Chapter 4
The Two Sample Location Problem
With Independent Samples
4.1 Wilcoxon Rank Sum Test
This test was first published by Wilcoxon in 1945. That is why some authors refer to this test as the
Wilcoxon Rank Sum Test (because it is based on the sum of ranks). Later, in 1947, Mann and
Whitney published the same test with more detail and some authors refer to it as the Wilcoxon,
MannWhitney Rank Sum Test.
•
Data:
We have a random sample of m observations from population 1, denoted by X
1
, • • •, X
m
and
another
independent
random sample of n observations from population 2, denoted by Y
1
, •
• •, Y
n
.
•
The total number of observations is N = m + n. Without loss of generality we take
n < m
.
•
Assumptions:
o
A1
: The observations X
1
, • • •, X
m
are a random sample from population 1, thus they are
independent
of one another and have the same distribution. The other sample, Y
1
, • • •,
Y
n
are a random sample from population 2, so they also are
independent
of one another
and have the same distribution.
o
A2
: The two samples are mutually independent of one another.
o
A3:
Population 1 and population 2 are both continuous populations.
o
A4
: Y
j
has the same distribution as X
i
+ ∆ for all i
= 1, • • •, m and j = 1, • ••, n.
o
The parameter ∆ is the
treatment effect
.
An Interpretation of the Parameter
∆
in the Two Sample Location Problem
• The parameter ∆ denotes the amount of shift, i.e. the separation between the two
populations. [Assumption A4 means the two populations have the same distribution, they
differ only in location.]
•
The tables that are contained in the text assume that
o
The sample size from population 2, n < m, the sample size from population 1.
o
That is,
"Population 2"
is the one from which the
smaller sample
is selected.
Procedure:
1.
Order the
combined sample
of N = m + n
X and Yvalues from smallest to largest.
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2.
Let S
1
denote the rank of Y
1
,
S
2
denote the rank of Y
2
,
.
.., and S
n
denote the rank of Y
n
in
this
joint ordering.
3.
The test statistic is W is the sum of the ranks assigned to the
Y
values. That is,
(4.3)
a.
OneSided UpperTail Test:
To
test Ho:
∆
= 0 vs. Ha:
∆
> 0
at α
level of significance, the decision rule is:
Reject Ho if W ≥ w
α
,
otherwise do not reject.
(4.4)
Here the constant w
α
is chosen to make the type I error probability equal to α.
Values of w
α
are given in Table A.6. The table is entered with the sum of the ranks corresponding to
the
smaller sample,
so when naming the samples,
call the smaller sample the Ysample
of size n
(
n < m).
If
n
=
m,
either sample can be designated the Ysample for use of Table A.6.
b.
OneSided LowerTail Test:
To
test Ho:
∆
= 0 vs. Ha:
∆
< 0
at the α level, the decision rule is
Reject Ho if W ≤
n(m + n + 1) – w
α
,
otherwise do not reject.
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 Summer '11
 YESILCAY
 Statistics, Normal Distribution, Statistical hypothesis testing

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