10.2478/v10048-009-0004-8
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Critical Values for Testing Location-Scale Hypothesis
František Rublı́k
Institute of Measurement Science, Slovak Academy of Sciences, Dúbravská cesta 9, 84104 Bratislava, Slovakia
E-mail: umerrubl@savba.sk
The exact critical points for selected sample sizes and significance levels are tabulated for the two-sample test statistic which
is a combination of the Wilcoxon and the Mood test statistic. This statistic serves for testing the null hypothesis that two sampled
populations have the same location and scale parameters.
Keywords: distribution-free rank test; combination of rank statistics; small sample sizes; location-scale hypothesis.
1. Introduction
S
better results than the statistics constructed especially for the
one type change of location or constructed especially for the
one type change of scale. Readers interested in further discussion of the need of testing the hypothesis (2) can found further
arguments in Section 1 of [6] or in [3].
We shall deal with the test based on the ranks of the observations. As is well known, the advantage of such a test is
that in the setting (1) with F continuous, the use of this rank
test guarantees that the significance level will have the chosen
value α. The null hypothesis (2) is against the alternative H1
that at least one of the equalities (2) does not hold, tested usually by means of the Lepage test from [4]. Critical constants
of this test for small sample sizes can be found in [5], the Lepage test is included also into the monograph [2]. The Lepage
test statistic is given by the formula
uppose that the random variables X, Y have the distribution functions
!
!
t − µY
t − µX
, P(Y ≤ t) = F
, (1)
P(X ≤ t) = F
σX
σY
where the real numbers µX , µY denote the location parameters, σX > 0, σY > 0 are scale parameters and F is a continuous distribution function defined on the real line. Assume that X1 , . . . , Xm is a random sample from the distribution of X, Y1 , . . . , Yn is a random sample from the distribution of Y and these random samples are independent. Let
(R1 , . . . , RN ), N = m+n, denote the ranks of the pooled sample
(X1 , . . . , Xm , Y1 , . . . , Yn ). The hypothesis
H0 : µX = µy ,
σ X = σY
T = T K + T B,
(2)
(S W − E(S W |H0 ))
(S B − E(S B |H0 ))2 (3)
, TB =
,
Var(S W |H0 )
Var(S B |H0 )
P
where S W = m
i=1 Ri is the Wilcoxon rank test statistic and
Pm
S B = i= aN (Ri ) is the Ansari-Bradley rank test statistic (i.e.,
the vector of scores aN = (1, 2, ..., k, k, . . . , 2, 1) if N = 2k and
aN = (1, 2, . . . , k, k + 1, k, . . . , 2, 1) if N = 2k + 1). An analogous statistic has been formulated in the multisample setting in [7], another test statistics for this problem have been
studied in [8], where also their non-centrality parameters for
testing the null location-scale hypothesis are for some situations computed. As shown on p. 283 of [8], the bounds
for the asymptotic efficiency in the case of there considered
sampled distributions and quadratic rank statistics do not depend on the number of sampled populations. As concluded
ibidem, taking into account computed values of the asymptotic efficiencies, one sees that a combination of the multisample Kruskal-Wallis statistic (in the two-sample case the
Wilcoxon test statistic) with the Mood test statistic appears
to be a good choice when one considers symmetric distributions whose type of tail weight is unknown. The statistics in
[8] are defined in the general multisample setting, but they
can be used also in the two-sample setting for small sample
sizes, when the tables of the critical constants can be computed. However, the only available tables of critical constants
2
TK =
is sometimes called also the null location-scale hypothesis.
If the populations X and Y differ in their location parameters only, then the tests designed for detecting this change
(e.g. Wilcoxon-Mann-Whitney or van der Waerden test) can
be used to detect this difference. Similarly, if X and Y differ in
the scale parameter only, then the tests designed for detecting
this change (e.g. Mood or Ansari-Bradley test) can be used.
However, the tests designed for detecting difference of the location usually do not perform well when the change occurs
in the scale parameter only. Similarly, the tests designed for
detecting difference of the scale usually do not perform well
when the change occurs in the location parameter only. But in
the real situation it is not known which type of change will occur and if the experimenter is interested both in the difference
of the location and also in the difference of scale, the use of
the test statistics mentioned before will not be advantageous.
This is due to the fact that such a difference cannot be simply
detected in the one step, and the two step use of the mentioned tests would lead to the increase of the nominal level of
significance (the probability of the error of type I). But what
is even more important is the fact, that in practice the change
in the location is very often accompanied by the change in
scale, and in such a case the statistics constructed especially
for testing the location-scale null hypothesis (2) usually yield
9
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
for testing the location scale-hypothesis are those published
in [5]. The mentioned table is computed for m=2(1)30, n=2;
m=3(1)26, n=3; m=4(1)18, n=4; m=5(1)13, n=5; m=6(1)10,
n=6; m=7,8, n=7 and the significance levels α = 0.01, 0.02,
0.05, 0.10 and 0.2. In this paper also the significance level
α = 0.005 is included, the extremely unbalanced sample
sizes like m = 2, n = 30 are not considered and instead of
these critical constants for some other small sample sizes for
m = 7(1)10 are computed.
2. Basic formulas
P
k2 ≤ (r + s)2 , then sj=1 i2j = k2 if and only if this equality
holds and (r + s) < {i1 , . . . , i s }.
(II) This assertion follows from the fact, that D(s, r, k1 , k2 )
is a union of the disjoint sets D(s, r − 1, k1 , k2 ) and
P
P
{(i1 , . . . , i s ) ∈ J(r, s); i s = s + r, sj=1 i j = k1 , sj=1 i2j = k2 }. If R = (R1 , . . . , RN ) is a random vector which is uniformly distributed over the set of all permutations of the set
{1, . . . , N}, then according to Theorem 1 on p. 167 of [1] for
any set A ⊂ {1, . . . , N} consisting of m distinct integers
!
N
P({R1 , . . . , Rm } = A) = 1
.
m
The topic of this paper is the computation of the critical values
for the two-sample test statistic (4). In accordance with [8] la- Combining this equality with Lemma 1 one can construct a
bel the combination of the Wilcoxon and the Mood statistic program for computation of critical values of the statistic (4).
by T S Q . Thus in the notation from the previous section
3. Tables of critical values
T S Q = T K + Q,
12
m(N + 1)
SW −
mn(N + 1)
2
!2
In this section we present the exact critical values of the statistic T S Q from (4), given in Table 2. First we describe the output
!2
of
the table.
2
(4)
m(N − 1)
180
Since the set V = V(m, n) of possible values of the statis,
S̃
−
Q=
12
mn(N + 1)(N 2 − 4)
tic
T
S Q is finite for all sample sizes m, n, generally one cannot
!2
m
X
find exact critical values for arbitrary prescribed probability α
N+1
S̃ =
Ri −
.
of the type I error. In the following table the number on the
2
i=1
intersection of the column for w with the row for significance
Since according to the assumptions the distribution function level α denotes the quantity
F in (1) is continuous, the statistic T S Q is distribution-free
w = min{t ∈ V; P(T S Q ≥ t) ≤ α},
whenever the null hypothesis (2) holds. The null hypothesis
(2) is rejected whenever T S Q ≥ wα . Values of wα = wα (m, n)
the entry corresponding to α and w is the quantity
can be found in the table presented in the next section, for the
sample sizes not included into the table instead of wα use the
w = max{t ∈ V; P(T S Q ≥ t) > α}.
(1 − α)th quantile of the chi-square distribution with 2 degrees
of freedom.
Further, for given α,
In the computation of the tables of this paper the following
p = P(T S Q ≥ w), p = P(T S Q ≥ w)
lemma is useful.
TK =
,
denote the corresponding probabilities of the type I error.
Thus p is the largest available significance level not exceeding α and p is the smallest available significance level greater
than α. The value of critical constant yielding the significance
level closer to the nominal level α is printed in boldface letter. If the difference in computed values exceeds the number
of decimal places used to describe the result of computation,
then the boldface symbol is used for the value corresponding
to p.
Lemma 1. Let J(m, n) denote the set of all m-tuples
(i1 , . . . , im ) consisting of integers such that 1 ≤ i1 < . . . <
im ≤ m + n. Suppose that
n
D(m, n, k1 , k2 ) = (i1 , . . . , im )
∈ J(m, n);
m
X
j=1
i j = k1 ,
m
X
j=1
o
i2j = k2 ,
and B(m, n, k1 , k2 ) denotes the number of elements of
D(m, n, k1 , k2 ).
4. Some simulation results
(I) Let s > 1. If at least one of the inequalities k1 ≤ (r + s),
k2 ≤ (r + s)2 holds, then B(s, r, k1 , k2 ) = B(s, r − 1, k1 , k2 ).
The aim of the simulation is to obtain a picture of the power
(II) If k1 > (s + r), k2 > (s + r)2 , then B(s, r, k1 , k2 ) = of tests based on the statistics (3) and (4) for small sample
B(s, r − 1, k1 , k2 ) + B(s − 1, r, k1 − (r + s), k2 − (r + s)2 ).
sizes covered by Table 2. To consider power of the concerned
tests in the case of distributions with various tail behaviour,
P
Proof. (I) If k1 ≤ r + s, then sj=1 i j = k1 if and only if the sampling from normal, logistic and Cauchy distribution is
this equality holds and (r + s) < {i1 , . . . , i s }. Similarly, if employed, in each case µX = 0, σX = 1. The simulations are
10
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Table 1: Simulation estimates of the power.
m = 3, n = 7
µY = 1.5, σY = 3
P(T S Q ≥ wα |Normal)
P(T ≥ tα|Normal)
P(T S Q ≥ wα |Logistic)
P(T ≥ tα|Logistic)
P(T S Q ≥ wα |Cauchy)
P(T ≥ tα|Cauchy)
α = 0.05
0.057
0.032
0.032
0.017
0.041
0.025
µY = 3.5, σY = 4
α = 0.10
0.073
0.141
0.041
0.131
0.058
0.114
α = 0.05
0.202
0.136
0.073
0.047
0.084
0.060
α = 0.10
0.224
0.251
0.086
0.170
0.104
0.165
µY = 7, σY = 5
α = 0.05
0.533
0.435
0.191
0.137
0.174
0.138
α = 0.10
0.547
0.539
0.208
0.247
0.197
0.233
m = 7, n = 8
µY = 1.5, σY = 3
P(T S Q ≥ wα |Normal)
P(T ≥ tα|Normal)
P(T S Q ≥ wα |Logistic)
P(T ≥ tα|Logistic)
P(T S Q ≥ wα |Cauchy)
P(T ≥ tα|Cauchy)
α = 0.05
0.448
0.415
0.363
0.326
0.228
0.219
µY = 3.5, σY = 4
α = 0.10
0.640
0.605
0.540
0.510
0.323
0.356
α = 0.05
0.745
0.730
0.586
0.547
0.431
0.419
α = 0.10
0.886
0.868
0.765
0.732
0.566
0.568
µ2 = 7, σ2 = 5
α = 0.05
0.950
0.952
0.796
0.774
0.659
0.643
α = 0.10
0.987
0.984
0.915
0.900
0.759
0.757
based in each case on 10000 trials. The critical constants wα Lepage test.
of the statistic T S Q defined in (4) are taken from Table 2, the
critical constants tα of the statistic (3) are those computed in
Acknowledgments
[5]. The results of simulations are in Table 1 .The power better of the two considered cases is emphasized by the boldface
The research was supported by the grants VEGA 1/0077/09,
type.
APVV-SK-AT-0003-09, and APVV-RPEU-0008-06.
5. Discussion and conclusions
References
As shown on p. 283 of [8], the bounds for the asymptotic efficiency of the test statistics (3) and (4) in the case of there
considered sampled distributions do not depend on the number of sampled populations, i.e., they are the same in the
two-sample case and in the multisample case. As concluded
in [8], taking into account computed values of the asymptotic efficiencies, one sees that a combination of the multisample Kruskal-Wallis statistic (in the two-sample case the
Wilcoxon test statistic) with the Mood test statistic appears to
be a good choice when one considers symmetric distributions
whose type of tail weight is unknown. However, these considerations are related to the asymptotic case, when both m and n
tend to infinity. The simulation results, given in Table 1 do not
contradict the mentioned asymptotic results. After inspecting
Table 1 it can be said that for small sample sizes and α = 0.05
testing based on (4) is preferable to (3), but for α = 0.1 it is
advisable to use the Lepage test. This suggests that the test
based on (4) can be considerered as a useful competitor to the
11
[1] Hájek, J., Šidák, Z. & Sen, P. K. (1999). Theory of Rank Tests.
Academic Press, San Diego.
[2] Hollander, M. & Wolfe, D. A. (1999). Nonparametric Statistical Methods. Wiley, New York.
[3] Hutchison, T. P. (2002). Should we routinely test for simultaneous location and scale changes? Ergonomics 45, 248-251.
[4] Lepage, Y. (1971). A combination of Wilcoxon’s and AnsariBradley’s statistics. Biometrika 58, 213-217.
[5] Lepage, Y. (1973). A table for a combined Wilcoxon AnsariBradley statistic. Biometrika 60, 113-116.
[6] Podgor, M. J. & Gastwirth, J. L. (1994). On non-parametric and
generalized tests for the two-sample problem with location and
scale change alternatives. Statistics in Medicine 13, 747-758.
[7] Rublı́k, F. (2005). The multisample version of the Lepage test.
Kybernetika 41, 713-733.
[8] Rublı́k, F. (2007). On the asymptotic efficiency of the multisample location-scale rank tests and their adjustment for ties.
Kybernetika 43, 279-306.
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Table 2: Critical values of the test statistic T S Q from (4).
α
w
p
m=3
0.200
0.100
0.050
0.020
0.010
0.005
3.8571
3.8571
.100
.100
m=3
0.200
0.100
0.050
0.020
0.010
0.005
3.5520
4.2077
6.3272
.179
.083
.048
m=3
0.200
0.100
0.050
0.020
0.010
0.005
3.2649
4.1147
5.5213
8.1098
8.9011
.191
.082
.045
.018
.009
m=3
0.200
0.100
0.050
0.020
0.010
0.005
3.0754
4.1233
5.3559
7.3751
9.3529
10.8733
.185
.099
.048
.018
.009
.0044
m=3
0.200
0.100
0.050
0.020
0.010
0.005
2.9745
4.1324
5.5500
7.7052
8.9403
10.4904
.196
.098
.049
.020
.010
.0049
m=3
0.200
0.100
0.050
0.020
0.010
0.005
2.9361
3.9796
5.3942
7.5072
9.0206
10.9565
.200
.100
.050
.020
.009
.0045
w
p
w
n=3
3.6428
3.6428
3.8571
3.8571
3.8571
3.8571
m=3
.300
.300
.100
.100
.100
.100
4.1666
n=6
3.2796
3.6121
4.7342
6.4519
6.4519
6.4519
.202
.107
.071
.024
.024
.024
3.1298
4.2532
5.5000
7.3181
.200
.100
.055
.027
.018
.009
3.0649
4.1558
5.7662
8.5714
9.6103
.207
.103
.052
.022
.013
.0088
3.0101
4.0495
5.4958
7.5350
9.6862
11.4342
.196
.100
.046
.018
.007
.0036
m=3
.201
.103
.051
.022
.012
.0074
2.9600
4.0403
5.3113
7.7315
9.4113
10.9618
n=18
2.9346
3.9680
5.3563
7.2937
9.0135
10.2827
.199
.094
.049
.014
.007
m=3
n=15
2.9535
4.0342
5.3508
7.2552
8.5929
10.2632
.2
.100
.0500
.017
m=3
n=12
3.0509
4.0607
5.3110
6.9951
8.8842
9.3529
.143
m=3
n=9
3.1184
4.0659
5.4434
6.9572
8.1098
8.9011
p
.198
.098
.050
.019
.008
.0041
m=3
.201
.102
.054
.021
.011
.0060
2.9725
4.0114
5.5537
7.5125
9.2757
11.1522
.197
.099
.049
.019
.009
.0039
12
w
p
w
n=4
3.1666
4.6666
4.6666
4.6666
4.6666
4.6666
m=3
.257
.114
.114
.114
.114
.114
3.9555
5.5555
n=7
3.0519
4.0454
4.2980
7.0000
7.3181
7.3181
.217
.117
.067
.033
.017
.017
3.3803
4.1880
6.2435
8.1367
.203
.108
.056
.021
.014
.007
3.0363
4.0621
5.1757
8.2787
10.2667
.204
.104
.050
.021
.011
.0071
2.9507
4.0935
5.5371
7.7827
9.9883
11.9532
.200
.099
.047
.018
.009
.0029
m=3
.200
.100
.052
.021
.010
.0062
2.94899
3.9981
5.5137
7.5665
9.5494
11.4046
n=19
2.9416
4.0034
5.5308
7.4759
9.1235
10.6854
.198
.099
.049
.016
.006
m=3
n=16
2.9512
3.9953
5.2542
7.3407
9.0836
10.5143
.176
.091
.036
.012
m=3
n=13
2.9532
4.0478
5.3165
7.4057
9.4535
9.6862
.143
.071
m=3
n=10
3.0026
3.8961
5.1532
7.6363
8.5714
9.6103
p
.198
.098
.049
.019
.009
.0035
m=3
.200
.100
.051
.022
.010
.0052
2.9438
4.0038
5.4752
7.7419
9.4466
11.0676
.199
.100
.050
.019
.009
.0045
w
p
2.8888
3.9555
5.5555
5.5555
5.5555
5.5555
.214
.143
.071
.071
.071
.071
n=5
n=8
3.2435
4.0213
4.9359
7.5897
8.1367
8.1367
.212
.103
.061
.024
.012
.012
n=11
2.9454
4.0606
5.0303
6.8015
8.9833
10.2667
.203
.104
.055
.022
.011
.0055
n=14
2.9239
3.9398
5.4970
7.6424
8.4444
9.9883
.206
.101
.050
.021
.012
.0059
n=17
2.94899
3.9879
5.4526
7.2454
8.7461
10.7446
.200
.100
.051
.021
.011
.0053
n=20
2.9419
3.9342
5.4142
7.7142
9.3609
10.8086
.202
.101
.051
.020
.010
.0056
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Table 2 (cont.)
α
w
p
m=3
0.200
0.100
0.050
0.020
0.010
0.005
2.9728
4.0028
5.4801
7.6875
9.5134
11.1861
.199
.100
.049
.020
.010
.0049
m=3
0.200
0.100
0.050
0.020
0.010
0.005
2.9346
3.9901
5.5686
7.6349
9.4075
10.8813
.198
.099
.050
.019
.010
.0048
m=4
0.200
0.100
0.050
0.020
0.010
0.005
4.0833
5.3333
.143
.057
m=4
0.200
0.100
0.050
0.020
0.010
0.005
3.3150
4.3809
5.4065
7.3589
7.8974
8.0769
.197
.094
.045
.015
.009
.0030
m=4
0.200
0.100
0.050
0.020
0.010
0.005
3.2362
4.2450
5.3450
7.0050
7.8862
9.6562
.199
.099
.049
.019
.009
.0050
m=4
0.200
0.100
0.050
0.020
0.010
0.005
3.1821
4.1970
5.3630
7.1794
8.2294
9.6545
.200
.100
.049
.019
.010
.0046
w
p
w
n=21
2.9421
3.9893
5.4174
7.4701
9.0867
10.1608
m=3
.201
.101
.050
.021
.011
.0059
2.9334
4.0059
5.4788
7.6777
9.4325
11.5446
n=24
2.9305
3.9696
5.5463
7.5029
9.1548
10.6064
.200
.100
.051
.021
.010
.0055
2.9230
3.9645
5.5424
7.6441
9.1838
10.9392
.257
.114
.057
.057
.057
.057
3.4088
4.0353
5.4285
6.3636
6.3636
.203
.100
.052
.021
.015
.0091
3.3681
4.2706
5.1442
6.3804
8.7012
.198
.099
.048
.018
.006
m=4
.201
.101
.051
.021
.013
.0070
3.2417
4.2284
5.2468
6.8384
8.2702
10.2907
n=13
3.1713
4.1781
5.3576
7.0958
8.1727
9.5330
.198
.087
.040
.008
.008
m=4
n=10
3.2050
4.2262
5.3362
6.5362
7.6762
9.0262
.199
.100
.049
.020
.010
.0049
m=4
n=7
3.2637
4.3260
5.3809
6.5897
7.3589
7.8974
.200
.100
.050
.0200
.010
.0043
m=3
n=4
3.9833
4.3333
5.3333
5.3333
5.3333
5.3333
p
.200
.099
.050
.020
.010
.0037
m=4
.200
.102
.050
.020
.011
.0055
3.1729
4.1513
5.2953
7.2302
8.4144
9.9476
.200
.100
.050
.020
.009
.0049
13
w
p
w
n=22
2.9296
3.9927
5.4576
7.5263
9.0514
10.5105
m=3
.201
.100
.050
.021
.010
.0052
2.9329
4.0080
5.5554
7.6948
9.2269
10.845
n=25
2.9196
3.9645
5.4941
7.6152
9.0769
10.4910
.200
.101
.050
.020
.010
.0055
2.9518
3.9828
5.6001
7.7392
9.1764
10.6898
.214
.103
.056
.024
.024
.008
3.5681
4.5284
4.8465
7.0000
7.2727
7.2727
.202
.103
.053
0.022
.010
.0061
3.2467
4.2337
5.5064
6.9610
8.2337
9.4285
.194
.099
.046
.018
.010
.0042
m=4
.201
.100
.051
.021
.011
.0051
3.2324
4.2535
5.3403
7.0168
8.7682
9.3060
n=14
3.1701
4.1393
5.2615
7.1506
8.3009
9.6938
.190
.095
.048
.014
.005
.0048
m=4
n=11
3.2056
4.1673
5.2347
6.7829
8.2625
9.7909
.200
.100
.050
.020
.010
.0049
m=4
n=8
3.3269
4.2417
5.0824
6.3804
8.1978
8.7912
.200
.099
.049
.019
.010
.0046
m=3
n=5
3.4088
3.9885
5.2322
6.1298
6.1298
6.3636
p
.200
.100
.050
.019
.009
.0049
m=4
.200
.100
.051
.020
.010
.0056
3.1781
4.1361
5.3610
7.3428
8.5902
10.1045
.200
.099
.050
.020
.010
.0049
w
p
2.9314
3.9974
5.5378
7.5875
9.0567
10.5965
.201
.100
.050
.020
.011
.0054
n=23
n=26
2.9288
3.9775
5.5736
7.6155
9.0829
10.6528
.200
.100
.050
.020
.010
.0055
n=6
3.4375
4.1363
4.8181
6.4375
7.0000
7.0000
.200
.105
.057
.024
.014
.014
n=9
3.2467
4.1991
5.3766
6.9350
7.1428
8.9610
.200
.105
.052
.021
.013
0. 070
n=12
3.1799
4.2156
5.2913
7.0035
8.3375
8.9698
.201
.101
.052
.020
.010
.0060
n=15
3.1734
4.1280
5.3381
7.3045
8.5122
9.7811
.201
.100
.050
.020
.010
.0054
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Table 2 (cont.)
α
w
p
m=4
0.200
0.100
0.050
0.020
0.010
0.005
3.1769
4.1680
5.3877
7.3336
8.5665
10.0103
0.199
.100
.050
.020
.010
.0047
m=4
0.200
0.100
0.050
0.020
0.010
0.005
3.1586
4.1541
5.3977
7.3112
8.7481
10.2226
.200
.100
.049
.020
.010
.0049
m=5
0.200
0.100
0.050
0.020
0.010
0.005
3.5290
4.4618
5.1818
6.8181
6.8181
.198
.095
.048
.008
.008
m=5
0.200
0.100
0.050
0.020
0.010
0.005
3.2961
4.4026
5.4935
6.7558
7.4415
8.3532
.197
.096
.046
.018
.009
.0047
m=5
0.200
0.100
0.050
0.020
0.010
0.005
3.2895
4.3676
5.4004
6.7731
8.0235
9.4096
.200
.100
.050
.020
.010
.0046
m=5
0.200
0.100
0.050
0.020
0.010
0.005
3.2444
4.3277
5.4038
7.0441
8.2403
9.5495
.200
.100
.050
.020
.010
.0050
w
p
w
n=16
3.1669
4.1661
5.3779
7.2881
8.5638
9.8625
m=4
.200
.101
.051
.020
.010
.0052
3.1482
4.2149
5.4040
7.3345
8.6124
9.9360
n=19
3.1398
4.1436
5.3751
7.2699
8.7225
10.1015
.200
.100
.050
.020
.010
.0051
3.1397
4.1493
5.3985
7.3358
8.7746
10.2820
.214
.111
.063
.024
.024
.008
3.4017
4.5094
5.0735
6.2940
7.4837
7.6068
.202
.106
.051
.021
.011
.0062
3.2900
4.4666
5.4444
6.6944
7.7777
8.2711
.200
.095
.050
.020
.010
.0050
m=5
.200
.100
.051
.020
.010
.0050
3.2467
4.3321
5.4362
6.9356
8.2432
9.1812
n=14
3.2438
4.3228
5.3989
7.0403
8.1971
9.4789
.197
.097
.050
.019
.009
.0043
m=5
n=11
3.2565
4.3590
5.3943
6.7034
7.9123
9.1162
.200
.100
.050
.020
.010
.0050
m=5
n=8
3.2961
4.3246
5.3532
6.4285
7.3168
8.2987
.200
.100
.050
.020
.010
.0048
m=4
n=5
3.2618
4.1836
4.8381
6.4090
6.4090
6.8181
p
.200
.100
.050
.020
.010
.0048
m=5
.200
.100
.050
.020
.010
.0052
3.2299
4.3187
5.4186
7.0269
8.4011
9.5897
.200
.100
.050
.020
.010
.0049
14
w
p
w
n=17
3.1464
4.2096
5.4038
7.2867
8.5756
9.7931
m=4
.201
.100
.050
.020
.010
.0052
3.1530
4.1820
5.3659
7.2789
8.7934
10.1821
n=20
3.1373
4.1479
5.3844
7.3185
8.7486
10.1472
.200
.100
.050
.020
.010
.0052
3.1371
4.1464
5.4134
7.4488
8.9051
10.2684
.201
.102
.058
.024
.011
.0087
3.3827
4.5054
5.4474
6.5852
7.4442
.201
.102
.051
.021
.011
.0060
3.3292
4.3595
5.3404
6.8762
7.6948
8.9092
.200
.100
.050
.020
.010
.0047
m=5
.200
.101
.050
.020
.010
.0052
3.2576
4.3295
5.4002
6.9724
8.2860
9.3724
n=15
3.2294
4.3173
5.4154
7.0255
8.3831
9.5786
.199
.098
.048
.018
.008
m=5
n=12
3.2444
4.3251
5.4339
6.8584
8.1988
8.9286
.200
.100
.050
.020
.010
.0049
m=5
n=9
3.2622
4.4100
5.4233
6.6400
7.5600
8.2400
.200
.099
.050
.0202
.010
.0048
m=4
n=6
3.3333
4.4273
5.0427
6.0683
6.8376
7.4837
p
.200
.100
.050
.020
.010
.0049
m=6
.200
.100
.050
.020
.010
.0050
3.4981
4.4981
5.4212
6.3553
7.4102
8.0109
.199
.100
.048
.019
.006
.0043
w
p
3.1413
4.1775
5.3559
7.2681
8.7173
10.1168
.200
.100
.050
.020
.010
.0051
n=18
n=21
3.1313
4.1330
5.4077
7.4122
8.8223
10.2359
.200
.100
.050
.020
.010
.0051
n=7
3.3450
4.4376
5.3871
6.4270
7.3538
8.4615
.202
.101
.051
.020
.010
.0051
n=10
3.3143
4.3405
5.3241
6.7239
7.6066
8.8310
.200
.101
.050
.020
.011
.0053
n=13
3.2576
4.3216
5.4002
6.9305
8.1773
9.2534
.200
.100
.050
.020
.010
.0051
n=6
3.2674
4.4835
5.4212
6.1648
7.2527
7.4102
.203
.104
.052
.024
.011
.0065
MEASUREMENT SCIENCE REVIEW, Volume 9, Section 1, No. 1, 2009
Table 2 (cont.)
α
w
p
m=6
w
p
w
n=7
p
w
m=6
p
n=8
3.3543
4.4749
5.4545
6.5677
7.2727
8.2300
.200
.100
.048
.019
.010
.0041
3.3321
4.4452
5.4545
6.3747
6.9758
8.0222
.203
.101
.050
.020
.011
.0051
3.3500
4.4468
5.4333
6.5302
7.3375
8.3218
.197
.100
.050
.020
.010
.0047
3.3375
4.4385
5.4218
6.5218
7.2708
8.1968
.201
.101
.051
.020
.011
.0053
0.200
0.100
0.050
0.020
0.010
0.005
3.3170
4.4683
5.4700
6.8313
7.7563
8.5019
m=6
.200
.100
.049
.020
.010
.0049
n=10
3.3086
4.4487
5.4683
6.8016
7.6605
8.4733
.201
.101
.050
.020
.010
.0054
3.2812
4.4572
5.4694
6.8474
7.8872
8.9718
m=6
.200
.100
.050
.020
.010
.0048
n=11
3.2748
4.4550
5.4630
6.8410
7.8745
8.8718
.202
.100
.051
.020
.010
.0050
0.200
0.100
0.050
0.020
0.010
0.005
3.3020
4.4653
5.5602
6.7183
7.2500
8.0500
.200
.100
.049
.020
.010
.0047
m=7
0.200
0.100
0.050
0.020
0.010
0.005
3.3223
4.5132
5.5739
6.8912
7.7453
8.5894
.199
.100
.050
.020
.010
.0050
m=8
0.200
0.100
0.050
0.020
0.010
0.005
3.3161
4.5215
5.5567
6.8413
7.6685
8.2352
.199
.099
.050
.020
.010
.0050
m=9
0.200
0.100
0.050
0.020
0.010
0.005
3.3001
4.5414
5.6218
6.8923
7.7602
8.5224
.200
.100
.050
.020
.010
.0050
n=7
3.2857
4.4500
5.5551
6.6163
7.0499
7.8622
m=7
.203
.101
.050
.021
.011
.0058
3.3223
4.4931
5.5029
6.7162
7.4921
8.0855
n=10
3.2982
4.4972
5.5719
6.8711
7.7052
8.5714
m=7
.201
.100
.050
.020
.010
.0051
3.2836
4.5140
5.5428
6.8373
7.7821
8.7792
n=8
3.3114
4.5063
5.5509
6.7904
7.5215
8.1060
.200
.100
.050
.020
.010
.0050
m=8
.200
.101
.050
.020
.010
.0053
3.3294
4.5126
5.6062
6.8323
7.6510
8.4580
n=9
3.2938
4.5321
5.6174
6.8849
7.7500
8.4931
.200
.100
.050
.020
.010
.0048
.199
.100
.050
.020
.010
.0050
m=9
.200
.100
.050
.020
.010
.0051
3.2831
4.5346
5.6481
6.9801
7.8680
8.7150
.200
.100
.050
.020
.010
.0050
15
m=6
.200
.100
.0050
.020
.010
.0054
n=12
3.2998
4.4561
5.5087
6.8837
7.9539
9.0235
.200
.100
.050
.020
.010
.0050
3.2998
4.4473
5.5065
6.8771
7.9100
8.9183
.200
.100
.050
.020
.010
.0051
3.3445
4.5012
5.5694
6.7635
7.5478
8.4740
3.2905
4.4804
5.6012
6.9289
7.9493
8.9306
n=9
3.3138
4.4990
5.6042
6.8245
7.6413
8.4541
.200
.100
.050
.020
.010
.0050
m=8
.200
.100
.050
.020
.010
.0051
3.2947
4.5401
5.6291
6.9243
7.7567
8.6375
n=10
3.2815
4.5308
5.6468
6.9779
7.8563
8.7101
.198
.100
.050
.020
.010
.0047
m=7
.200
.100
.050
.020
.010
.0050
.200
.100
.050
.020
.010
.0050
m=10
.200
.100
.100
.020
.010
.0050
n=9
3.2843
4.5020
5.4296
6.7569
7.4612
8.2782
n=11
3.2831
4.5140
5.5428
6.8373
7.7805
8.7648
p
.199
.100
.050
.020
.010
.0050
m=7
.200
.100
.050
.020
.010
.0051
w
3.2953
4.5037
5.4449
6.7825
7.5588
8.2953
n=8
3.3189
4.4921
5.5029
6.6980
7.4686
7.9873
p
m=6
0.200
0.100
0.050
0.020
0.010
0.005
m=7
w
3.2938
4.5354
5.6922
7.0358
7.9567
8.7702
.200
.100
.050
.020
.010
.0050
n=9
3.3328
4.4756
5.5480
6.7581
7.5424
8.4441
.200
.100
.050
.020
.010
.0051
n=12
3.2871
4.4789
5.6000
6.9159
7.9373
8.9087
.201
.101
.050
.020
.010
.0050
n=10
3.2922
4.5401
5.6236
6.9159
7.7552
8.6138
.200
.100
.050
.020
.010
.0050
n=10
3.2922
4.5224
5.6862
7.0311
7.9475
8.7657
.200
.100
.050
.020
.010
.0050