If
we have a sample
then
we find
and
rank them from smallest to largest.
University Maths Notes: Probability and Statistics – The Wilcoxon Signed Rank Test
The Wilcoxon signed rank test assumes only a continuous and
symmetric distribution with mean =median =If
we have a samplethen
we find
and
rank them from smallest to largest.
The null hypothesis is
The test statistic is
the
sum of the ranks of those
with
positive.
The alternative hypothesis may be stated as one of those below,
with associated rejection region for a level
test,
where
and
are
obtained from tables.
|
|
|
|
|
|
|
|
Either |
Example: A manufacturer of electric irons, wishing to test the accuracy of the thermostat control at the 500 degree fahrenheit setting, obtains actual temperatures at that setting for fifteen irons. They are
494.6, 510.8, 487.5, 493.2, 502.6, 485, 495.9, 498.2, 501.6, 497.3, 492.0, 504.3, 499.2, 493.5, 505.8
Assuming a symmetric distribution for the temperature, we can apply the Wilcoxon signed rank test. Subtracting 500 from each gives
-5.4, 10.8, -12.5, -6.8, 2.6, -15, -4.1, -1.8, 1.6, -2.7, -8.0, 4.3,-0.8, -6.5,5.8
The ranks are obtained by ordering these from smallest to largest, obtaining,
|
Absolute Value |
0.8 |
1.6 |
1.8 |
2.6 |
2.7 |
4.1 |
4.3 |
5.6 |
5.8 |
6.8 |
6.8 |
8 |
10.8 |
12.5 |
15 |
|
Rank |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
|
Sign |
- |
+ |
- |
+ |
- |
- |
+ |
- |
+ |
- |
- |
- |
+ |
- |
- |
Thus
From
the Wilcoxon tables,
when
is
true sois
rejected if either
or
neither
of which apply here sois
not rejected. There is no evidence at this level to suppose the
thermostat is defective.
or