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, H0and H1.
(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 and B’s, write under each letter A the number of B’s that precede it in the sequence (or vice-versa).
(v) Add together the numbers obtained from (iv) and denote total by U. U is defined as whichever type of count would be expected to be smallest when H1is true.
(vi) Use Table 63.5 on pages 622 and 623 for given values of n1 and n2, and α1 or α2 to read the critical region of U. For example, if, say, n1 = 10 and n2=16 andα2=5%, then from Table 63.5, U≤42. If U in part (v) is greater than 42 we accept the null hypothesis H0, and if U is equal or less than 42, we accept the alternative hypothesis H1.
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 7 12 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:
H0: Equal proportions of British and non- British cars have breakdowns.
H1: A higher proportion of British cars have breakdowns.
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(ii) Level of significanceα1=1%.
(iii) Let the sizes of the samples be nP and nQ, where nP=8 and nQ=10. The Mann-Whitney test compares every item in sample P in turn with every item in sample Q, a record being kept of the number of times, say, that the item from P is greater than Q, or vice-versa. In this case there are nPnQ, i.e. (8)(10)=80 compar- isons to be made. All the data is arranged into ascending order whilst retaining their separate identities—an easy way is to arrange a linear scale as shown in Fig. 63.1, on page 624.
From Fig. 63.1, a list of P’s and Q’s can be ranked giving:
P P Q P P Q P Q P P P Q Q Q Q Q Q Q
(iv) Write under each letter P the number of Q’s that precede it in the sequence, giving:
P P Q P P Q P Q P P P Q
0 0 1 1 2 3 3 3
Q Q Q Q Q Q
(v) Add together these 8 numbers, denoting the sum by U, i.e.
U=0+0+1+1+2+3+3+3=13 (vi) The critical regions are of the form U≤critical
region.
From Table 63.5, for a sample size 8 and 10 at significance levelα1=1% the critical regions is U≤13.
The value of U in our case, from (v), is 13 which is significant at 1% significance level.
The Mann-Whitney test has therefore confirmed that there is evidence that the non-British cars have better reliability than the British cars in the first 10 000 miles, i.e. the alternative hypothesis applies.
Problem 12. Two machines, A and B, are used to measure vibration in a particular rubber prod- uct. The data given below are the vibrational forces, in kilograms, of random samples from each machine:
A 9.7 10.2 11.2 12.4 14.1 22.3 29.6 31.7 33.0 33.2 33.4 46.2 50.7 52.5 55.4
B 20.6 25.3 29.2 35.2 41.9 48.5 54.1 57.1 59.8 63.2 68.5
Use the Mann-Whitney test at a significance level of 5% to determine if there is any evidence of the two machines producing different results.
Using the procedure:
(i) H0: There is no difference in results from the machines, on average.
H1: The results from the two machines are different, on average.
(ii) α2=5%.
(iii) Arranging the data in order gives:
9.7 10.2 11.2 12.4 14.1 20.6 22.3
A A A A A B A
25.3 29.2 29.6 31.7 33.0 33.2 33.4
B B A A A A A
35.2 41.9 46.2 48.5 50.7 52.5 54.1
B B A B A A B
55.4 57.1 59.8 63.2 68.5
A B B B B
(iv) The number of B’s preceding the A’s in the sequence is as follows:
A A A A A B A B B
0 0 0 0 0 1
A A A A A B B A B
3 3 3 3 3 5
A A B A B B B B
6 6 7
(v) Adding the numbers from (iv) gives:
U =0+0+0+0+0+1+3+3+3+3 +3+5+6+6+7=40
(vi) From Table 63.5, for n1=11 and n2=15, and α2=5%, U≤44.
Since our value of U from (v) is less than 44, H0 is rejected and H1 accepted, i.e. the results from the two machines are different.
622 STATISTICS AND PROBABILITY
Table 63.5 Critical values for the Mann-Whitney test
α1=5% 212% 1% 12% α1=5% 212% 1% 12%
n1 n2 α2=10% 5% 2% 1% n1 n2 α2=10% 5% 2% 1%
2 2 — — — — 4 17 15 11 8 6
2 3 — — — — 4 18 16 12 9 6
2 4 — — — — 4 19 17 13 9 7
2 5 0 — — — 4 20 18 14 10 8
2 6 0 — — —
2 7 0 — — — 5 5 4 2 1 0
2 8 1 0 — — 5 6 5 3 2 1
2 9 1 0 — — 5 7 6 5 3 1
2 10 1 0 — — 5 8 8 6 4 2
2 11 1 0 — — 5 9 9 7 5 3
2 12 2 1 — — 5 10 11 8 6 4
2 13 2 1 0 — 5 11 12 9 7 5
2 14 3 1 0 — 5 12 13 11 8 6
2 15 3 1 0 — 5 13 15 12 9 7
2 16 3 1 0 — 5 14 16 13 10 7
2 17 3 2 0 — 5 15 18 14 11 8
2 18 4 2 0 — 5 16 19 15 12 9
2 19 4 2 1 0 5 17 20 17 13 10
2 20 4 2 1 0 5 18 22 18 14 11
5 19 23 19 15 12
3 3 0 — — — 5 20 25 20 16 13
3 4 0 — — —
3 5 1 0 — — 6 6 7 5 3 2
3 6 2 1 — — 6 7 8 6 4 3
3 7 2 1 0 — 6 8 10 8 6 4
3 8 3 2 0 — 6 9 12 10 7 5
3 9 4 2 1 0 6 10 14 11 8 6
3 10 4 3 1 0 6 11 16 13 9 7
3 11 5 3 1 0 6 12 17 14 11 9
3 12 5 4 2 1 6 13 19 16 12 10
3 13 6 4 2 1 6 14 21 17 13 11
3 14 7 5 2 1 6 15 23 19 15 12
3 15 7 5 3 2 6 16 25 21 16 13
3 16 8 6 3 2 6 17 26 22 18 15
3 17 9 6 4 2 6 18 28 24 19 16
3 18 9 7 4 2 6 19 30 25 20 17
3 19 10 7 4 3 6 20 32 27 22 18
3 20 11 8 5 3
7 7 11 8 6 4
4 4 1 0 — — 7 8 13 10 7 6
4 5 2 1 0 — 7 9 15 12 9 7
4 6 3 2 1 0 7 10 17 14 11 9
4 7 4 3 1 0 7 11 19 16 12 10
4 8 5 4 2 1 7 12 21 18 14 12
4 9 6 4 3 1 7 13 24 20 16 13
4 10 7 5 3 2 7 14 26 22 17 15
4 11 8 6 4 2 7 15 28 24 19 16
4 12 9 7 5 3 7 16 30 26 21 18
4 13 10 8 5 3 7 17 33 28 23 19
4 14 11 9 6 4 7 18 35 30 24 21
4 15 12 10 7 5 7 19 37 32 26 22
4 16 14 11 7 5 7 20 39 34 28 24
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Table 63.5 (Continued)
α1=5% 212% 1% 12% α1=5% 212% 1% 12%
n1 n2 α2=10% 5% 2% 1% n1 n2 α2=10% 5% 2% 1%
8 8 15 13 9 7 12 14 51 45 38 34
8 9 18 15 11 9 12 15 55 49 42 37
8 10 20 17 13 11 12 16 60 53 46 41
8 11 23 19 15 13 12 17 64 57 49 44
8 12 26 22 17 15 12 18 68 61 53 47
8 13 28 24 20 17 12 19 72 65 56 51
8 14 31 26 22 18 12 20 77 69 60 54
8 15 33 29 24 20
8 16 36 31 26 22 13 13 51 45 39 34
8 17 39 34 28 24 13 14 56 50 43 38
8 18 41 36 30 26 13 15 61 54 47 42
8 19 44 38 32 28 13 16 65 59 51 45
8 20 47 41 34 30 13 17 70 63 55 49
13 18 75 67 59 53
9 9 21 17 14 11 13 19 80 72 63 57
9 10 24 20 16 13 13 20 84 76 67 60
9 11 27 23 18 16
9 12 30 26 21 18 14 14 61 55 47 42
9 13 33 28 23 20 14 15 66 59 51 46
9 14 36 31 26 22 14 16 71 64 56 50
9 15 39 34 28 24 14 17 77 69 60 54
9 16 42 37 31 27 14 18 82 74 65 58
9 17 45 39 33 29 14 19 87 78 69 63
9 18 48 42 36 31 14 20 92 83 73 67
9 19 51 45 38 33
9 20 54 48 40 36 15 15 72 64 56 51
15 16 77 70 61 55
10 10 27 23 19 16 15 17 83 75 66 60
10 11 31 26 22 18 15 18 88 80 70 64
10 12 34 29 24 21 15 19 94 85 75 69
10 13 37 33 27 24 15 20 100 90 80 73
10 14 41 36 30 26
10 15 44 39 33 29 16 16 83 75 66 60
10 16 48 42 36 31 16 17 89 81 71 65
10 17 51 45 38 34 16 18 95 86 76 70
10 18 55 48 41 37 16 19 101 92 82 74
10 19 58 52 44 39 16 20 107 98 87 79
10 20 62 55 47 42
17 17 96 87 77 70
11 11 34 30 25 21 17 18 102 92 82 75
11 12 38 33 28 24 17 19 109 99 88 81
11 13 42 37 31 27 17 20 115 105 93 86
11 14 46 40 34 30
11 15 50 44 37 33 18 18 109 99 88 81
11 16 54 47 41 36 18 19 116 106 94 87
11 17 57 51 44 39 18 20 123 112 100 92
11 18 61 55 47 42
11 19 65 58 50 45 19 19 123 112 101 93
11 20 69 62 53 48 19 20 130 119 107 99
12 12 42 37 31 27 20 20 138 127 114 105
12 13 47 41 35 31
624 STATISTICS AND PROBABILITY
8 7 5
4 10 12 1415
20 21 25 28 34
30 20
10 0
16 18 11
9 6 SAMPLE P
SAMPLE Q
Figure 63.1
Now try the following exercise.
Exercise 230 Further problems on the Mann-Whitney test
1. The tar content of two brands of cigarettes (in mg) was measured as follows:
Brand P 22.6 4.1 3.9 0.7 3.2 Brand Q 3.4 6.2 3.5 4.7 6.3 Brand P 6.1 1.7 2.3 5.6 2.0 Brand Q 5.5 3.8 2.1
Use the Mann-Whitney test at a 0.05 level of significance to determine if the tar contents of the two brands are equal.
⎡
⎢⎢
⎣
H0: TA =TB, H1: TA=TB, U =30.From Table 63.5, U ≤17, hence accept H0, i.e. there is no difference between the brands
⎤
⎥⎥
⎦
2. A component is manufactured by two pro- cesses. Some components from each process are selected at random and tested for breaking strength to determine if there is a difference between the processes. The results are:
Process A 9.7 10.5 10.1 11.6 9.8 Process B 11.3 8.6 9.6 10.2 10.9 Process A 8.9 11.2 12.0 9.2 Process B 9.4 10.8
At a level of significance of 10%, use the Mann-Whitney test to determine if there is a difference between the mean breaking
strengths of the components manufactured by the two processes.
⎡
⎢⎢
⎢⎢
⎢⎢
⎣
H0: B.S.A =B.S.B, H1: B.S.A=B.S.B, α2=10%, U=28.From Table 63.5, U≤15, hence accept H0, i.e. there is no difference between the processes
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
3. An experiment, designed to compare two pre- ventive methods against corrosion gave the following results for the maximum depths of pits (in mm) in metal strands:
Method
A 143 106 135 147 139 132 153 140 Method
B 98 105 137 94 112 103
Use the Mann-Whitney test, at a level of significance of 0.05, to determine whether the two tests are equally effective.
⎡
⎢⎢
⎢⎢
⎣
H0: A=B, H1: A=B, α2=5%, U=4.From Table 63.5,U≤8, hence null hypothesis is rejected, i.e. the two methods are not equally effective
⎤
⎥⎥
⎥⎥
⎦
4. Repeat Problem 3 of Exercise 228, page 616 using the Mann-Whitney test.
⎡
⎢⎢
⎢⎢
⎣
H0: meanA =meanB, H1: meanA=meanB, α2=5%, U =90
From Table 63.5, U≤99, hence H0is rejected and H1accepted
⎤
⎥⎥
⎥⎥
⎦
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Statistics and probability
Assignment 17
This assignment covers the material contained in Chapters 61 to 63.
The marks for each question are shown in brackets at the end of each question.
1. 1200 metal bolts have a mean mass of 7.2 g and a standard deviation of 0.3 g. Determine the standard error of the means. Calculate also the probability that a sample of 60 bolts chosen at random, without replacement, will have a mass of (a) between 7.1 g and 7.25 g, and (b) more
than 7.3 g. (12)
2. A sample of 10 measurements of the length of a component are made and the mean of the sample is 3.650 cm. The standard deviation of the samples is 0.030 cm. Determine (a) the 99%
confidence limits, and (b) the 90% confidence limits for an estimate of the actual length of the
component. (10)
3. An automated machine produces metal screws and over a period of time it is found that 8%
are defective. Random samples of 75 screws are drawn periodically.
(a) If a decision is made that production con- tinues until a sample contains more than 8 defective screws, determine the type I error based on this decision for a defect rate of 8%.
(b) Determine the magnitude of the type II error when the defect rate has risen to 12%.
The above sample size is now reduced to 55 screws. The decision now is to stop the machine for adjustment if a sample contains 4 or more defective screws.
(c) Determine the type I error if the defect rate remains at 8%.
(d) Determine the type II error when the defect
rate rises to 9%. (22)
4. In a random sample of 40 similar light bulbs drawn from a batch of 400 the mean lifetime is found to be 252 hours. The standard deviation of the lifetime of the sample is 25 hours. The batch is classed as inferior if the mean lifetime of the batch is less than the population mean of 260 hours. As a result of the sample data, determine whether the batch is considered to be inferior at a level of significance of (a) 0.05, and (b) 0.01. (9) 5. The lengths of two products are being compared.
Product 1: sample size=50, mean value of sample=6.5 cm, standard devia- tion of whole of batch=0.40 cm.
Product 2: sample size=60, mean value of sample=6.65 cm, standard devia- tion of whole of batch=0.35 cm.
Determine if there is any significant difference between the two products at a level of significance of (a) 0.05, and (b) 0.01. (7) 6. The resistance of a sample of 400 resistors pro- duced by an automatic process have the following resistance distribution.
Resistance Frequency ()
50.11 9
50.15 35
50.19 61
50.23 102
50.27 89
50.31 83
50.35 21
Calculate for the sample: (a) the mean, and (b) the standard deviation. (c) Test the null hypothesis that the resistance of the resistors are normally distributed at a level of significance of 0.05, and
626 STATISTICS AND PROBABILITY
determine if the distribution gives a ‘too good’ fit at a level of confidence of 90%. (25) 7. A fishing line is manufactured by two processes, A and B. To determine if there is a difference in the mean breaking strengths of the lines, 8 lines by each process are selected and tested for breaking strength. The results are as follows:
Process A 8.6 7.1 6.9 6.5 7.9 6.3 7.8 8.1 Process B 6.8 7.6 8.2 6.2 7.5 8.9 8.0 8.7
Determine if there is a difference between the mean breaking strengths of the lines manufac- tured by the two processes, at a significance level of 0.10, using (a) the sign test, (b) the Wilcoxon signed-rank test, (c) the Mann-Whitney test.
(15)
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Laplace transforms
K
64
Introduction to Laplace transforms