Empirische Wirtschaftsforschung Prof. Dr. Ralf Runde
Tabellen mit Quantilen zu
Normalverteilung
t-Verteilung
2-Verteilung
F-Verteilung
Binomialverteilung
Kolmogorov-Smirnov- Anpassungstest
Wilcoxon-Einstichprobentest
Wilcoxon-Rangsummentest (Mann-Whitney-U-Test)
Van-der-Waerden-Test
Mood-Test
Differenz-Test (Unabhängigkeit)
32
0A
NH
AN
GB
.T
AB
EL
LE
N
Tabelle
7:Q
uaiitileder
F-V
erteilungm
it(ni,n2)
Freiheitsgraden
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32
1
Tabelle
7;Q
uantileder
F-V
erteilungm
it(ni,722)
Freiheitsgraden
Fn^,nr,a(Forts.)
ni
15
7.5595.2693.9382.8716.3144.5683.5112.6325.5154.1013.2182.4644.9623.7693.0062.3404.5583.5222.8452.244
7.6055.2973.9562.8816.3594.5963.5292.6435.5594.1303.2372.4755.0053.7983.0252.3514.6013.5502.8652.255
CO
7.6585.3293.9762.8926.4104.6283.5502.6545.6094.1623.2592.4885.0553.8313.0482.3644.6503.5832.8872.269
CM
7.7185.3664.0002.9056.4694.6663.5752.6685.6674.2003.2842.5025.1113.8683.0732.3794.7063.6212.9132.284
11
7.7905.4104.0272.9196.5384.7093.6032.6845.7344.2433.3132.5195.1783.9123.1022.3964.7723.6652.9432.302
0
7.8745.4614.0602.9376.6204.7613.6372.7035.8144.2953.347
12.5385.2573.9643.1372.1464.8493.7172.9782.323
0
7.9765.5234.0992.9586.7194.8233.6772.7255.9114.3573.3882.5615.3514.0263.1792.4404.9423.7793.0202.347
00
8.1025.6004.1472.9836.8404.8993.7262.7526.0294.4333.4382.5895.4674.1023.2302.4695.0573.8553.0722.377
t-
8.2605.6954.2073.0146.9934.9953.7872.7856.1784.5293.5002.6245.6134.1973.2932.5055.2003.9503.1352.414
CD
8.4665.8204.2843.0557.1915.1193.8662.8276.3714.6523.5812.6685.8024.3203.3742.5515.3864.0723.2172.461
10
8.7465.9884.3873.1087.4605.2853.9722.8836.6324.8173.6872.7266.0574.4843.4822.6115.6364.2363.3262.522
rr
9.1486.2274.5343.1817.8475.5234.1202.9617.0065.0533.8382.8066.4224.7183.6332.6935.9944.4683.4782.605
CO
9.7806.5994.7573.2898.4515.8904.3473.0747.5915.4164.0662.9246.9925.0783.8632.8136.5524.8263.7082.728
CM
10.927.2605.1433.4639.5476.5424.7373.2578.6496.0594.4593.1138.0225.7154.2563.0067.5595.4564.1032.924
13.758.8135.9873.77612.258.0735.5913.58911.267.5715.3183.45810.567.2095.1173.36010.046.9374.9653.285
ö
0.9900.9750.9500.9900.9900.9750.9500.9800.9900.9750.9500.9000.9900.9750.9500.9000.9900.9750.9500.900
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32
2A
NH
AN
GB
.T
AB
EL
LE
N
Tabelle
7:Q
uantilederF-V
erteilungm
it(rii,«2)Freiheitsgraden
F'm.nj;«
(Forts.)lO
10
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32
3
Tabelle
7:Q
uantileder
F-Verteilung
mit
(ni,712)Freiheitsgraden(Forts.)
00
PO
CM
CO
CM
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10
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00
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00
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CM
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M
95
CM
00
95
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95
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95
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95
00
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CO
95
95
95
95
95
32
4A
NH
AN
GB
.T
AB
EL
LE
N
Tabelle
7:Q
uantilederF-V
erteilungm
it(«1,722)Freiheitsgraden
(Forts.)
CM
CM
CM
10
<0
O)
p10
CO
CM
CM
CO
CM
00
CO
CM
CM
CM
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00
O)
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00
CO
CM
CM
^
CM
CO
03
CM
pp
rrp
CO
CM
CM
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es
CO
CM
CO
CO
CM
CM
10
iO0
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f-HC
OC
MC
M
pp
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00
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oo
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CO
CM
CM
CM
CM
CM
10
OD
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10
CM
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CM
CM
00
CO
"T
CO
00
CM
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CO
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CM
CM
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32
5
Tabelle
7:Q
uantileder
F-V
erteilungm
it(«1,722)
Freiheitsgraden
F,n\,n
2;Q
r
c
MO
2.2231.968.1.768
1.5572.160
1.9221.734
1.5332.129
1.9001.717
1.5222.075
1.8591.686
1.5012.0391.833
1.6661.487
1—1
2.2652.000
1.7921.573
2.2031.955
1.7581.550
2.1721.9321.742
1.5392.117
1.8921.712
1.5182.081
1.8661.692
1.505
CO
2.3132.036
1.8191.592
2.2511.991
1.7861.569
2.2201.9691.769
1.5582.166
1.9291.740
1.5372.1301.903
1.7201.524
CM
2.3672.077
1.8501.612
2.3052.033
1.8171.590
2.2752.010
1.8011.5792.220
1.9711.772
1.5592.185
1.9451.752
1.546
•"H
2.4302.124
1.8861.636
2.3682.0801.853
1.6142.338
2.0581.837
1.6032.283
2.0191.808
1.5842.248
1.9931.789
1.571
0f-H
2.5032.179
1.9271.663
2.4412.135
1.8941.642
2.4112.113
1.8781.631
2.3572.074
1.8501.612
2.3212.048
1.8311.599
a>
2.5902.244
1.9751.695
2.5282.200
1.9431.674
2.4972.1781.927
1.6632.443
2.1391.899
1.6442.407
2.1141.880
1.632
00
2.6942.321
2.0321.732
2.6322.2782.001
1.7122.601
2.2561.9851.701
2.5472.2171.957
1.6832.511
2.1921.938
1.670
2.8232.417
2.1031.778
2.7612.373
2.0711.757
2.7302.3512.056
1.7472.675
2.3132.028
1.7292.6392.288
2.0101.717
CO
2.9882.537
2.191i1.8342.924
2.4942.160
1.8142.893
2.4722.1441.804
2.8382.434
2.1171.786
2.8022.4082.099
1.774
MO
3.2062.696
2.3051.906
3.1422.6522.275
1.8863.110
2.6302.2591.876
3.0542.592
2.2321.859
3.0172.567
2.2141.847
3.5132.917
2.4632.0023.4472.872
2.4321.983
3.4142.8502.417
1.9733.357
2.8112.390
1.9563.319
2.7862.372
1.945
CO
3.9842.250
2.6962.139
3.9153.204
2.6652.121
3.8813.182
2.6502.111
3.8213.142
2.6232.095
3.7823.116
2.6052.084
CM
4.8243.828
3.0872.3564.750
3.7813.056
2.3384.713
3.7583.0412.329
4.6483.7163.0142.313
4.6053.689
2.9962.303
♦H
6.8955.179
3.9362.756
6.8075.126
3.9042.739
6.7635.100
3.8882.731
6.6865.054
3.8602.716
6.6355.024
3.8412.706
ö
0.9900.9750.950
0.9000.990
0.9750.950
0.9000.990
0.9750.9500.900
0.9900.975
0.9500.9000.9900.975
0.9500.900
«s
ff
00
150
200
500
00
32
6A
NH
AN
GB
.T
AB
EL
LE
N
Tabelle
7:Q
uaiitileder
FV
erteilungm
it(ni,n-j)
FreiheitsgradenFn,.«!;«
(Forts.)
lOlO
04
CO
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co
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lo
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lOo
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cqCMco
oo
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32
7
Tabelle
7;Q
uantileder
F-V
erteilungm
it(ni,712)
Freilieitsgraden
(Forts.)
00
00
CD
O0
0C
D
CO
00
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00
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t-
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CO
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00
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co
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00
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co
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32
8A
NH
AN
Gß
.T
AB
EL
LE
N
Tabelle
7:Q
uantileder
F-V
erteiluugm
it(ni,712)
Frciheitsgraden
Fin.njia(F
orts.)
CO
(M0
1
CO
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es
CM
CO
00
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co
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0>
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co
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lO0
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es
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to
co
00
00
00
co
es
es
co
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es
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to
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r--es
co
es
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co
es
10
es
co
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co
es
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^
00
co
05
00
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co
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es
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ko
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10
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pes
co
es
es
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00
00
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es
00
co
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es
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es
r-l
00
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00
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pp
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es
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rH
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Oi
oi
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32
9
Tabelle
7:Q
uantileder
F-V
erteilungm
it(ni,712)
Freiheitsgraden
F'ni,nj;a(Forts.)
e
00
2.7532.316
2.0101.718
2.6532.247
1.9601.686
2.5662.187
1.9171.657
2.4892.133
1.8781.631
2.4212.085
1.8431.607
500
2.7752.333
2.0221.726
2.6762.2641.973
1.6942.589
2.2041.929
1.6652.5122.150
1.8911.639
2.4452.103
1.8561.616
200
2.8082.3572.039
1.7382.709
2.2891.991
1.7062.623
2.2291.948
1.6782.547
2.1761.910
1.6522.4792.128
1.8751.629
150
2.8272.370
2.0491.744
2.7282.3022.001
1.7132.641
2.2421.958
1.6842.565
2.1901.920
1.6592.498
2.1421.886
1.636
100
2.8632.396
2.06811.7572.7642.329
2.0201.726
2.6782.2691.978
1.6982.602
2.2171.940
1.6732.5352.170
1.9071.650
75
2.8982.4222.087
1.7692.8002.3552.040
1.7382.7142.2961.998
1.7112.639
2.2431.9601.6862.5722.1971.927
1.664
50
2.9672.472
2.1241.793
2.8692.405
2.0771.763
2.7842.3472.035
1.7362.709
2.2951.999
1.7112.643
2.2491.966
1.690
40
3.0182.5092.151
1.8112.9202.442
2.1041.781
2.8352.3842.063
1.7542.761
2.3332.0261.730
2.6952.287
1.9941.708
30
3.1012.568
2.1941.839
3.0032.5022.148
1.8092.919
2.4442.1071.783
2.8442.394
2.0711.759
2.7782.349
2.0391.738
25
3.1652.614
2.2271.860
3.0682.5482.181
1.8312.983
2.4912.1411.805
2.9092.441
2.1061.782
2.8432.3962.074
1.761
20
3.2592.681
2.2761.891
3.1622.616
2.2301.862
3.0772.5592.191
1.8373.003
2.5092.155
1.8142.938
2.4642.124
1.794
o>
3.2832.698
2.2881.899
3.1862.633
2.2431.870
3.1012.5762.203
1.8453.027
2.5262.168
1.8222.962
2.4822.137
1.802
00
3.3102.717
2.3021.908
3.2122.652
2.2571.879
3.1282.596
2.2171.854
3.0542.546
2.1821.831
2.9892.501
2.1511.811
r-
3.3392.738
2.3171.917
3.2422.6732.2721.889
3.1582.6172.233
1.8643.084
2.5672.198
1.8413.0182.5232.167
1.821
co
3.3722.761
2.3331.928
3.2752.697
2.2891.900
3.1902.6402.250
1.8753.116
2.5912.215
1.8523.0512.5472.184
1.833
ö
0.9900.975
0.9500.9000.990
0.9750.950
0.9000.990
0.9750.9500.900
0.9900.9750.9500.900
0.9900.9750.9500.900
es
eC
O1
-H
18
19
20
33
0A
NH
AN
GB
.T
AB
EL
LE
N
Tabelle
7:Q
uaiitileder
F-V
erteilungm
it(ni,«2)
Freiheitsgraden
(Forts.)
00
CM
Oi
<0
Oi
to
CO
00
CM
CO
CM
O0
>
OC
OIC
00
(O
CM
O<
0
CM
CM
CM
CM
CM
CM
00
^IC
CM
CM
CM
o>
o
CO
O0
0
CO
00
05
00
to
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O»-H
00
cc
0>
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M
CM
CM
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CM
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f-H
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o
00
co
10
CO
co
o>
<0
oC
O1
0C
M0
00
0^
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co
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lO
00
1-
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IC
co
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00
IC
co
<0
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f-H^
O0
0o
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rr^
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CM
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IC
CM
00
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CM
<-1
^
0>
O0
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)
00
CM
0>
CO
IC
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00
CM
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CM
00
00
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cc
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cc
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CM
00
00
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co
05
co
co
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00
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00
00
to
IC
co
00
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CC
00
CM
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CM
00
00
CO
33
1
Tabelle
7:Q
uantilederF-V
erteilungm
it(ni,712)FreiheitsgradenFn^^„^.Q
(Forts.)
c
00
1.4271.347
1.2831.214
1.3311.271
1.2221.169
1.2791.229
1.1891.144
1.1641.137
1.1131.087
1.0001.0001.0001.000
500
1.4661.378
1.3081.232
1.3761.307
1.2521.191
1.3281.269
1.2211.168
1.2321.192
1.1591.122
1.1531.128
1.1061.082
200
1.5181.420
1.3421.2571.435
1.3551.290
1.2191.391
1.3201.263
1.1991.308
1.2541.210
1.1601.247
1.2051.170
1.130
0IC
1.5461.442
1.3591.270
1.4651.3791.3091.234
1.4231.3461.2831.214
1.3441.2841.2331.178
1.2881.2391.1971.151
00
1.5981.483
1.3921.293
1.5201.423
1.3451.260
1.4811.393
1.3211.242
1.4081.336
1.2751.209
1.3581.296
1.2431.850
IC
1.6461.522
1.4221.315
1.5721.464
1.3771.283
1.5341.435
1.3541.267
1.4651.381
1.3121.236
1.4131.3401.279
1.654
50
1.7351.592
1.4771.355
1.6651.538
1.4361.325
1.6291.511
1.4151.310
1.5661.462
1.3761.282
1.5231.428
1.3501.263
40
1.7971.6401.515j1.3821
1.7291.588
1.4751.353
1.6941.562
1.4551.339
1.6331.515
1.4191.313
1.5921.484
1.3941.295
30
1.8931.715
1.5731.423
1.8271.665
1.5351.396
1.7941.640
1.5161.383
1.7351.596
1.4821.358
1.6961.588
1.4761.342
25
1.9651.770
1.6161.453
1.9001.722
1.5801.427
1.8681.698
1.5611.414
1.8121.655
1.5281.391
1.7741.626
1.5061.375
20
2.0671.849
1.6761.494
2.0031.801
1.6411.470
1.9711.778
1.6231.458
1.9151.736
1.5921.435
1.8781.708
1.7511.421
0
2.0921.868
1.6911.505
2.0291.821
1.6561.480
1.9971.798
1.6391.468
1.9421.757
1.6071.446
1.9051.729
1.5871.432
00
2.1201.890
1.7081.516
2.0571.843
1.6731.492
2.0261.820
1.6561.480
1.9701.779
1.6251.458
1.9341.752
1.6041.444
h-
2.1511.913
1.7261.528
2.0881.867
1.6911.505
2.0571.844
1.6741.493
2.0021.803
1.6431.471
1.9651.776
1.6231.457
<0
»-H
2.1851.939
1.7461.542
2.1221.893
1.7111.518
2.0911.8701.694
1.5072.036
1.8301.664
1.4852.000
1.8031.644
1.471
ö
0.9900.975
0.9500.900
0.9900.9750.950
0.9000.9900.9750.950
0.9000.990
0.9750.9500.9000.9900.9750.9500.900
e
00
150
200
500
00
Hinweis: Für p > 0.5 erhält man die Werte der Verteilungsfunktion aus der Beziehung Fn,p(k) = 1 - Fn,1-p(n-k-1), wobei Fn,p die Verteilungsfunktion der B(n, p)-Verteilung bezeichnet.
Quantile Kn;1−α/2 für den zweiseitigenKolmogorov-Smirnov-Anpassungstest
n α = 0, 2 α = 0, 1 α = 0, 05 α = 0, 02 α = 0, 01 α = 0, 0051 0,9000 0,9500 0,9750 0,9900 0,9950 0,99752 0,6838 0,7764 0,8419 0,8100 0,9293 0,94103 0,5648 0,6360 0,7076 0,7845 0,8290 0,86424 0,4927 0,5652 0,6239 0,6888 0,7342 0,77635 0,4470 0,5094 0,5633 0,6272 0,6685 0,70536 0,4104 0,4680 0,5193 0,5774 0,6166 0,65287 0,3815 0,4361 0,4834 0,5384 0,5758 0,60978 0,3583 0,4096 0,4543 0,5065 0,5417 0,57429 0,3391 0,3875 0,4300 0,4796 0,5133 0,5444
10 0,3226 0,3687 0,4092 0,4566 0,4889 0,518611 0,3083 0,3524 0,3912 0,4367 0,4677 0,496312 0,2957 0,3381 0,3754 0,4192 0,4490 0,476613 0,2847 0,3255 0,3614 0,4036 0,4324 0,459114 0,2748 0,3142 0,3489 0,3897 0,4176 0,443515 0,2689 0,3040 0,3376 0,3771 0,4042 0,429316 0,2577 0,2947 0,3273 0,3657 0,3920 0,416417 0,2504 0,2863 0,3180 0,3553 0,3808 0,404618 0,2436 0,2785 0,3094 0,3457 0,3706 0,393719 0,2373 0,2714 0,3014 0,3368 0,3611 0,383720 0,2315 0,2647 0,2941 0,3286 0,3524 0,374521 0,2261 0,2586 0,2872 0,3210 0,3442 0,365822 0,2211 0,2528 0,2809 0,3139 0,3366 0,357823 0,2164 0,2475 0,2749 0,3073 0,3295 0,350224 0,2120 0,2424 0,2693 0,3010 0,3228 0,343125 0,2079 0,2377 0,2640 0,2952 0,3165 0,336526 0,2040 0,2332 0,2591 0,2896 0,3106 0,330227 0,2003 0,2290 0,2544 0,2844 0,3050 0,324228 0,1968 0,2250 0,2500 0,2794 0,2999 0,318629 0,1934 0,2212 0,2457 0,2747 0,2946 0,313230 0,1903 0,2176 0,2417 0,2702 0,2898 0,308131 0,1873 0,2141 0,2379 0,2660 0,2853 0,303332 0,1844 0,2108 0,2342 0,2619 0,2809 0,298733 0,1817 0,2077 0,2308 0,2580 0,2768 0,294234 0,1791 0,2047 0,2274 0,2543 0,2728 0,290035 0,1766 0,2018 0,2242 0,2507 0,2690 0,2860
Grobe Approximation für n > 35 : Kn;1−α/2 =
√ln( 2
α)
√2n
.
Wilcoxons -Test nW
Die Tabelle gibt kritische Werte der
-Statistik für den linksseitigen Test C mit nm ≤ an. Für den rechtsseitigen Test B gilt:
αw
nW
ααα− −µ=−= wwWEw N 2)(21
Ist , so wird durch Umbenennung die x-Stichprobe zur y-Stichprobe und damit Test C zu Test B.
nm >
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