616 STATISTICS AND PROBABILITY
each as shown below:(A−B) + 1. 1 − 0. 9 + 1. 7 + 0. 8
− 0. 5 + 0. 4 − 0. 5 − 0. 1
+ 0. 4 − 1. 6 + 0. 6 + 0. 3(iv) There are 7 ‘+signs’ and 5 ‘−signs’. Taking
the smallest number,S= 5.(v) From Table 63.3, withn=12 andα 2 =5%,
S≤ 2.Since from (iv),Sis not equal or less than 2,
the null hypothesis cannot be rejected, i.e.
the two metering devices produce the same
fuel mileage performance.Now try the following exercise.
Exercise 228 Further problems on the sign
test- The following data represent the number of
hours of flight training received by 16 trainee
pilots prior to their first solo flight:
11.5 h 20 h 9 h 12.5 h 15 h 19 h
11 h 10.5 h 13 h 22 h 14.5 h 16.5 h
17 h 18 h 14 h 12 hUse the sign test at a significance level of 2%
to test the claim that, on average, the trainees
solo after 15 hours of flight training.
⎡⎣H 0 :t=15 h,H 1 :t=15 h
S= 6 .From Table 63.3,
S≤2, hence acceptH 0⎤⎦- In a laboratory experiment, 18 measurements
of the coefficient of friction,μ, between metal
and leather gave the following results:
0.60 0.57 0.51 0.55 0.66 0.56
0.52 0.59 0.58 0.48 0.59 0.63
0.61 0.69 0.57 0.51 0.58 0.54Use the sign test at a level of significance
of 5% to test the null hypothesisμ= 0. 56
against an alternative hypothesisμ= 0 .56.
[
S= 6 .From Fig. 63.3,S≤4, hence
null hypothesis accepted]- 18 random samples of two types of 9 V batter-
ies are taken and the mean lifetime (in hours)
of each are:
TypeA 8.2 7.0 11.3 13.9 9.0
13.8 16.2 8.6 9.4 3.6
7.5 6.5 18.0 11.5 13.4
6.9 14.2 12.4
TypeB 15.3 15.4 11.2 16.1 18.1
17.1 17.7 8.4 13.5 7.8
9.8 10.6 16.4 12.7 16.8
9.9 12.9 14.7
Use the sign test, at a level of significance of
5%, to test the null hypothesis that the two
samples come from the same population.
⎡⎢
⎣H 0 : meanA=meanB,
H 1 : meanA=meanB,S= 4
From Table 63.3,S≤4,
henceH 1 is accepted⎤⎥
⎦63.5 Wilcoxon signed-rank test
The sign test represents data by using only plus and
minus signs, all other information being ignored. The
Wilcoxon signed-rank test does make some use of
the sizes of the differences between the observed
values and the hypothesized median. However, the
distribution needs to be continuous and reasonably
symmetric.Procedure(i) State for the data the null and alternative
hypotheses,H 0 andH 1.
(ii) Know whether the stated significance level,α,
is for a one-tailed or a two-tailed test (see (ii)
in the procedure for the sign test on page 614).
(iii) Find the difference of each piece of data
compared with the null hypothesis (see Prob-
lems 8 and 9) or assign plus and minus signs
to the difference for paired observations (see
Problem 10).
(iv) Rank the differences, ignoring whether they are
positive or negative.
(v) The Wilcoxon signed-rank statistic T is
calculated as the sum of the ranks of
either the positive differences or the negative
differences—whichever is the smaller for a
two-tailed test, and the one which would be