Higher Engineering Mathematics

CHI-SQUARE AND DISTRIBUTION-FREE TESTS 619

J

hypothesis is accepted, i.e. the average oper-

ating time is not significantly different from

1.7 h.

[Note that if, say, a piece of the given data was 1.7 h,
such that the difference was zero, that data is ignored
andnwould be 10 instead of 11 in this case.]

Problem 10. An engineer is investigating two

different types of metering devices,AandB, for

an electronic fuel injection system to determine

if they differ in their fuel mileage performance.

The system is installed on 12 different cars, and

a test is run with each metering system in turn

on each car. The observed fuel mileage data (in

miles/gallon) is shown below:

A 18.7 20.3 20.8 18.3 16.4 16.8

B 17.6 21.2 19.1 17.5 16.9 16.4

A 17.2 19.1 17.9 19.8 18.2 19.1

B 17.7 19.2 17.5 21.4 17.6 18.8

Use the Wilcoxon signed-rank test, at a level

of significance of 5%, to determine whether

there is any difference between the two

systems.

(This is the same as Problem 7 where the sign test
was used)

Using the procedure:

(i)H 0 :FA=FBandH 1 :FA=FBwhereFAand

FBare the fuels in miles/gallon for systemsA

andBrespectively.

(ii)α 2 =5%(since it is a two-tailed test).

(iii) The difference between the observations is

determined and a+or a−sign assigned to

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) The differences are now ranked from 1 to 12

(ignoring whether they are positive or nega-

tive). When ordered, 0.4 occupies positions 3

and 4; their average is 3.5 and both are assigned

this value when ranked. Similarly 0.5 occupies

positions 5 and 6 and their average of 5.5 is

assigned to each when ranked.

Rank 1 2 3.5 3.5

Difference −0.1 +0.3 +0.4 +0.4

Rank 5.5 5.5 7 8

Difference −0.5 −0.5 +0.6 +0.8

Rank 9 101112

Difference −0.9 +1.1 −1.6 +1.7

(v) There are 7 ‘+signs’ and 5 ‘−signs’. Taking

the smaller number, the negative signs have

rankings of 1, 5.5, 5.5, 9 and 11.

Summing the negative ranks gives:

T= 1 + 5. 5 + 5. 5 + 9 + 11 = 32.

(vi) From Table 63.4, whenn=12 andα 2 =5%,

T≤ 13.

Since from (iv),Tis not equal or less than 13,

the null hypothesis cannot be rejected, i.e.

the two metering devices produce the same

fuel mileage performance.

Now try the following exercise.

Exercise 229 Further problems on the

Wilcoxon signed-rank test

1. The time to repair an electronic instrument is
   a random variable. The repair times (in hours)
   for 16 instruments are as follows:

218 275 264 210 161 374 178 265

150 360 185 171 215 100 474 248

Use the Wilcoxon signed-rank test, at a 5%

level of significance, to test the hypothesis

that the mean repair time is 220 hours.

[

H 0 :t=220 h,H 1 :t=220 h,

T= 74 .From Table 63.4,

T≤29, henceH 0 is accepted

]

2. 18 samples of serum are analyzed for their
   sodium content. The results, expressed as
   ppm are as follows:

169 151 166 155 149 154

164 151 147 142 168 152

149 129 153 154 149 143