Higher Engineering Mathematics

614 STATISTICS AND PROBABILITY

with such distributions are calleddistribution-free

tests; since they do not involve the use of parameters,

they are known asnon-parametric tests. Three such

tests are explained in this chapter—thesign testin

Section 63.4 following, theWilcoxon signed-rank

testin Section 63.5 and theMann-Whitney testin

Section 63.6.

63.4 The sign test

The sign test is the simplest, quickest and oldest of
all non-parametric tests.

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. Let, for

example,H 0 :x=φ, then ifH 1 :x=φthen a

two-tailed test is suggested becausexcould

be less than or more thanφ(thus useα 2 in

Table 63.3 Critical values for the sign test

α 1 =5% 212 %1%^12 % α 1 =5% 212 %1%^12 %

n α 2 =10% 5% 2% 1% n α 2 =10% 5% 2% 1%

1———— 26 8 7 6 6

2———— 27 8 7 7 6

3———— 28 9 8 7 6

4———— 29 9 8 7 7

50——— 30 10 9 8 7

600—— 31 10 9 8 7

7000— 32 10 9 8 8

8100033 11 10 9 8

9110034 11 10 9 9

10 1 1 0 0 35 12 11 10 9

11 2 1 1 0 36 12 11 10 9

12 2 2 1 1 37 13 12 10 10

13 3 2 1 1 38 13 12 11 10

14 3 2 2 1 39 13 12 11 11

15 3 3 2 2 40 14 13 12 11

16 4 3 2 2 41 14 13 12 11

17 4 4 3 2 42 15 14 13 12

18 5 4 3 3 43 15 14 13 12

19 5 4 4 3 44 16 15 13 13

20 5 5 4 3 45 16 15 14 13

21 6 5 4 4 46 16 15 14 13

22 6 5 5 4 47 17 16 15 14

23 7 6 5 4 48 17 16 15 14

24 7 6 5 5 49 18 17 15 15

25 7 7 6 5 50 18 17 16 15

Table 63.3), but if sayH 1 :x<φorH 1 :x>φ

then a one-tailed test is suggested (thus useα 1

in Table 63.3).

(iii) Assign plus or minus signs to each piece of

data—compared withφ(see Problems 5 and 6)

or assign plus and minus signs to the difference

for paired observations (see Problem 7).

(iv) Sum either the number of plus signs or the

number of minus signs. For the two-tailed test,

whichever is the smallest is taken; for a one-

tailed test, the one which would be expected to

have the smaller value whenH 1 is true is used.

The sum decided upon is denoted byS.

(v) Use Table 63.3 for given values ofn, andα 1 or

α 2 to read the critical region ofS. For exam-

ple, if, say,n=16 andα 1 =5%, then from

Table 63.3,S≤4. Thus ifSin part (iv) is greater

than 4 we accept the null hypothesisH 0 and

ifSis less than or equal to 4 we accept the

alternative hypothesisH 1.

This procedure for the sign test is demonstrated in

the following Problems.