Higher Engineering Mathematics

620 STATISTICS AND PROBABILITY

At a level of significance of 5%, use the

Wilcoxon signed-rank test to test the null

hypothesis that the average value for the

method of analysis used is 150 ppm.

⎡

⎢

⎣

H 0 :s=150,H 1 :s=150,

T= 38 .From Table 63.4,

T≤40, hence alternative

hypothesisH 1 is accepted

⎤

⎥

⎦

3. A paint supplier claims that a new additive
   will reduce the drying time of their acrylic
   paint. To test his claim, 12 pieces of wood
   are painted, one half of each piece with paint
   containing the regular additive and the other
   half with paint containing the new additive.
   The drying time (in hours) were measured as
   follows:

New

additive 4.5 5.5 3.9 3.6 4.1 6.3

Regular

additive 4.7 5.9 3.9 3.8 4.4 6.5

New

additive 5.9 6.7 5.1 3.6 4.0 3.0

Regular

additive 6.9 6.5 5.3 3.6 3.9 3.9

Use the Wilcoxon signed-rank test at a sig-

nificance level of 5% to test the hypothesis

that there is no difference, on average, in the

drying times of the new and regular additive

paints.

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

H 0 :N=R,H 1 :N=R,T= 5

From Table 63.4, withn= 10

(since two differences are zero),

T≤8, Hence there is a

significant difference in the

drying times

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

63.6 The Mann-Whitney test

As long as the sample sizes are not too large, for
tests involving two samples, the Mann-Whitney test
is easy to apply, is powerful and is widely used.

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) Arrange all the data in ascending order whilst

retaining their separate identities.

(iv) If the data is now a mixture of, say,A’s andB’s,

write under each letterAthe number ofB’s that

precede it in the sequence (or vice-versa).

(v) Add together the numbers obtained from

(iv) and denote total byU.Uis defined as

whichever type of count would be expected to

be smallest whenH 1 is true.

(vi) Use Table 63.5 on pages 622 and 623 for given

values ofn 1 andn 2 , andα 1 orα 2 to read

the critical region ofU. For example, if, say,

n 1 =10 andn 2 =16 andα 2 =5%, then from

Table 63.5,U≤42. IfUin part (v) is greater

than 42 we accept the null hypothesisH 0 , and

ifUis equal or less than 42, we accept the

alternative hypothesisH 1.

The procedure for the Mann-Whitney test is demon-

strated in the following problems.

Problem 11. 10 British cars and 8 non-British

cars are compared for faults during their first

10 000 miles of use. The percentage of cars of

each type developing faults were as follows:

Non-British

cars,P 5 8 14 10 15

British

cars,Q 18 9 25 6 21

Non-British

cars,P 712 4

British

cars,Q 20 28 11 16 34

Use the Mann-Whitney test, at a level of sig-

nificance of 1%, to test whether non-British

cars have better average reliability than British

models.

Using the above procedure:

(i) The hypotheses are:

H 0 : Equal proportions of British and non-

British cars have breakdowns.

H 1 : A higher proportion of British cars have

breakdowns.