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)

Verteilungsfunktion )(zΦ der Standardnormalverteilung

)(zΦ für 0.40 <≤ z

Hinweis: )(1)( zz Φ−=−Φ

Quantile der t-Verteilung nt ;1 α−

(n = Anzahl der Freiheitsgrade)

Quantile 2

;1 nα−χ der 2χ -Verteilung

(n = Anzahl der Freiheitsgrade)

Quantile der F-Verteilung mit

n1 und n2 Freiheitsgraden ;, 21 nnF

Es gilt:

;,1;,

21

12

1

nnnn F

F

32

0A

NH

AN

GB

.T

AB

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Tabelle

7:Q

uaiitileder

F-V

erteilungm

it(ni,n2)

Freiheitsgraden

a>

<o

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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

c«

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32

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Tabelle

7:Q

uantilederF-V

erteilungm

it(rii,«2)Freiheitsgraden

F'm.nj;«

(Forts.)lO

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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

o>

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00

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05

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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

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CO

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00

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00

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00

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^

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03

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pp

rrp

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es

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10

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f-HC

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pp

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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

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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

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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

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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

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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

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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

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150

200

500

00

32

6A

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AB

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N

Tabelle

7:Q

uaiitileder

FV

erteilungm

it(ni,n-j)

FreiheitsgradenFn,.«!;«

(Forts.)

lOlO

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CO

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32

7

Tabelle

7;Q

uantileder

F-V

erteilungm

it(ni,712)

Freilieitsgraden

(Forts.)

00

00

CD

O0

0C

D

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32

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AB

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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

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00

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es

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00

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00

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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

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0

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00

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CM

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f-H

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00

co

10

CO

co

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M0

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co

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00

1-

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00

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f-H^

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rr^

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CM

00

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CM

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0>

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)

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00

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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

Verteilungsfunktion der Binomialverteilung ini

k

ipp

in

kXP −

=

−

=≤ ∑ )1()(

0

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 +nW -Test

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 >

Van der Waerdens -Test nX

Die Tabelle gibt kritische Werte der -Statistik an. nX

Moods -Test nM

Die Tabelle gibt kritische Werte der -Statistik nach dem folgenden Schema an:

αc nM

-1-

Quantile dn;α für den Unabhängigkeitstest D

-2-