:
T = min(T+ , T- ) = 3
 Use Appendix Table 10 with n = 9 to find the critical value:
The null hypothesis is rejected if T ≤ 4
 Since T = 3 < 4, we reject the null hypothesis
(continued)
Signed Rank Test Example

Wilcoxon Signed Rank Test
Normal Approximation
A normal approximation can be used when
 Paired samples are observed
 The sample size is large
 The hypothesis test is that the population
distribution of differences is centered at zero

 The table in Appendix 10 includes T values
only for sample sizes from 4 to 20
 The T statistic approaches a normal
distribution as sample size increases
 If the number of paired values is larger than 20,
a normal approximation can be used
Wilcoxon Signed Rank Test
Normal Approximation
(continued)

 The mean and standard deviation for
Wilcoxon T :
4
1)
n(n
μ
E(T) T



24
1)
1)(2n
(n)(n
σ
Var(T) 2
T




where n is the number of paired values
Wilcoxon Matched Pairs Test
for Large Samples

 Normal approximation for the Wilcoxon T Statistic:
(continued)
24
1)
1)(2n
n(n
4
1)
n(n
T
σ
μ
T
z
T
T







 If the alternative hypothesis is one-sided, reject the null
hypothesis if
 If the alternative hypothesis is two-sided, reject the null
hypothesis if
α
T
T
z
σ
μ
T



α/2
T
T
z
σ
μ
T



Wilcoxon Matched Pairs Test
for Large Samples

Mann-Whitney U-Test
Used to compare two samples from two populations
Assumptions:
 The two samples are independent and random
 The value measured is a continuous variable
 The two distributions are identical except for a possible
difference in the central location
 The sample size from each population is at least 10

 Consider two samples
 Pool the two samples (combine into a singe list) but
keep track of which sample each value came from
 rank the values in the combined list in ascending
order
 For ties, assign each the average rank of the tied values
 sum the resulting rankings separately for each
sample
 If the sum of rankings from one sample differs enough
from the sum of rankings from the other sample, we
conclude there is a difference in the population
medians
Mann-Whitney U-Test
(continued)

Mann-Whitney U Statistic
 Consider n1 observations from the first population and
n2 observations from the second
 Let R1 denote the sum of the ranks of the observations
from the first population
 The Mann-Whitney U statistic is
1
1
1
2
1 R
2
1)
(n
n
n
n
U 




Mann-Whitney U Statistic
 The null hypothesis is that the central locations of the
two population distributions are the same
 The Mann-Whitney U statistic has mean and variance
 Then for large sample sizes (both at least 10), the
distribution of the random variable
is approximated by the normal distribution
(continued)
2
n
n
μ
E(U) 2
1
U 

12
1)
n
(n
n
n
σ
Var(U) 2
1
2
1
2
U




U
U
σ
μ
U
z



Decision Rules for
Mann-Whitney Test
The decision rule for the null hypothesis that the two
populations have the same central location:
 For a one-sided upper-tailed alternative hypothesis:
 For a one-sided lower-tailed hypothesis:
 For a two-sided alternative hypothesis:
α
U
U
0 z
σ
μ
U
z
if
H
Reject 



α
U
U
0 z
σ
μ
U
z
if
H
Reject 


α
U
U
0
α
U
U
0 z
σ
μ
U
z
if
H
Reject
or
z
σ
μ
U
z
if
H
Reject 







Mann-Whitney U-Test Example
Claim: Median class size for Math is larger
than the median class size for English
A random sample of 10 Math and 10 English
classes is selected (samples do not have to
be of equal size)
Rank the combined values and then
determine rankings by original sample

 Suppose the results are:
Class size (Math, M) Class size (English, E)
23
45
34
78
34
66
62
95
81
99
30
47
18
34
44
61
54
28
40
96
(continued)

Size Rank
18 1
23 2
28 3
30 4
34 6
34 6
34 6
40 8
44 9
45 10
Size Rank
47 11
54 12
61 13
62 14
66 15
78 16
81 17
95 18
96 19
99 20
Ranking for combined samples
tied
(continued)

 Rank by
original
sample:
Class size
(Math, M)
Rank
Class size
(English, E)
Rank
23
45
34
78
34
66
62
95
81
99
2
10
6
16
6
15
14
18
17
20
30
47
18
34
44
61
54
28
40
96
4
11
1
6
9
13
12
3
8
19
 = 124  = 86
(continued)

H0: MedianM ≤ MedianE
(Math median is not greater
than English median)
HA: MedianM > MedianE
(Math median is larger)
Claim: Median class size for
Math is larger than the
median class size for English
31
124
2
(10)(11)
(10)(10)
R
2
1)
(n
n
n
n
U 1
1
1
2
1 






 
(continued)

(continued)
1.436
12
1)
10
0
(10)(10)(1
2
(10)(10)
31
12
1)
n
(n
n
n
2
n
n
U
σ
μ
U
z
2
1
2
1
2
1
U
U












H0: MedianM ≤ MedianE
HA: MedianM > MedianE
 The decision rule for this one-sided upper-tailed alternative
hypothesis:
 For  = 0.05, -z = -1.645
 The calculated z value is not in the rejection region, so we
conclude that there is not sufficient evidence of difference in class
size medians
α
U
U
0 z
σ
μ
U
z
if
H
Reject 




Wilcoxon Rank Sum Test
 Similar to Mann-Whitney U test
 Results will be the same for both tests

 n1 observations from the first population
 n2 observations from the second population
 Pool the samples and rank the observations in
ascending order
 Let T denote the sum of the ranks of the
observations from the first population
 (T in the Wilcoxon Rank Sum Test is the same as
R1 in the Mann-Whitney U Test)
(continued)

 The Wilcoxon Rank Sum Statistic, T, has mean
 And variance
 Then, for large samples (n1  10 and n2  10) the
distribution of the random variable
is approximated by the normal distribution
(continued)
2
1)
n
(n
n
μ
E(T) 2
1
1
T




12
1)
n
(n
n
n
σ
Var(T) 2
1
2
1
2
T




T
T
σ
μ
T
Z



Wilcoxon Rank Sum Example
 We wish to test
 Use  = 0.05
 Suppose two samples are obtained:
 n1 = 40 , n2 = 50
 When rankings are completed, the sum of ranks
for sample 1 is R1 = 1475 = T
 When rankings are completed, the sum of ranks
for sample 2 is R2 = 2620
H0: Median1  Median2
H1: Median1 < Median2

 Using the normal approximation:
(continued)
2.80
12
1)
50
0
(40)(50)(4
2
1)
50
(40)(40
1475
12
1)
n
(n
n
n
2
1)
n
(n
n
T
σ
μ
T
z
2
1
2
1
2
1
1
T
T

















Since z = -2.80 < -1.645, we reject H0 and conclude
that median 1 is less than median 2 at the 0.05 level
of significance
645
.
1
z 


Reject H0
 = .05
Do not reject H0
0
(continued)
2.80
σ
μ
T
z
T
T




H0: Median1  Median2
H1: Median1 < Median2

Spearman Rank Correlation
 Consider a random sample (x1 , y1), . . .,(xn, yn) of n pairs of
observations
 Rank xi and yi each in ascending order
 Calculate the sample correlation of these ranks
 The resulting coefficient is called Spearman’s Rank Correlation
Coefficient.
 If there are no tied ranks, an equivalent formula for computing this
coefficient is
where the di are the differences of the ranked pairs
1)
n(n
d
6
1
r 2
n
1
i
2
i
S






 Consider the null hypothesis
H0: no association in the population
 To test against the alternative of positive association,
the decision rule is
 To test against the alternative of negative association,
the decision rule is
 To test against the two-sided alternative of some
association, the decision rule is
Spearman Rank Correlation
(continued)
α
S,
S
0 r
r
if
H
Reject 
α
S,
S
0 r
r
if
H
Reject 

/2
S,
S
/2
S,
S
0 r
r
or
r
r
if
H
Reject α
α 



Chapter Summary
 Used the sign test for paired or matched samples, and
the normal approximation for the sign test
 Developed and applied the Wilcoxon signed rank test,
and the large sample normal approximation
 Developed and applied the Mann-Whitney U-test for
two population medians
 Used the Wilcoxon rank-sum test
 Examined Spearman rank correlation for tests of
association

Newbold_chap15.ppt

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