Higher Engineering Mathematics

CHI-SQUARE AND DISTRIBUTION-FREE TESTS 617

J

Table 63.4 Critical values for the Wilcoxon signed-rank 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———— 26 110 98 84 75

2———— 27 119 107 92 83

3———— 28 130 116 101 91

4———— 29 140 126 110 100

50——— 30 151 137 120 109

620—— 31 163 147 130 118

7320— 32 175 159 140 128

8531033 187 170 151 138

9853134 200 182 162 148

10 10 8 5 3 35 213 195 173 159

11 13 10 7 5 36 227 208 185 171

12 17 13 9 7 37 241 221 198 182

13 21 17 12 9 38 256 235 211 194

14 25 21 15 12 39 271 249 224 207

15 30 25 19 15 40 286 264 238 220

16 35 29 23 19 41 302 279 252 233

17 41 34 27 23 42 319 294 266 247

18 47 40 32 27 43 336 310 281 261

19 53 46 37 32 44 353 327 296 276

20 60 52 43 37 45 371 343 312 291

21 67 58 49 42 46 389 361 328 307

22 75 65 55 48 47 407 378 345 322

23 83 73 62 54 48 426 396 362 339

24 91 81 69 61 49 446 415 379 355

25 100 89 76 68 50 466 434 397 373

expected to have the smaller value whenH 1

is true for a one-tailed test.

(vi) Use Table 63.4 for given values ofn, andα 1 or

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

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

Table 63.4,T≤35. Thus ifTin part (v) is

greater than 35 we accept the null hypothesis

H 0 and ifTis less than or equal to 35 we accept

the alternative hypothesisH 1.

This procedure for the Wilcoxon signed-rank test is
demonstrated in the following Problems.

Problem 8. A manager of a manufacturer is

concerned about suspected slow progress in

dealing with orders. He wants at least half of the

orders received to be processed within a work-

ing day (i.e. 7 hours). A little later he decides to

time 17 orders selected at random, to check if

his request had been met. The times spent by the

17 orders being processed were as follows:

434 h9^34 h15^12 h11h 8^14 h6^12 h

9h 834 h10^34 h3^12 h8^12 h9^12 h

1514 h 13h 8h 734 h6^34 h

Use the Wilcoxon signed-rank test at a signif-

icance level of 5% to check if the managers

request for quicker processing is being met.

(This is the same as Problem 5 where the sign test

was used).

Using the procedure:

(i) The hypotheses areH 0 :t=7handH 1 :t>7h,

wheretis time.

(ii) SinceH 1 ist>7 h, a one-tail test is assumed,

i.e.α 1 =5%.