normal table

726Appendix A Statistical Tables and Proofs

Table A.1 Binomial Probability Table727rTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0

Table A.1 Binomial Probability Sums

r b(x; n, p)

x=0

p

nr0.100.200.250.300.400.500.600.700.800.90

100.90000.80000.75000.70000.60000.50000.40000.30000.20000.1000

11.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

200.81000.64000.56250.49000.36000.25000.16000.09000.04000.0100

10.99000.96000.93750.91000.84000.75000.64000.51000.36000.1900

21.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

300.72900.51200.42190.34300.21600.12500.06400.02700.00800.0010

10.97200.89600.84380.78400.64800.50000.35200.21600.10400.0280

20.99900.99200.98440.97300.93600.87500.78400.65700.48800.2710

31.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

400.65610.40960.31640.24010.12960.06250.02560.00810.00160.0001

10.94770.81920.73830.65170.47520.31250.17920.08370.02720.0037

20.99630.97280.94920.91630.82080.68750.52480.34830.18080.0523

30.99990.99840.99610.99190.97440.93750.87040.75990.59040.3439

41.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

500.59050.32770.23730.16810.07780.03130.01020.00240.00030.0000

10.91850.73730.63280.52820.33700.18750.08700.03080.00670.0005

20.99140.94210.89650.83690.68260.50000.31740.16310.05790.0086

30.99950.99330.98440.96920.91300.81250.66300.47180.26270.0815

41.00000.99970.99900.99760.98980.96880.92220.83190.67230.4095

51.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

600.53140.26210.17800.11760.04670.01560.00410.00070.00010.0000

10.88570.65540.53390.42020.23330.10940.04100.01090.00160.0001

20.98420.90110.83060.74430.54430.34380.17920.07050.01700.0013

30.99870.98300.96240.92950.82080.65630.45570.25570.09890.0159

40.99990.99840.99540.98910.95900.89060.76670.57980.34460.1143

51.00000.99990.99980.99930.99590.98440.95330.88240.73790.4686

61.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000

700.47830.20970.13350.08240.02800.00780.00160.00020.0000

10.85030.57670.44490.32940.15860.06250.01880.00380.00040.0000

20.97430.85200.75640.64710.41990.22660.09630.02880.00470.0002

30.99730.96670.92940.87400.71020.50000.28980.12600.03330.0027

40.99980.99530.98710.97120.90370.77340.58010.35290.14800.0257

51.00000.99960.99870.99620.98120.93750.84140.67060.42330.1497

61.00000.99990.99980.99840.99220.97200.91760.79030.5217

71.00001.00001.00001.00001.00001.00001.00001.0000

p

nr0.100.200.250.300.400.500.600.700.800.90

800.43050.16780.10010.05760.01680.00390.00070.00010.0000

10.81310.50330.36710.25530.10640.03520.00850.00130.0001

20.96190.79690.67850.55180.31540.14450.04980.01130.00120.0000

30.99500.94370.88620.80590.59410.36330.17370.05800.01040.0004

40.99960.98960.97270.94200.82630.63670.40590.19410.05630.0050

51.00000.99880.99580.98870.95020.85550.68460.44820.20310.0381

60.99990.99960.99870.99150.96480.89360.74470.49670.1869

71.00001.00000.99990.99930.99610.98320.94240.83220.5695

81.00001.00001.00001.00001.00001.00001.0000

900.38740.13420.07510.04040.01010.00200.00030.0000

10.77480.43620.30030.19600.07050.01950.00380.00040.0000

20.94700.73820.60070.46280.23180.08980.02500.00430.00030.0000

30.99170.91440.83430.72970.48260.25390.09940.02530.00310.0001

40.99910.98040.95110.90120.73340.50000.26660.09880.01960.0009

50.99990.99690.99000.97470.90060.74610.51740.27030.08560.0083

61.00000.99970.99870.99570.97500.91020.76820.53720.26180.0530

71.00000.99990.99960.99620.98050.92950.80400.56380.2252

81.00001.00000.99970.99800.98990.95960.86580.6126

91.00001.00001.00001.00001.00001.0000

1000.34870.10740.05630.02820.00600.00100.00010.0000

10.73610.37580.24400.14930.04640.01070.00170.00010.0000

20.92980.67780.52560.38280.16730.05470.01230.00160.0001

30.98720.87910.77590.64960.38230.17190.05480.01060.00090.0000

40.99840.96720.92190.84970.63310.37700.16620.04730.00640.0001

50.99990.99360.98030.95270.83380.62300.36690.15030.03280.0016

61.00000.99910.99650.98940.94520.82810.61770.35040.12090.0128

70.99990.99960.99840.98770.94530.83270.61720.32220.0702

81.00001.00000.99990.99830.98930.95360.85070.62420.2639

91.00000.99990.99900.99400.97180.89260.6513

101.00001.00001.00001.00001.00001.0000

1100.31380.08590.04220.01980.00360.00050.0000

10.69740.32210.19710.11300.03020.00590.00070.0000

20.91040.61740.45520.31270.11890.03270.00590.00060.0000

30.98150.83890.71330.56960.29630.11330.02930.00430.0002

40.99720.94960.88540.78970.53280.27440.09940.02160.00200.0000

50.99970.98830.96570.92180.75350.50000.24650.07820.01170.0003

61.00000.99800.99240.97840.90060.72560.46720.21030.05040.0028

70.99980.99880.99570.97070.88670.70370.43040.16110.0185

81.00000.99990.99940.99410.96730.88110.68730.38260.0896

91.00001.00000.99930.99410.96980.88700.67790.3026

101.00000.99950.99640.98020.91410.6862

111.00001.00001.00001.00001.0000

p

nr0.100.200.250.300.400.500.600.700.800.90

1200.28240.06870.03170.01380.00220.00020.0000

10.65900.27490.15840.08500.01960.00320.00030.0000

20.88910.55830.39070.25280.08340.01930.00280.00020.0000

30.97440.79460.64880.49250.22530.07300.01530.00170.0001

40.99570.92740.84240.72370.43820.19380.05730.00950.00060.0000

50.99950.98060.94560.88220.66520.38720.15820.03860.00390.0001

60.99990.99610.98570.96140.84180.61280.33480.11780.01940.0005

71.00000.99940.99720.99050.94270.80620.56180.27630.07260.0043

80.99990.99960.99830.98470.92700.77470.50750.20540.0256

91.00001.00000.99980.99720.98070.91660.74720.44170.1109

101.00000.99970.99680.98040.91500.72510.3410

111.00000.99980.99780.98620.93130.7176

121.00001.00001.00001.00001.0000

1300.25420.05500.02380.00970.00130.00010.0000

10.62130.23360.12670.06370.01260.00170.00010.0000

20.86610.50170.33260.20250.05790.01120.00130.0001

30.96580.74730.58430.42060.16860.04610.00780.00070.0000

40.99350.90090.79400.65430.35300.13340.03210.00400.0002

50.99910.97000.91980.83460.57440.29050.09770.01820.00120.0000

60.99990.99300.97570.93760.77120.50000.22880.06240.00700.0001

71.00000.99880.99440.98180.90230.70950.42560.16540.03000.0009

80.99980.99900.99600.96790.86660.64700.34570.09910.0065

91.00000.99990.99930.99220.95390.83140.57940.25270.0342

101.00000.99990.99870.98880.94210.79750.49830.1339

111.00000.99990.99830.98740.93630.76640.3787

121.00000.99990.99870.99030.94500.7458

131.00001.00001.00001.00001.0000

1400.22880.04400.01780.00680.00080.00010.0000

10.58460.19790.10100.04750.00810.00090.0001

20.84160.44810.28110.16080.03980.00650.00060.0000

30.95590.69820.52130.35520.12430.02870.00390.0002

40.99080.87020.74150.58420.27930.08980.01750.00170.0000

50.99850.95610.88830.78050.48590.21200.05830.00830.0004

60.99980.98840.96170.90670.69250.39530.15010.03150.00240.0000

71.00000.99760.98970.96850.84990.60470.30750.09330.01160.0002

80.99960.99780.99170.94170.78800.51410.21950.04390.0015

91.00000.99970.99830.98250.91020.72070.41580.12980.0092

101.00000.99980.99610.97130.87570.64480.30180.0441

111.00000.99940.99350.96020.83920.55190.1584

120.99990.99910.99190.95250.80210.4154

131.00000.99990.99920.99320.95600.7712

141.00001.00001.00001.00001.0000

730Appendix A Statistical Tables and ProofsrTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0

Table A.1 Binomial Probability Table731rTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0

p

nr0.100.200.250.300.400.500.600.700.800.90

1500.20590.03520.01340.00470.00050.0000

10.54900.16710.08020.03530.00520.00050.0000

20.81590.39800.23610.12680.02710.00370.00030.0000

30.94440.64820.46130.29690.09050.01760.00190.0001

40.98730.83580.68650.51550.21730.05920.00930.00070.0000

50.99780.93890.85160.72160.40320.15090.03380.00370.0001

60.99970.98190.94340.86890.60980.30360.09500.01520.0008

71.00000.99580.98270.95000.78690.50000.21310.05000.00420.0000

80.99920.99580.98480.90500.69640.39020.13110.01810.0003

90.99990.99920.99630.96620.84910.59680.27840.06110.0022

101.00000.99990.99930.99070.94080.78270.48450.16420.0127

111.00000.99990.99810.98240.90950.70310.35180.0556

121.00000.99970.99630.97290.87320.60200.1841

131.00000.99950.99480.96470.83290.4510

141.00000.99950.99530.96480.7941

151.00001.00001.00001.0000

1600.18530.02810.01000.00330.00030.0000

10.51470.14070.06350.02610.00330.00030.0000

20.78920.35180.19710.09940.01830.00210.0001

30.93160.59810.40500.24590.06510.01060.00090.0000

40.98300.79820.63020.44990.16660.03840.00490.0003

50.99670.91830.81030.65980.32880.10510.01910.00160.0000

60.99950.97330.92040.82470.52720.22720.05830.00710.0002

70.99990.99300.97290.92560.71610.40180.14230.02570.00150.0000

81.00000.99850.99250.97430.85770.59820.28390.07440.00700.0001

90.99980.99840.99290.94170.77280.47280.17530.02670.0005

101.00000.99970.99840.98090.89490.67120.34020.08170.0033

111.00000.99970.99510.96160.83340.55010.20180.0170

121.00000.99910.98940.93490.75410.40190.0684

130.99990.99790.98170.90060.64820.2108

141.00000.99970.99670.97390.85930.4853

151.00000.99970.99670.97190.8147

161.00001.00001.00001.0000

p

nr0.100.200.250.300.400.500.600.700.800.90

1700.16680.02250.00750.00230.00020.0000

10.48180.11820.05010.01930.00210.00010.0000

20.76180.30960.16370.07740.01230.00120.0001

30.91740.54890.35300.20190.04640.00640.00050.0000

40.97790.75820.57390.38870.12600.02450.00250.0001

50.99530.89430.76530.59680.26390.07170.01060.00070.0000

60.99920.96230.89290.77520.44780.16620.03480.00320.0001

70.99990.98910.95980.89540.64050.31450.09190.01270.0005

81.00000.99740.98760.95970.80110.50000.19890.04030.00260.0000

90.99950.99690.98730.90810.68550.35950.10460.01090.0001

100.99990.99940.99680.96520.83380.55220.22480.03770.0008

111.00000.99990.99930.98940.92830.73610.40320.10570.0047

121.00000.99990.99750.97550.87400.61130.24180.0221

131.00000.99950.99360.95360.79810.45110.0826

140.99990.99880.98770.92260.69040.2382

151.00000.99990.99790.98070.88180.5182

161.00000.99980.99770.97750.8332

171.00001.00001.00001.0000

1800.15010.01800.00560.00160.00010.0000

10.45030.09910.03950.01420.00130.0001

20.73380.27130.13530.06000.00820.00070.0000

30.90180.50100.30570.16460.03280.00380.0002

40.97180.71640.51870.33270.09420.01540.00130.0000

50.99360.86710.71750.53440.20880.04810.00580.0003

60.99880.94870.86100.72170.37430.11890.02030.00140.0000

70.99980.98370.94310.85930.56340.24030.05760.00610.0002

81.00000.99570.98070.94040.73680.40730.13470.02100.0009

90.99910.99460.97900.86530.59270.26320.05960.00430.0000

100.99980.99880.99390.94240.75970.43660.14070.01630.0002

111.00000.99980.99860.97970.88110.62570.27830.05130.0012

121.00000.99970.99420.95190.79120.46560.13290.0064

131.00000.99870.98460.90580.66730.28360.0282

140.99980.99620.96720.83540.49900.0982

151.00000.99930.99180.94000.72870.2662

160.99990.99870.98580.90090.5497

171.00000.99990.99840.98200.8499

181.00001.00001.00001.0000

p

nr0.100.200.250.300.400.500.600.700.800.90

1900.13510.01440.00420.00110.0001

10.42030.08290.03100.01040.00080.0000

20.70540.23690.11130.04620.00550.00040.0000

30.88500.45510.26310.13320.02300.00220.0001

40.96480.67330.46540.28220.06960.00960.00060.0000

50.99140.83690.66780.47390.16290.03180.00310.0001

60.99830.93240.82510.66550.30810.08350.01160.0006

70.99970.97670.92250.81800.48780.17960.03520.00280.0000

81.00000.99330.97130.91610.66750.32380.08850.01050.0003

90.99840.99110.96740.81390.50000.18610.03260.0016

100.99970.99770.98950.91150.67620.33250.08390.00670.0000

111.00000.99950.99720.96480.82040.51220.18200.02330.0003

120.99990.99940.98840.91650.69190.33450.06760.0017

131.00000.99990.99690.96820.83710.52610.16310.0086

141.00000.99940.99040.93040.71780.32670.0352

150.99990.99780.97700.86680.54490.1150

161.00000.99960.99450.95380.76310.2946

171.00000.99920.98960.91710.5797

180.99990.99890.98560.8649

191.00001.00001.00001.0000

2000.12160.01150.00320.00080.0000

10.39170.06920.02430.00760.00050.0000

20.67690.20610.09130.03550.00360.0002

30.86700.41140.22520.10710.01600.00130.0000

40.95680.62960.41480.23750.05100.00590.0003

50.98870.80420.61720.41640.12560.02070.00160.0000

60.99760.91330.78580.60800.25000.05770.00650.0003

70.99960.96790.89820.77230.41590.13160.02100.00130.0000

80.99990.99000.95910.88670.59560.25170.05650.00510.0001

91.00000.99740.98610.95200.75530.41190.12750.01710.0006

100.99940.99610.98290.87250.58810.24470.04800.00260.0000

110.99990.99910.99490.94350.74830.40440.11330.01000.0001

121.00000.99980.99870.97900.86840.58410.22770.03210.0004

131.00000.99970.99350.94230.75000.39200.08670.0024

141.00000.99840.97930.87440.58360.19580.0113

150.99970.99410.94900.76250.37040.0432

161.00000.99870.98400.89290.58860.1330

170.99980.99640.96450.79390.3231

181.00000.99950.99240.93080.6083

191.00000.99920.98850.8784

201.00001.00001.0000


Table A.2 Poisson Probability Table732rTable A.2 (continued) Poisson Probability Sums p(x; )

Table A.2 Poisson Probability Sums

r p(x; )

x=0

r0.10.20.30.40.50.60.70.80.9

00.90480.81870.74080.67030.60650.54880.49660.44930.4066

10.99530.98250.96310.93840.90980.87810.84420.80880.7725

20.99980.99890.99640.99210.98560.97690.96590.95260.9371

31.00000.99990.99970.99920.99820.99660.99420.99090.9865

41.00001.00000.99990.99980.99960.99920.99860.9977

51.00001.00001.00000.99990.99980.9997

61.00001.00001.0000

r1.01.52.02.53.03.54.04.55.0

00.36790.22310.13530.08210.04980.03020.01830.01110.0067

10.73580.55780.40600.28730.19910.13590.09160.06110.0404

20.91970.80880.67670.54380.42320.32080.23810.17360.1247

30.98100.93440.85710.75760.64720.53660.43350.34230.2650

40.99630.98140.94730.89120.81530.72540.62880.53210.4405

50.99940.99550.98340.95800.91610.85760.78510.70290.6160

60.99990.99910.99550.98580.96650.93470.88930.83110.7622

71.00000.99980.99890.99580.98810.97330.94890.91340.8666

81.00000.99980.99890.99620.99010.97860.95970.9319

91.00000.99970.99890.99670.99190.98290.9682

100.99990.99970.99900.99720.99330.9863

111.00000.99990.99970.99910.99760.9945

121.00000.99990.99970.99920.9980

131.00000.99990.99970.9993

141.00000.99990.9998

151.00000.9999

161.0000

x=0

r5.56.06.57.07.58.08.59.09.5

00.00410.00250.00150.00090.00060.00030.00020.00010.0001

10.02660.01740.01130.00730.00470.00300.00190.00120.0008

20.08840.06200.04300.02960.02030.01380.00930.00620.0042

30.20170.15120.11180.08180.05910.04240.03010.02120.0149

40.35750.28510.22370.17300.13210.09960.07440.05500.0403

50.52890.44570.36900.30070.24140.19120.14960.11570.0885

60.68600.60630.52650.44970.37820.31340.25620.20680.1649

70.80950.74400.67280.59870.52460.45300.38560.32390.2687

80.89440.84720.79160.72910.66200.59250.52310.45570.3918

90.94620.91610.87740.83050.77640.71660.65300.58740.5218

100.97470.95740.93320.90150.86220.81590.76340.70600.6453

110.98900.97990.96610.94670.92080.88810.84870.80300.7520

120.99550.99120.98400.97300.95730.93620.90910.87580.8364

130.99830.99640.99290.98720.97840.96580.94860.92610.8981

140.99940.99860.99700.99430.98970.98270.97260.95850.9400

150.99980.99950.99880.99760.99540.99180.98620.97800.9665

160.99990.99980.99960.99900.99800.99630.99340.98890.9823

171.00000.99990.99980.99960.99920.99840.99700.99470.9911

181.00000.99990.99990.99970.99930.99870.99760.9957

191.00001.00000.99990.99970.99950.99890.9980

200.99990.99980.99960.9991

211.00000.99990.99980.9996

221.00000.99990.9999

231.00000.9999

241.0000

Table A.2 (continued) Poisson Probability Sums

r p(x; )

x=0

r10.011.012.013.014.015.016.017.018.0

00.00000.00000.0000

10.00050.00020.00010.00000.0000

20.00280.00120.00050.00020.00010.00000.0000

30.01030.00490.00230.00110.00050.00020.00010.00000.0000

40.02930.01510.00760.00370.00180.00090.00040.00020.0001

50.06710.03750.02030.01070.00550.00280.00140.00070.0003

60.13010.07860.04580.02590.01420.00760.00400.00210.0010

70.22020.14320.08950.05400.03160.01800.01000.00540.0029

80.33280.23200.15500.09980.06210.03740.02200.01260.0071

90.45790.34050.24240.16580.10940.06990.04330.02610.0154

100.58300.45990.34720.25170.17570.11850.07740.04910.0304

110.69680.57930.46160.35320.26000.18480.12700.08470.0549

120.79160.68870.57600.46310.35850.26760.19310.13500.0917

130.86450.78130.68150.57300.46440.36320.27450.20090.1426

140.91650.85400.77200.67510.57040.46570.36750.28080.2081

150.95130.90740.84440.76360.66940.56810.46670.37150.2867

160.97300.94410.89870.83550.75590.66410.56600.46770.3751

170.98570.96780.93700.89050.82720.74890.65930.56400.4686

180.99280.98230.96260.93020.88260.81950.74230.65500.5622

190.99650.99070.97870.95730.92350.87520.81220.73630.6509

200.99840.99530.98840.97500.95210.91700.86820.80550.7307

210.99930.99770.99390.98590.97120.94690.91080.86150.7991

220.99970.99900.99700.99240.98330.96730.94180.90470.8551

230.99990.99950.99850.99600.99070.98050.96330.93670.8989

241.00000.99980.99930.99800.99500.98880.97770.95940.9317

250.99990.99970.99900.99740.99380.98690.97480.9554

261.00000.99990.99950.99870.99670.99250.98480.9718

270.99990.99980.99940.99830.99590.99120.9827

281.00000.99990.99970.99910.99780.99500.9897

291.00000.99990.99960.99890.99730.9941

300.99990.99980.99940.99860.9967

311.00000.99990.99970.99930.9982

321.00000.99990.99960.9990

330.99990.99980.9995

341.00000.99990.9998

351.00000.9999

360.9999

371.0000

Table A.3 Areas under the Normal Curve 0 z

z.00.01.02.03.04.05.06.07.08.09

3.40.00030.00030.00030.00030.00030.00030.00030.00030.00030.0002

3.30.00050.00050.00050.00040.00040.00040.00040.00040.00040.0003

3.20.00070.00070.00060.00060.00060.00060.00060.00050.00050.0005

3.10.00100.00090.00090.00090.00080.00080.00080.00080.00070.0007

3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.49600.49200.48800.48400.48010.47610.47210.46810.4641


Table A.3 Normal Probability Table736Area

Table A.3 (continued) Areas under the Normal Curve

z.00.01.02.03.04.05.06.07.08.09

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

3.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993

3.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995

3.30.99950.99950.99950.99960.99960.99960.99960.99960.99960.9997

3.40.99970.99970.99970.99970.99970.99970.99970.99970.99970.9998

Table A.4 Critical Values of the t-Distribution

0 t


Table A.4 Student t-Distribution Probability Table738

v0.400.300.200.150.100.050.025

10.3250.7271.3761.9633.0786.31412.706

20.2890.6171.0611.3861.8862.9204.303

30.2770.5840.9781.2501.6382.3533.182

40.2710.5690.9411.1901.5332.1322.776

50.2670.5590.9201.1561.4762.0152.571

60.2650.5530.9061.1341.4401.9432.447

70.2630.5490.8961.1191.4151.8952.365

80.2620.5460.8891.1081.3971.8602.306

90.2610.5430.8831.1001.3831.8332.262

100.2600.5420.8791.0931.3721.8122.228

110.2600.5400.8761.0881.3631.7962.201

120.2590.5390.8731.0831.3561.7822.179

130.2590.5380.8701.0791.3501.7712.160

140.2580.5370.8681.0761.3451.7612.145

150.2580.5360.8661.0741.3411.7532.131

160.2580.5350.8651.0711.3371.7462.120

170.2570.5340.8631.0691.3331.7402.110

180.2570.5340.8621.0671.3301.7342.101

190.2570.5330.8611.0661.3281.7292.093

200.2570.5330.8601.0641.3251.7252.086

210.2570.5320.8591.0631.3231.7212.080

220.2560.5320.8581.0611.3211.7172.074

230.2560.5320.8581.0601.3191.7142.069

240.2560.5310.8571.0591.3181.7112.064

250.2560.5310.8561.0581.3161.7082.060

260.2560.5310.8561.0581.3151.7062.056

270.2560.5310.8551.0571.3141.7032.052

280.2560.5300.8551.0561.3131.7012.048

290.2560.5300.8541.0551.3111.6992.045

300.2560.5300.8541.0551.3101.6972.042

400.2550.5290.8511.0501.3031.6842.021

600.2540.5270.8481.0451.2961.6712.000

1200.2540.5260.8451.0411.2891.6581.980

0.2530.5240.8421.0361.2821.6451.960

Table A.4 (continued) Critical Values of the t-Distribution

v0.020.0150.010.00750.0050.00250.0005

115.89421.20531.82142.43363.656127.321636.578

24.8495.6436.9658.0739.92514.08931.600

33.4823.8964.5415.0475.8417.45312.924

42.9993.2983.7474.0884.6045.5988.610

52.7573.0033.3653.6344.0324.7736.869

62.6122.8293.1433.3723.7074.3175.959

72.5172.7152.9983.2033.4994.0295.408

82.4492.6342.8963.0853.3553.8335.041

92.3982.5742.8212.9983.2503.6904.781

102.3592.5272.7642.9323.1693.5814.587

112.3282.4912.7182.8793.1063.4974.437

122.3032.4612.6812.8363.0553.4284.318

132.2822.4362.6502.8013.0123.3724.221

142.2642.4152.6242.7712.9773.3264.140

152.2492.3972.6022.7462.9473.2864.073

162.2352.3822.5832.7242.9213.2524.015

172.2242.3682.5672.7062.8983.2223.965

182.2142.3562.5522.6892.8783.1973.922

192.2052.3462.5392.6742.8613.1743.883

202.1972.3362.5282.6612.8453.1533.850

212.1892.3282.5182.6492.8313.1353.819

222.1832.3202.5082.6392.8193.1193.792

232.1772.3132.5002.6292.8073.1043.768

242.1722.3072.4922.6202.7973.0913.745

252.1672.3012.4852.6122.7873.0783.725

262.1622.2962.4792.6052.7793.0673.707

272.1582.2912.4732.5982.7713.0573.689

282.1542.2862.4672.5922.7633.0473.674

292.1502.2822.4622.5862.7563.0383.660

302.1472.2782.4572.5812.7503.0303.646

402.1232.2502.4232.5422.7042.9713.551

602.0992.2232.3902.5042.6602.9153.460

1202.0762.1962.3582.4682.6172.8603.373

2.054 2.170 2.326 2.432 2.576 2.807 3.290

2Table A.5 Critical Values of the Chi-Squared Distribution 0

v0.9950.990.980.9750.950.900.800.750.700.50

10.04 3930.03 1570.03 6280.03 9820.003930.01580.06420.1020.1480.455

20.01000.02010.04040.05060.1030.2110.4460.5750.7131.386

30.07170.1150.1850.2160.3520.5841.0051.2131.4242.366

40.2070.2970.4290.4840.7111.0641.6491.9232.1953.357

50.4120.5540.7520.8311.1451.6102.3432.6753.0004.351

60.6760.8721.1341.2371.6352.2043.0703.4553.8285.348

70.9891.2391.5641.6902.1672.8333.8224.2554.6716.346

81.3441.6472.0322.1802.7333.4904.5945.0715.5277.344

91.7352.0882.5322.7003.3254.1685.3805.8996.3938.343

102.1562.5583.0593.2473.9404.8656.1796.7377.2679.342

112.6033.0533.6093.8164.5755.5786.9897.5848.14810.341

123.0743.5714.1784.4045.2266.3047.8078.4389.03411.340

133.5654.1074.7655.0095.8927.0418.6349.2999.92612.340

144.0754.6605.3685.6296.5717.7909.46710.16510.82113.339

154.6015.2295.9856.2627.2618.54710.30711.03711.72114.339

165.1425.8126.6146.9087.9629.31211.15211.91212.62415.338

175.6976.4087.2557.5648.67210.08512.00212.79213.53116.338

186.2657.0157.9068.2319.39010.86512.85713.67514.44017.338

196.8447.6338.5678.90710.11711.65113.71614.56215.35218.338

207.4348.2609.2379.59110.85112.44314.57815.45216.26619.337

218.0348.8979.91510.28311.59113.24015.44516.34417.18220.337

228.6439.54210.60010.98212.33814.04116.31417.24018.10121.337

239.26010.19611.29311.68913.09114.84817.18718.13719.02122.337

249.88610.85611.99212.40113.84815.65918.06219.03719.94323.337

2510.52011.52412.69713.12014.61116.47318.94019.93920.86724.337

2611.16012.19813.40913.84415.37917.29219.82020.84321.79225.336

2711.80812.87814.12514.57316.15118.11420.70321.74922.71926.336

2812.46113.56514.84715.30816.92818.93921.58822.65723.64727.336

2913.12114.25615.57416.04717.70819.76822.47523.56724.57728.336

3013.78714.95316.30616.79118.49320.59923.36424.47825.50829.336

4020.70722.16423.83824.43326.50929.05132.34533.6634.87239.335

5027.99129.70731.66432.35734.76437.68941.44942.94244.31349.335

6035.53437.48539.69940.48243.18846.45950.64152.29453.80959.335


Table A.5 Chi-Squared Distribution Probability Table740

Table A.5 (continued) Critical Values of the Chi-Squared Distribution

v0.300.250.200.100.050.0250.020.010.0050.001

11.0741.3231.6422.7063.8415.0245.4126.6357.87910.827

22.4082.7733.2194.6055.9917.3787.8249.21010.59713.815

33.6654.1084.6426.2517.8159.3489.83711.34512.83816.266

44.8785.3855.9897.7799.48811.14311.66813.27714.86018.466

56.0646.6267.2899.23611.07012.83213.38815.08616.75020.515

67.2317.8418.55810.64512.59214.44915.03316.81218.54822.457

78.3839.0379.80312.01714.06716.01316.62218.47520.27824.321

89.52410.21911.03013.36215.50717.53518.16820.09021.95526.124

910.65611.38912.24214.68416.91919.02319.67921.66623.58927.877

1011.78112.54913.44215.98718.30720.48321.16123.20925.18829.588

1112.89913.70114.63117.27519.67521.92022.61824.72526.75731.264

1214.01114.84515.81218.54921.02623.33724.05426.21728.30032.909

1315.11915.98416.98519.81222.36224.73625.47127.68829.81934.527

1416.22217.11718.15121.06423.68526.11926.87329.14131.31936.124

1517.32218.24519.31122.30724.99627.48828.25930.57832.80137.698

1618.41819.36920.46523.54226.29628.84529.63332.00034.26739.252

1719.51120.48921.61524.76927.58730.19130.99533.40935.71840.791

1820.60121.60522.76025.98928.86931.52632.34634.80537.15642.312

1921.68922.71823.90027.20430.14432.85233.68736.19138.58243.819

2022.77523.82825.03828.41231.41034.17035.02037.56639.99745.314

2123.85824.93526.17129.61532.67135.47936.34338.93241.40146.796

2224.93926.03927.30130.81333.92436.78137.65940.28942.79648.268

2326.01827.14128.42932.00735.17238.07638.96841.63844.18149.728

2427.09628.24129.55333.19636.41539.36440.27042.98045.55851.179

2528.17229.33930.67534.38237.65240.64641.56644.31446.92852.619

2629.24630.43531.79535.56338.88541.92342.85645.64248.29054.051

2730.31931.52832.91236.74140.11343.19544.14046.96349.64555.475

2831.39132.62034.02737.91641.33744.46145.41948.27850.99456.892

2932.46133.71135.13939.08742.55745.72246.69349.58852.33558.301

3033.53034.80036.25040.25643.77346.97947.96250.89253.67259.702

4044.16545.61647.26951.80555.75859.34260.43663.69166.76673.403

5054.72356.33458.16463.16767.50571.42072.61376.15479.49086.660

6065.22666.98168.97274.39779.08283.29884.5888.37991.95299.608

Table A.6 Critical Values of the F-Distribution 0 f

f0.05 (v1 , v2 )

v1

v2123456789

1161.45199.50215.71224.58230.16233.99236.77238.88240.54

218.5119.0019.1619.2519.3019.3319.3519.3719.38

310.139.559.289.129.018.948.898.858.81

47.716.946.596.396.266.166.096.046.00

56.615.795.415.195.054.954.884.824.77

65.995.144.764.534.394.284.214.154.10

75.594.744.354.123.973.873.793.733.68

85.324.464.073.843.693.583.503.443.39

95.124.263.863.633.483.373.293.233.18

104.964.103.713.483.333.223.143.073.02

114.843.983.593.363.203.093.012.952.90

124.753.893.493.263.113.002.912.852.80

134.673.813.413.183.032.922.832.772.71

144.603.743.343.112.962.852.762.702.65

154.543.683.293.062.902.792.712.642.59

164.493.633.243.012.852.742.662.592.54

174.453.593.202.962.812.702.612.552.49

184.413.553.162.932.772.662.582.512.46

194.383.523.132.902.742.632.542.482.42

204.353.493.102.872.712.602.512.452.39

214.323.473.072.842.682.572.492.422.37

224.303.443.052.822.662.552.462.402.34

234.283.423.032.802.642.532.442.372.32

244.263.403.012.782.622.512.422.362.30

254.243.392.992.762.602.492.402.342.28

264.233.372.982.742.592.472.392.322.27

274.213.352.962.732.572.462.372.312.25

284.203.342.952.712.562.452.362.292.24

294.183.332.932.702.552.432.352.282.22

304.173.322.922.692.532.422.332.272.21

404.083.232.842.612.452.342.252.182.12

604.003.152.762.532.372.252.172.102.04

1203.923.072.682.452.292.182.092.021.96

3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88Reproduced from Table 18 of Biometrika Tables for Statisticians, Vol. I, by permission of E.S. Pearson and the Biometrika Trustees.

744Appendix A Statistical Tables and ProofsTable A.6 (continued) Critical Values of the F-Distribution

Table A.6 F-Distribution Probability Table744

f0.05 (v1 , v2 )v1 v2 10 12 15 20 24 30 40 60 120 1 241.88 243.91 245.95 248.01 249.05 250.10 251.14 252.20 253.25 254.312 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.503 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.534 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.635 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.366 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.677 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.238 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.939 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.7110 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.5411 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.4012 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.3013 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.2114 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.1315 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.0716 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.0117 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.9618 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.9219 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.8820 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.8421 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.8122 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.7823 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.7624 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.7325 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.7126 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.6927 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.6728 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.6529 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.6430 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.6240 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.5160 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39120 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

Table A.6 (continued) Critical Values of the F-Distributionf0.01 (v1 , v2 )v1

v2123456789

14052.184999.505403.355624.585763.655858.995928.365981.076022.47

298.5099.0099.1799.2599.3099.3399.3699.3799.39

334.1230.8229.4628.7128.2427.9127.6727.4927.35

421.2018.0016.6915.9815.5215.2114.9814.8014.66

516.2613.2712.0611.3910.9710.6710.4610.2910.16

613.7510.929.789.158.758.478.268.107.98

712.259.558.457.857.467.196.996.846.72

811.268.657.597.016.636.376.186.035.91

910.568.026.996.426.065.805.615.475.35

1010.047.566.555.995.645.395.205.064.94

119.657.216.225.675.325.074.894.744.63

129.336.935.955.415.064.824.644.504.39

139.076.705.745.214.864.624.444.304.19

148.866.515.565.044.694.464.284.144.03

158.686.365.424.894.564.324.144.003.89

168.536.235.294.774.444.204.033.893.78

178.406.115.184.674.344.103.933.793.68

188.296.015.094.584.254.013.843.713.60

198.185.935.014.504.173.943.773.633.52

208.105.854.944.434.103.873.703.563.46

218.025.784.874.374.043.813.643.513.40

227.955.724.824.313.993.763.593.453.35

237.885.664.764.263.943.713.543.413.30

247.825.614.724.223.903.673.503.363.26

257.775.574.684.183.853.633.463.323.22

267.725.534.644.143.823.593.423.293.18

277.685.494.604.113.783.563.393.263.15

287.645.454.574.073.753.533.363.233.12

297.605.424.544.043.733.503.333.203.09

307.565.394.514.023.703.473.303.173.07

407.315.184.313.833.513.293.122.992.89

607.084.984.133.653.343.122.952.822.72

1206.854.793.953.483.172.962.792.662.56

6.634.613.783.323.022.802.642.512.41

f0.01 (v1 , v2 )v1 v2 10 12 15 20 24 30 40 60 120 1 6055.85 6106.32 6157.28 6208.73 6234.63 6260.65 6286.78 6313.03 6339.39 6365.862 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.503 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.134 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.465 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.026 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.887 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.658 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.869 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.3110 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.9111 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.6012 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.3613 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.1714 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.0015 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.8716 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.7517 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.6518 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.5719 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.4920 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.4221 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.3622 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.3123 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.2624 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.2125 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.1726 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.1327 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.1028 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.0629 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.0330 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.0140 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.8060 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

745Table A.7 Tolerance Factors for Normal Distributions Table A.7 Tolerance Factors for Normal Distributions Two-Sided Intervals One-Sided Intervals = 0.05 = 0.01 = 0.05 = 0.01 1 1 1 1 n 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.992 32.019 37.674 48.430 160.193 188.491 242.300 20.581 26.260 37.094 103.029 131.426 185.6173 8.380 9.916 12.861 18.930 22.401 29.055 6.156 7.656 10.553 13.995 17.170 23.8964 5.369 6.370 8.299 9.398 11.150 14.527 4.162 5.144 7.042 7.380 9.083 12.3875 4.275 5.079 6.634 6.612 7.855 10.260 3.407 4.203 5.741 5.362 6.578 8.9396 3.712 4.414 5.775 5.337 6.345 8.301 3.006 3.708 5.062 4.411 5.406 7.3357 3.369 4.007 5.248 4.613 5.488 7.187 2.756 3.400 4.642 3.859 4.728 6.4128 3.136 3.732 4.891 4.147 4.936 6.468 2.582 3.187 4.354 3.497 4.285 5.8129 2.967 3.532 4.631 3.822 4.550 5.966 2.454 3.031 4.143 3.241 3.972 5.38910 2.839 3.379 4.433 3.582 4.265 5.594 2.355 2.911 3.981 3.048 3.738 5.07411 2.737 3.259 4.277 3.397 4.045 5.308 2.275 2.815 3.852 2.898 3.556 4.82912 2.655 3.162 4.150 3.250 3.870 5.079 2.210 2.736 3.747 2.777 3.410 4.63313 2.587 3.081 4.044 3.130 3.727 4.893 2.155 2.671 3.659 2.677 3.290 4.47214 2.529 3.012 3.955 3.029 3.608 4.737 2.109 2.615 3.585 2.593 1.189 4.33715 2.480 2.954 3.878 2.945 3.507 4.605 2.068 2.566 3.520 2.522 3.102 4.22216 2.437 2.903 3.812 2.872 3.421 4.492 2.033 2.524 3.464 2.460 3.028 4.12317 2.400 2.858 3.754 2.808 3.345 4.393 2.002 2.486 3.414 2.405 2.963 4.03718 2.366 2.819 3.702 2.753 3.279 4.307 1.974 2.453 3.370 2.357 2.905 3.96019 2.337 2.784 3.656 2.703 3.221 4.230 1.949 2.423 3.331 2.314 2.854 3.89220 2.310 2.752 3.615 2.659 3.168 4.161 1.926 2.396 3.295 2.276 2.808 1.83225 2.208 2.631 3.457 2.494 2.972 3.904 1.838 2.292 3.158 2.129 2.633 3.00130 2.140 2.549 3.350 2.385 2.841 3.733 1.777 2.220 3.064 2.030 2.516 3.44735 2.090 2.490 3.272 2.306 2.748 3.611 1.732 2.167 2.995 1.957 2.430 3.33440 2.052 2.445 3.213 2.247 2.677 3.518 1.697 2.126 2.941 1.902 2.364 3.24945 2.021 2.408 3.165 2.200 2.621 3.444 1.669 2.092 2.898 1.857 2.312 3.18050 1.996 2.379 3.126 2.162 2.576 3.385 1.646 2.065 2.863 1.821 2.269 3.12560 1.958 2.333 3.066 2.103 2.506 3.293 1.609 2.022 2.807 1.764 2.202 3.03870 1.929 2.299 3.021 2.060 2.454 3.225 1.581 1.990 2.765 1.722 2.153 2.97480 1.907 2.272 2.986 2.026 2.414 3.173 1.559 1.965 2.733 1.688 2.114 2.92490 1.889 2.251 2.958 1.999 2.382 3.130 1.542 1.944 2.706 1.661 2.082 2.883100 1.874 2.233 2.934 1.977 2.355 3.096 1.527 1.927 2.684 1.639 2.056 2.850150 1.825 2.175 2.859 1.905 2.270 2.983 1.478 1.870 2.611 1.566 1.971 2.741200 1.798 2.143 2.816 1.865 2.222 2.921 1.450 1.837 2.570 1.524 1.923 2.679250 1.780 2.121 2.788 1.839 2.191 2.880 1.431 1.815 2.542 1.496 1.891 2.638300 1.767 2.106 2.767 1.820 2.169 2.850 1.417 1.800 2.522 1.476 1.868 2.608 1.645 1.960 2.576 1.645 1.960 2.576 1.282 1.645 2.326 1.282 1.645 2.326Adapted from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Analysis, Chapter 2, McGraw-Hill Book Company, New York, 1947. Used with permission of McGraw-Hill Book Company.



Table A.8 Sample Size for the t-Test of the Mean

Level of t-Test

Single-Sided TestDouble-Sided Test = 0.005 = 0.01 = 0.01 = 0.02 = 0.025 = 0.05 = 0.05 = 0.1

= 0.1.01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2 .5 .01 .05.1.2.5

0.05

0.10

0.15122

0.201399970

0.2511090128 6413910145

0.3013478115631199045122977132

0.351259958109854710988673490725224

0.401159777451018566371178468512610170554019

0.45927762371108168533093675441218055443315

0.5010075635130906655432576544434186545362713

0.558363534226755546362163453728155438302211

0.60715345362263473931185338322413463226199

0.65614639312055413427164633272112392822178

0.70534034281747353024144029241910342419158

0.7547363025164231272113352621169302117137

0.8041322722143728241912312219159271915126

0.8537292420133325211711282117138241714116

0.9034262218122923191610251916127211513105

Value of0.95312420171127211814923171411719141195

= ||/1.00282219161025191613921161310618131185

1.124191614921161412818131196151197

1.221161412818141210715121085131086

1.31815131181613119614109711876

1.4161312107141110961298710875

1.515121197131098611876976

1.613111086121097510876866

1.71210986119879765865

1.812109861087787676

1.91198761087686675

2.010887597767656

2.1108778766766

2.298768765766

2.39776866655

2.487767666

2.587667666

3.076656555

3.56555

4.06

Reproduced with permission from O. L. Davies, ed., Design and Analysis of Industrial Experi- ments, Oliver & Boyd, Edinburgh, 1956.

Table A.9 Table of Sample Sizes for the Test of the Dierence between Two Means 747

Table A.9 Sample Size for the t-Test of the Dierence between Two MeansLevel of t-Test

Single-Sided TestDouble-Sided Test = 0.005 = 0.01 = 0.01 = 0.02 = 0.025 = 0.05 = 0.05 = 0.1

= 0.1.01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2.5

0.05

0.10

0.15

0.20137

0.2512488

0.301238761

0.35110906410245

0.408570100501087835

0.4511868101551057939 108866228

0.5096551068245 106866432 88705123

0.551017946 106886838 87715327 112 73584219

0.60101856739 90745832 104 74604523 89 61493616

0.6587735734 104 77644927 88 63513920 76 52423014

0.7010075635029906655432476554434176645362612

0.758866554426795848382167483929155740322311

0.807758493923705143331959423426145035282110

0.85695143352162463830175237312312453125189

0.90624639311955413427154734272111402822168

Value of0.95554235281750373124144230251910362520157

= ||/1.0050383226154533282213382723179332318147

1.142322722133828231911322319148271915126

1.23627231811322420169272016127231613105

1.3312320161028211714823171411620141195

1.427201714924181512820151210617121084

1.5241815138211614117181311951511974

1.6211614117191412106161210851410864

1.7191513107171311961411974129763

1.817137110615121085131086411875

1.916121196141198512976410765

2.01411108613109751187649764

2.113109851298751086538654

2.2121087511976497658654

2.311987510876497657554

2.411986510876486547544

2.51087649765486546543

3.086654765436544543

3.5655436544544343

4.0654454434434

Reproduced with permission from O. L. Davies, ed., Design and Analysis of Industrial Experi- ments, Oliver & Boyd, Edinburgh, 1956.

Table A.10 Critical Values for Bartletts Testbk (0.01; n)Number of Populations, k

n2345678910

30.14110.1672

40.28430.31650.34750.37290.39370.4110

50.39840.43040.46070.48500.50460.52070.53430.54580.5558

60.48500.51490.54300.56530.5832O.59780.61000.62040.6293

70.55120.57870.60450.62480.64100.65420.66520.67440.6824

80.60310.62820.65180.67040.68510.69700.70690.71530.7225

90.64450.66760.68920.70620.71970.73050.73950.74710.7536

100.67830.69960.71950.73520.74750.75750.76570.77260.7786

110.70630.72600.74450.75900.77030.77950.78710.79350.7990

120.72990.74830.76540.77890.78940.79800.80500.81090.8160

130.75010.76720.78320.79580.80560.81350.82010.82560.8303

140.76740.78350.79850.81030.81950.82690.83300.83820.8426

150.78250.79770.81180.82290.83150.83850.84430.84910.8532

160.79580.81010.82350.83390.84210.84860.85410.85860.8625

170.80760.82110.83380.84360.85140.85760.86270.86700.8707

180.81810.83090.84290.85230.85960.86550.87040.87450.8780

190.82750.83970.85120.86010.86700.87270.87730.88110.8845

200.83600.84760.85860.86710.87370.87910.88350.88710.8903

210.84370.85480.86530.87340.87970.88480.88900.89260.8956

220.85070.86140.87140.87910.88520.89010.89410.89750.9004

230.85710.86730.87690.88440.89020.89490.89880.90200.9047

240.86300.87280.88200.88920.89480.89930.90300.90610.9087

250.86840.87790.88670.89360.89900.90340.90690.90990.9124

260.87340.88250.89110.89770.90290.90710.91050.91340.9158

270.87810.88690.89510.90150.90650.91050.91380.91660.9190

280.88240.89090.89880.90500.90990.91380.91690.91960.9219

290.88640.89460.90230.90830.91300.91670.91980.92240.9246

300.89020.89810.90560.91140.91590.91950.92250.92500.9271

400.91750.92350.92910.93350.93700.93970.94200.94390.9455

500.93390.93870.94330.94680.94960.95180.95360.95510.9564

600.94490.94890.95270.95570.95800.95990.96140.96260.9637

800.95860.96170.96460.96680.96850.96990.97110.97200.9728

1000.96690.96930.97160.97340.97480.97590.97690.97760.9783

Reproduced from D. D. Dyer and J. P. Keating, On the Determination of Critical Values forBartletts Test, J. Am. Stat. Assoc., 75, 1980, by permission of the Board of Directors.



Table A.10 Table for Bartletts Test 749

Table A.10 (continued) Critical Values for Bartletts Testbk (0.05; n)Number of Populations, k

n2345678910

30.31230.30580.31730.3299

40.47800.46990.48030.49210.50280.51220.52040.52770.5341

50.58450.57620.58500.59520.60450.61260.61970.62600.6315

60.65630.64830.65590.66460.67270.67980.68600.69140.6961

70.70750.70000.70650.71420.72130.72750.73290.73760.7418

80.74560.73870.74440.75120.75740.76290.76770.77190.7757

90.77510.76860.77370.77980.78540.79030.79460.79840.8017

100.79840.79240.79700.80250.80760.81210.81600.81940.8224

110.81750.81180.81600.82100.82570.82980.83330.83650.8392

120.83320.82800.83170.83640.84070.84440.84770.85060.8531

130.84650.84150.84500.84930.85330.85680.85980.86250.8648

140.85780.85320.85640.86040.86410.86730.87010.87260.8748

150.86760.86320.86620.86990.87340.87640.87900.88140.8834

160.87610.87190.87470.87820.88150.88430.88680.88900.8909

170.88360.87960.88230.88560.88860.89130.89360.89570.8975

180.89020.88650.88900.89210.89490.89750.89970.90160.9033

190.89610.89260.89490.89790.90060.90300.90510.90690.9086

200.90150.89800.90030.90310.90570.90800.91000.91170.9132

210.90630.90300.90510.90780.91030.91240.91430.91600.9175

220.91060.90750.90950.91200.91440.91650.91830.91990.9213

230.91460.91160.91350.91590.91820.92020.92190.92350.9248

240.91820.91530.91720.91950.92170.92360.92530.92670.9280

250.92160.91870.92050.92280.92490.92670.92830.92970.9309

260.92460.92190.92360.92580.92780.92960.93110.93250.9336

270.92750.92490.92650.92860.93050.93220.93370.93500.9361

280.93010.92760.92920.93120.93300.93470.93610.93740.9385

290.93260.93010.93160.93360.93540.93700.93830.93960.9406

300.93480.93250.93400.93580.93760.93910.94040.94160.9426

400.95130.94950.95060.95200.95330.95450.95550.95640.9572

500.96120.95970.96060.96170.96280.96370.96450.96520.9658

600.96770.96650.96720.96810.96900.96980.97050.97100.9716

800.97580.97490.97540.97610.97680.97740.97790.97830.9787

1000.98070.97990.98040.98090.98150.98190.98230.98270.9830

Table A.11 Critical Values for Cochrans Test

= 0.01n

Appendix A Statistical Tables and Proofs750 k 2 3 4 5 6 7 8 9 10 11 17 37 145 2 0.9999 0.9950 0.9794 0.9586 0.9373 0.9172 0.8988 0.8823 0.8674 0.8539 0.7949 0.7067 0.6062 0.50003 0.9933 0.9423 0.8831 0.8335 0.7933 0.7606 0.7335 0.7107 0.6912 0.6743 0.6059 0.5153 0.4230 0.33334 0.9676 0.8643 0.7814 0.7212 0.6761 0.6410 0.6129 0.5897 0.5702 0.5536 0.4884 0.4057 0.3251 0.25005 0.9279 0.7885 0.6957 0.6329 0.5875 0.5531 0.5259 0.5037 0.4854 0.4697 0.4094 0.3351 0.2644 0.20006 0.8828 0.7218 0.6258 0.5635 0.5195 0.4866 0.4608 0.4401 0.4229 0.4084 0.3529 0.2858 0.2229 0.16677 0.8376 0.6644 0.5685 0.5080 0.4659 0.4347 0.4105 0.3911 0.3751 0.3616 0.3105 0.2494 0.1929 0.14298 0.7945 0.6152 0.5209 0.4627 0.4226 0.3932 0.3704 0.3522 0.3373 0.3248 0.2779 0.2214 0.1700 0.12509 0.7544 0.5727 0.4810 0.4251 0.3870 0.3592 0.3378 0.3207 0.3067 0.2950 0.2514 0.1992 0.1521 0.111110 0.7175 0.5358 0.4469 0.3934 0.3572 0.3308 0.3106 0.2945 0.2813 0.2704 0.2297 0.1811 0.1376 0.100012 0.6528 0.4751 0.3919 0.3428 0.3099 0.2861 0.2680 0.2535 0.2419 0.2320 0.1961 0.1535 0.1157 0.083315 0.5747 0.4069 0.3317 0.2882 0.2593 0.2386 0.2228 0.2104 0.2002 0.1918 0.1612 0.1251 0.0934 0.066720 0.4799 0.3297 0.2654 0.2288 0.2048 0.1877 0.1748 0.1646 0.1567 0.1501 0.1248 0.0960 0.0709 0.050024 0.4247 0.2871 0.2295 0.1970 0.1759 0.1608 0.1495 0.1406 0.1338 0.1283 0.1060 0.0810 0.0595 0.041730 0.3632 0.2412 0.1913 0.1635 0.1454 0.1327 0.1232 0.1157 0.1100 0.1054 0.0867 0.0658 0.0480 0.033340 0.2940 0.1915 0.1508 0.1281 0.1135 0.1033 0.0957 0.0898 0.0853 0.0816 0.0668 0.0503 0.0363 0.025060 0.2151 0.1371 0.1069 0.0902 0.0796 0.0722 0.0668 0.0625 0.0594 0.0567 0.0461 0.0344 0.0245 0.0167120 0.1225 0.0759 0.0585 0.0489 0.0429 0.0387 0.0357 0.0334 0.0316 0.0302 0.0242 0.0178 0.0125 0.0083 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Reproduced from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Analysis, Chapter 15, McGraw- Hill Book Company, New, York, 1947. Used with permission of McGraw-Hill Book Company.

751Table A.11 Table for Cochrans TestTable A.11 (continued) Critical Values for Cochrans Test = 0.05n k 2 3 4 5 6 7 8 9 10 11 17 37 145 2 0.9985 0.9750 0.9392 0.9057 0.8772 0.8534 0.8332 0.8159 0.8010 0.7880 0.7341 0.6602 0.5813 0.50003 0.9669 0.8709 0.7977 0.7457 0.7071 0.6771 0.6530 0.6333 0.6167 0.6025 0.5466 0.4748 0.4031 0.33334 0.9065 0.7679 0.6841 0.6287 0.5895 0.5598 0.5365 0.5175 0.5017 0.4884 0.4366 0.3720 0.3093 0.25005 0.8412 0.6838 0.5981 0.5441 0.5065 0.4783 0.4564 0.4387 0.4241 0.4118 0.3645 0.3066 0.2513 0.20006 0.7808 0.6161 0.5321 0.4803 0.4447 0.4184 0.3980 0.3817 0.3682 0.3568 0.3135 0.2612 0.2119 0.16677 0.7271 0.5612 0.4800 0.4307 0.3974 0.3726 0.3535 0.3384 0.3259 0.3154 0.2756 0.2278 0.1833 0.14298 0.6798 0.5157 0.4377 0.3910 0.3595 0.3362 0.3185 0.3043 0.2926 0.2829 0.2462 0.2022 0.1616 0.12509 0.6385 0.4775 0.4027 0.3584 0.3286 0.3067 0.2901 0.2768 0.2659 0.2568 0.2226 0.1820 0.1446 0.111110 6.6020 0.4450 0.3733 0.3311 0.3029 0.2823 0.2666 0.2541 0.2439 0.2353 0.2032 0.1655 0.1308 0.100012 0.5410 0.3924 0.3264 0.2880 0.2624 0.2439 0.2299 0.2187 0.2098 0.2020 0.1737 0.1403 0.1100 0.083315 0.4709 0.3346 0.2758 0.2419 0.2195 0.2034 0.1911 0.1815 0.1736 0.1671 0.1429 0.1144 0.0889 0.066720 0.3894 0.2705 0.2205 0.1921 0.1735 0.1602 0.1501 0.1422 0.1357 0.1303 0.1108 0.0879 0.0675 0.050024 0.3434 0.2354 0.1907 0.1656 0.1493 0.1374 0.1286 0.1216 0.1160 0.1113 0.0942 0.0743 0.0567 0.041730 0.2929 0.1980 0.1593 0.1377 0.1237 0.1137 0.1061 0.1002 0.0958 0.0921 0.0771 0.0604 0.0457 0.033340 0.2370 0.1576 0.1259 0.1082 0.0968 0.0887 0.0827 0.0780 0.0745 0.0713 0.0595 0.0462 0.0347 0.025060 0.1737 0.1131 0.0895 0.0765 0.0682 0.0623 0.0583 0.0552 0.0520 0.0497 0.0411 0.0316 0.0234 0.0167120 0.0998 0.0632 0.0495 0.0419 0.0371 0.0337 0.0312 0.0292 0.0279 0.0266 0.0218 0.0165 0.0120 0.0083 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table A.12 Upper Percentage Points of the Studentized Range Distribution: Values ofq(0.05; k, v)

Degrees ofNumberof Treatmentsk

Freedom, v2345678910

118.027.032.837.240.543.115.147.149.1

26.095.339.8010.8911.7312.4313.0313.5413.99

34.505.916.837.518.048.478.859.189.46

43.935.045.766.296.717.067.357.607.83

53.644.605.225.676.036.336.586.806.99

63.464.344.905.315.635.896.126.326.49

73.344.164.685.065.355.595.805.996.15

83.264.044.534.895.175.405.605.775.92

93.203.954.424.765.025.245.435.605.74

103.153.884.334.664.915.125.305.465.60

113.113.824.264.584.825.035.205.355.49

123.083.774.204.514.754.955.125.275.40

133.063.734.154.464.694.885.055.195.32

143.033.704.114.414.654.834.995.135.25

153.013.674.084.374.594.784.945.085.20

163.003.654.054.344.564.744.905.035.05

172.983.624.024.314.524.704.864.995.11

182.973.614.004.284.494.674.834.965.07

192.963.593.984.264.474.644.794.925.04

202.953.583.964.244.454.624.774.905.01

242.923.533.904.174.374.544.684.814.92

302.893.483.844.114.304.464.604.724.83

402.863.443.794.044.234.394.524.634.74

602.833.403.743.984.164.314.444.554.65

1202.803.363.693.924.104.244.364.474.56

2.77 3.32 3.63 3.86 4.03 4.17 4.29 4.39 4.47



Table A.13 Table for Duncans Test 753

Table A.13 Least Signicant Studentized Ranges rp (0.05; p, v) = 0.05

p

v2345678910

117.9717.9717.9717.9717.9717.9717.9717.9717.97

26.0856.0856.0856.0856.0856.0856.0856.0856.085

34.5014.5164.5164.5164.5164.5164.5164.5164.516

43.9274.0134.0334.0334.0334.0334.0334.0334.033

53.6353.7493.7973.8143.8143.8143.8143.8143.814

63.4613.5873.6493.683.6943.6973.6973.6973.697

73.3443.4773.5483.5883.6113.6223.6263.6263.626

83.2613.3993.4753.5213.5493.5663.5753.5793.579

93.1993.3393.4203.4703.5023.5233.5363.5443.547

103.1513.2933.3763.4303.4653.4893.5053.5163.522

113.1133.2563.3423.3973.4353.4623.483.4933.501

123.0823.2253.3133.3703.4103.4393.4593.4743.484

133.0553.2003.2893.3483.3893.4193.4423.4583.470

143.0333.1783.2683.3293.3723.4033.4263.4443.457

153.0143.1603.253.3123.3563.3893.4133.4323.446

162.9983.1443.2353.2983.3433.3763.4023.4223.437

172.9843.1303.2223.2853.3313.3663.3923.4123.429

182.9713.1183.2103.2743.3213.3563.3833.4053.421

192.9603.1073.1993.2643.3113.3473.3753.3973.415

202.9503.0973.1903.2553.3033.3393.3683.3913.409

242.9193.0663.1603.2263.2763.3153.3453.3703.390

302.8883.0353.1313.1993.2503.2903.3223.3493.371

402.8583.0063.1023.1713.2243.2663.3003.3283.352

602.8292.9763.0733.1433.1983.2413.2773.3073.333

1202.8002.9473.0453.1163.1723.2173.2543.2873.314

2.772 2.918 3.017 3.089 3.146 3.193 3.232 3.265 3.294 Abridged from H. L. Harter, Critical Values for Duncans New Multiple Range Test,Biometrics, 16, No. 4, 1960, by permission of the author and the editor.

Table A.13 (continued) Least Signicant Studentized Ranges rp (0.01; p, v) = 0.01

p

v2345678910

190.0390.0390.0390.0390.0390.0390.0390.0390.03

214.0414.0414.0414.0414.0414.0414.0414.0414.04

38.2618.3218.3218.3218.3218.3218.3218.3218.321

46.5126.6776.7406.7566.7566.7566.7566.7566.756

55.7025.8935.9896.0406.0656.0746.0746.0746.074

65.2435.4395.5495.6145.6555.6805.6945.7015.703

74.9495.1455.2605.3345.3835.4165.4395.4545.464

84.7464.9395.0575.1355.1895.2275.2565.2765.291

94.5964.7874.9064.9865.0435.0865.1185.1425.160

104.4824.6714.7904.8714.9314.9755.0105.0375.058

114.3924.5794.6974.7804.8414.8874.9244.9524.975

124.3204.5044.6224.7064.7674.8154.8524.8834.907

134.2604.4424.5604.6444.7064.7554.7934.8244.850

144.2104.3914.5084.5914.6544.7044.7434.7754.802

154.1684.3474.4634.5474.6104.6604.7004.7334.760

164.1314.3094.4254.5094.5724.6224.6634.6964.724

174.0994.2754.3914.4754.5394.5894.6304.6644.693

184.0714.2464.3624.4454.5094.5604.6014.6354.664

194.0464.2204.3354.4194.4834.5344.5754.6104.639

204.0244.1974.3124.3954.4594.5104.5524.5874.617

243.9564.1264.2394.3224.3864.4374.4804.5164.546

303.8894.0564.1684.2504.3144.3664.4094.4454.477

403.8253.9884.0984.1804.2444.2964.3394.3764.408

603.7623.9224.0314.1114.1744.2264.2704.3074.340

1203.7023.8583.9654.0444.1074.1584.2024.2394.272

3.643 3.796 3.900 3.978 4.040 4.091 4.135 4.172 4.205



Table A.14 Table for Dunnetts Two-Sided Test 755

Table A.14 Values of d/2 (k, v) for Two-Sided Comparisons between k Treatments and a Control = 0.05k = Number of Treatment Means (excluding control)

v123456789

52.573.033.293.483.623.733.823.903.97

62.452.863.103.263.393.493.573.643.71

72.362.752.973.123.243.333.413.473.53

82.312.672.883.023.133.223.293.353.41

92.262.612.812.953.053.143.203.263.32

102.232.572.762.892.993.073.143.193.24

112.202.532.722.842.943.023.083.143.19

122.182.502.682.812.902.983.043.093.14

132.162.482.652.782.872.943.003.063.10

142.142.462.632.752.842.912.973.023.07

152.132.442.612.732.822.892.953.003.04

162.122.422.592.712.802.872.922.973.02

172.112.412.582.692.782.852.902.953.00

182.102.402.562.682.762.832.892.942.98

192.092.392.552.662.752.812.872.922.96

202.092.382.542.652.732.802.862.902.95

242.062.352.512.612.702.762.812.862.90

302.042.322.472.582.662.722.772.822.86

402.022.292.442.542.622.682.732.772.81

602.002.272.412.512.582.642.692.732.77

1201.982.242.382.472.552.602.652.692.73

1.96 2.21 2.35 2.44 2.51 2.57 2.61 2.65 2.69Reproduced from Charles W. Dunnett, New Tables for Multiple Comparison with a Con- trol, Biometrics, 20, No. 3, 1964, by permission of the author and the editor.

Table A.14 (continued) Values of d/2 (k, v) for Two-Sided Comparisons between k Treat- ments and a Control = 0.01k = Number of Treatment Means (excluding control)

v123456789

54.034.634.985.225.415.565.695.805.89

63.714.214.514.714.875.005.105.205.28

73.503.954.214.394.534.644.744.824.89

83.363.774.004.174.294.404.484.564.62

93.253.633.854.014.124.224.304.374.43

103.173.533.743.883.994.084.164.224.28

113.113.453.653.793.893.984.054.114.16

123.053.393.583.713.813.893.964.024.07

133.013.333.523.653.743.823.893.943.99

142.983.293.473.593.693.763.833.883.93

152.953.253.433.553.643.713.783.833.88

162.923.223.393.513.603.673.733.783.83

172.903.193.363.473.563.633.693.743.79

182.883.173.333.443.533.603.663.713.75

192.863.153.313.423.503.573.633.683.72

202.853.133.293.403.483.553.603.653.69

242.803.073.223.323.403.473.523.573.61

302.753.013.153.253.333.393.443.493.52

402.702.953.093.193.263.323.373.413.44

602.662.903.033.123.193.253.293.333.37

1202.622.852.973.063.123.183.223.263.29

2.58 2.79 2.92 3.00 3.06 3.11 3.15 3.19 3.22



Table A.15 Table for Dunnetts One-Sided Test 757

Table A.15 Values of d (k, v) for One-Sided Comparisons between k Treatments and a Control = 0.05k = Number of Treatment Means (excluding control)

v123456789

52.022.442.682.852.983.083.163.243.30

61.942.342.562.712.832.923.003.073.12

71.892.272.482.622.732.822.892.953.01

81.862.222.422.552.662.742.812.872.92

91.832.182.372.502.602.682.752.812.86

101.812.152.342.472.562.642.702.762.81

111.802.132.312.442.532.602.672.722.77

121.782.112.292.412.502.582.642.692.74

131.772.092.272.392.482.552.612.662.71

141.762.082.252.372.462.532.592.642.69

151.752.072.242.362.442.512.572.622.67

161.752.062.232.342.432.502.562.612.65

171.742.052.222.332.422.492.542.592.64

181.732.042.212.322.412.482.532.582.62

191.732.032.202.312.402.472.522.572.61

201.722.032.192.302.392.462.512.562.60

241.712.012.172.282.362.432.482.532.57

301.701.992.152.252.332.402.452.502.54

401.681.972.132.232.312.372.422.472.51

601.671.952.102.212.282.352.392.442.48

1201.661.932.082.182.262.322.372.412.45

1.64 1.92 2.06 2.16 2.23 2.29 2.34 2.38 2.42Reproduced from Charles W. Dunnett, A Multiple Comparison Procedure for Compar- ing Several Treatments with a Control, J. Am. Stat. Assoc., 50, 1955, 10961121, by permission of the author and the editor.

Table A.15 (continued) Values of d (k, v) for One-Sided Comparisons between k Treat- ments and a Control = 0.01k = Number of Treatment Means (excluding control)

v123456789

53.373.904.214.434.604.734.854.945.03

63.143.613.884.074.214.334.434.514.59

73.003.423.663.833.964.074.154.234.30

82.903.293.513.673.793.883.964.034.09

92.823.193.403.553.663.753.823.893.94

102.763.113.313.453.563.643.713.783.83

112.723.063.253.383.483.563.633.693.74

122.683.013.193.323.423.503.563.623.67

132.652.973.153.273.373.443.513.563.61

142.622.943.113.233.323.403.463.513.56

152.602.913.083.203.293.363.423.473.52

162.582.883.053.173.263.333.393.443.48

172.572.863.033.143.233.303.363.413.45

182.552.843.013.123.213.273.333.383.42

192.542.832.993.103.183.253.313.363.40

202.532.812.973.083.173.233.293.343.38

242.492.772.923.033.113.173.223.273.31

302.462.722.872.973.053.113.163.213.24

402.422.682.822.922.993.053.103.143.18

602.392.642.782.872.943.003.043.083.12

1202.362.602.732.822.892.942.993.033.06

2.33 2.56 2.68 2.77 2.84 2.89 2.93 2.97 3.00



Table A.16 Table for the Signed-Rank Test 759

Table A.16 Critical Values for the Signed-Rank Test

nOne-Sided = 0.01Two-Sided = 0.02One-Sided = 0.025Two-Sided = 0.05One-Sided = 0.05Two-Sided = 0.1

51

612

7024

8246

9368

105811

1171114

12101417

13131721

14162126

15202530

16243036

17283541

18334047

19384654

20435260

21495968

22566675

23627383

24698192

257790101

268598110

2793107120

28102117130

29111127141

30120137152

Reproduced from F. Wilcoxon and R. A. Wilcox, Some Rapid Approximate Statistical Procedures, American Cyanamid Company, Pearl River, N.Y., 1964, by permission of the American Cyanamid Company.

Table A.17 Critical Values for the Wilcoxon Rank-Sum TestOne-Tailed Test at = 0.001 or Two-Tailed Test at = 0.002n2

n167891011121314151617181920

1

2

30000

400011122333

500112233455677

6012234456789101112

723356789101113141516

8556891112141517182021

97810121415171921232526

101012141719212325272932

1115172022242729323437

12202325283134374042

132629323538424548

1432363943465054

15404347515559

164852566065

1757616670

18667176

197782

20 88

One-Tailed Test at = 0.01 or Two-Tailed Test at = 0.02

n2

n1567891011121314151617181920

1

200000011

300111222334445

401123345567789910

512345678910111213141516

6346789111213151618192022

76891112141617192123242628

810111315172022242628303234

9141618212326283133363840

101922242730333638414447

1125283134374144475053

12313538424649535660

133943475155596367

1447515660656973

15566166707580

166671768287

1777828893

188894100

19101107

20 114Based in part on Tables 1, 3, 5, and 7 of D. Auble, Extended Tables for the Mann-Whitney Statistic, Bulletin of the Institute of Educational Research at Indiana University, 1, No. 2, 1953, by permission of the director.



Table A.17 Table for the Rank-Sum Test 761

Table A.17 (continued) Critical Values for the Wilcoxon Rank-Sum TestOne-Tailed Test at = 0.025 or Two-Tailed Test at = 0.05n2

n14567891011121314151617181920

1

20000111112222

30112233445566778

401234456789101111121313

52356789111213141517181920

6568101113141617192122242527

7810121416182022242628303234

813151719222426293134363841

9172023262831343739424548

102326293336394245485255

1130333740444751555862

12374145495357616569

134550545963677276

1455596467747883

15647075808590

167581869298

17879399105

1899106112

19113119

20 127

One-Tailed Test at = 0.05 or Two-Tailed Test at = 0.1n2

n134567891011121314151617181920

100

20001111223333444

300122344556778991011

41234567891011121415161718

5456891112131516181920222325

67810121416171921232526283032

71113151719212426283033353739

815182023262831333639414447

9212427303336394245485154

102731343741444851555862

1134384246505457616569

12424751556064687277

135156616570758084

1461667177828792

157277838894100

16838995101107

1796102109115

18109116123

19123130

20 138

Table A.18 P (V v when H0 is true) in the Runs Test v

(n1 , n2 )2345678910

(2, 3)0.2000.5000.9001.000

(2, 4)0.1330.4000.8001.000

(2, 5)0.0950.3330.7141.000

(2, 6)0.0710.2860.6431.000

(2, 7)0.0560.2500.5831.000

(2, 8)0.0440.2220.5331.000

(2, 9)0.0360.2000.4911.000

(2, 10)0.0300.1820.4551.000

(3, 3)0.1000.3000.7000.9001.000

(3, 4)0.0570.2000.5430.8000.9711.000

(3, 5)0.0360.1430.4290.7140.9291.000

(3, 6)0.0240.1070.3450.6430.8811.000

(3, 7)0.0170.0830.2830.5830.8331.000

(3, 8)0.0120.0670.2360.5330.7881.000

(3, 9)0.0090.0550.2000.4910.7451.000

(3, 10)0.0070.0450.1710.4550.7061.000

(4, 4)0.0290.1140.3710.6290.8860.9711.000

(4, 5)0.0160.0710.2620.5000.7860.9290.9921.000

(4, 6)0.0100.0480.1900.4050.6900.8810.9761.000

(4, 7)0.0060.0330.1420.3330.6060.8330.9541.000

(4, 8)0.0040.0240.1090.2790.5330.7880.9291.000

(4, 9)0.0030.0180.0850.2360.4710.7450.9021.000

(4, 10)0.0020.0140.0680.2030.4190.7060.8741.000

(5, 5)0.0080.0400.1670.3570.6430.8330.9600.9921.000

(5, 6)0.0040.0240.1100.2620.5220.7380.9110.9760.998

(5, 7)0.0030.0150.0760.1970.4240.6520.8540.9550.992

(5, 8)0.0020.0100.0540.1520.3470.5760.7930.9290.984

(5, 9)0.0010.0070.0390.1190.2870.5100.7340.9020.972

(5, 10)0.0010.0050.0290.0950.2390.4550.6780.8740.958

(6, 6)0.0020.0130.0670.1750.3920.6080.8250.9330.987

(6, 7)0.0010.0080.0430.1210.2960.5000.7330.8790.966

(6, 8)0.0010.0050.0280.0860.2260.4130.6460.8210.937

(6, 9)0.0000.0030.0190.0630.1750.3430.5660.7620.902

(6, 10)0.0000.0020.0130.0470.1370.2880.4970.7060.864

(7, 7)0.0010.0040.0250.0780.2090.3830.6170.7910.922

(7, 8)0.0000.0020.0150.0510.1490.2960.5140.7040.867

(7, 9)0.0000.0010.0100.0350.1080.2310.4270.6220.806

(7, 10)0.0000.0010.0060.0240.0800.1820.3550.5490.743

(8, 8)0.0000.0010.0090.0320.1000.2140.4050.5950.786

(8, 9)0.0000.0010.0050.0200.0690.1570.3190.5000.702

(8, 10)0.0000.0000.0030.0130.0480.1170.2510.4190.621

(9, 9)0.0000.0000.0030.0120.0440.1090.2380.3990.601

(9, 10)0.0000.0000.0020.0080.0290.0770.1790.3190.510

(10, 10) 0.000 0.000 0.001 0.004 0.019 0.051 0.128 0.242 0.414Reproduced from C. Eisenhart and R. Swed, Tables for Testing Randomness of Group-ing in a Sequence of Alternatives, Ann. Math. Stat., 14, 1943, by permission of theeditor.



Table A.18 Table for the Runs Test 763

Table A.18 (continued) P (V v when H0 is true) in the Runs Test v

(n1 , n2 )11121314151617181920

(2, 3)

(2, 4)

(2, 5)

(2, 6)

(2, 7)

(2, 8)

(2, 9)

(2, 10)

(3, 3)

(3, 4)

(3, 5)

(3, 6)

(3, 7)

(3, 8)

(3, 9)

(3, 10)

(4, 4)

(4, 5)

(4, 6)

(4, 7)

(4, 8)

(4, 9)

(4, 10)

(5, 5)

(5, 6)1.000

(5, 7)1.000

(5, 8)1.000

(5, 9)1.000

(5, 10)1.000

(6, 6)0.9981.000

(6, 7)0.9920.9991.000

(6, 8)0.9840.9981.000

(6, 9)0.9720.9941.000

(6, 10)0.9580.9901.000

(7, 7)0.9750.9960.9991.000

(7, 8)0.9490.9880.9981.0001.000

(7, 9)0.9160.9750.9940.9991.000

(7, 10)0.8790.9570.9900.9981.000

(8, 8)0.9000.9680.9910.9991.0001.000

(8, 9)0.8430.9390.9800.9960.9991.0001.000

(8, 10)0.7820.9030.9640.9900.9981.0001.000

(9, 9)0.7620.8910.9560.9880.9971.0001.0001.000

(9, 10)0.6810.8340.9230.9740.9920.9991.0001.0001.000

(10, 10) 0.586 0.758 0.872 0.949 0.981 0.996 0.999 1.000 1.000 1.000

Table A.19 Sample Size for Two-Sided Nonparametric Tolerance Limits 1 1 0.50 0.70 0.90 0.95 0.99 0.995

0.99533648877794713251483

0.99168244388473662740

0.9534497793130146

0.90172438466472

0.85111625304247

0.8091218223134

0.7571015182427

0.706812142022

0.60469101416

0.5035781112

Reproduced from Table A25d of Wilfrid J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis, 3rd ed. McGraw-Hill, New York, 1969. Used with permission of McGraw-Hill Book Company.

Table A.20 Sample Size for One-Sided Nonparametric Tolerance Limits 1 1 0.50 0.70 0.95 0.99 0.995

0.9951392415989191379

0.9969120299459688

0.9514245990135

0.90712294466

0.8558192943

0.8046142131

0.753511725

0.702491320

0.602361014

0.50125710

Reproduced from Table A25e of Wilfrid J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis, 3rd ed. McGraw-Hill, New York, 1969. Used with permission of McGraw-Hill Book Company.



Table A.21 Table for Spearmans Rank Correlation Coecients 765

Table A.21 Critical Values for Spearmans Rank Correlation Coecients

n = 0.05 = 0.025 = 0.01 = 0.005

50.900

60.8290.8860.943

70.7140.7860.893

80.6430.7380.8330.881

90.6000.6830.7830.833

100.5640.6480.7450.794

110.5230.6230.7360.818

120.4970.5910.7030.780

130.4750.5660.6730.745

140.4570.5450.6460.716

150.4410.5250.6230.689

160.4250.5070.6010.666

170.4120.4900.5820.645

180.3990.4760.5640.625

190.3880.4620.5490.608

200.3770.4500.5340.591

210.3680.4380.5210.576

220.3590.4280.5080.562

230.3510.4180.4960.549

240.3430.4090.4850.537

250.3360.4000.4750.526

260.3290.3920.4650.515

270.3230.3850.4560.505

280.3170.3770.4480.496

290.3110.3700.4400.487

300.3050.3640.4320.478

Reproduced from E. G. Olds, Distribution of Sums of Squares of Rank Dierences for Small Samples, Ann. Math. Stat., 9, 1938, by permission of the editor.

Appendix A Statistical Tables and Proofs766Chart forAveragesChart for Standard DeviationsChart for RangesObs. inSampleFactors forControl LimitsFactors for Factors forCenterline Control LimitsFactors for Factors forCenterline Control LimitsnA2 A3c4 1/c4 B3 B4 B5B6d2 1/d2 d3 D3 D421.8802.6590.79791.253303.26702.6061.1280.88650.85303.26731.0231.9540.88621.128402.56802.2761.6930.59070.88802.57440.7291.6280.92131.085402.26602.0882.0590.48570.88002.28250.5771.4270.94001.063802.08901.9642.3260.42990.86402.11460.4831.2870.95151.05100.0301.9700.0291.8742.5340.39460.84802.00470.4191.1820.95941.04230.1181.8820.1131.8062.7040.36980.8330.0761.92480.3731.0990.96501.03630.1851.8150.1791.7512.8470.35120.8200.1361.86490.3371.0320.96931.03170.2391.7610.2321.7072.9700.33670.8080.1841.816100.3080.9750.97271.02810.2841.7160.2761.6693.0780.32490.7970.2231.777110.2850.9270.97541.02520.3211.6790.3131.6373.1730.31520.7870.2561.744120.2660.8860.97761.02290.3541.6460.3461.6103.2580.30690.7780.2831.717130.2490.8500.97941.02100.3821.6180.3741.5853.3360.29980.7700.3071.693140.2350.8170.98101.01940.4061.5940.3991.5633.4070.29350.7630.3281.672150.2230.7890.98231.01800.4281.5720.4211.5443.4720.28800.7560.3471.653160.2120.7630.98351.01680.4481.5520.4401.5263.5320.28310.7500.3631.637170.2030.7390.98451.01570.4661.5340.4581.5113.5880.27870.7440.3781.622180.1940.7180.98541.01480.4821.5180.4751.4963.6400.27470.7390.3911.608190.1870.6980.98621.01400.4971.5030.4901.4833.6890.27110.7340.4031.597200.1800.6800.98691.01330.5101.4900.5041.4703.7350.26770.7290.4151.585210.1730.6630.98761.01260.5231.4770.5161.4593.7780.26470.7240.4251.575220.1670.6470.98821.01190.5341.4660.5281.4483.8190.26180.7200.4341.566230.1620.6330.98871.01140.5451.4550.5391.4383.8580.25920.7160.4431.557240.1570.6190.98921.01090.5551.4450.5491.4293.8950.25670.7120.4511.548250.1530.6060.98961.01050.5651.4350.5591.4203.9310.25440.7080.4594.541Table A.22 Factors for Constructing Control Charts

Section A.24 Proof of Mean of the Hypergeometric Distribution 767

0 () Table A.23 The Incomplete Gamma Function: F (x; )= x 1 y1 ey dy

x12345678910

10.63200.26400.08000.01900.00400.00100.00000.00000.00000.0000

20.86500.59400.32300.14300.05300.01700.00500.00100.00000.0000

30.95000.80100.57700.35300.18500.08400.03400.01200.00400.0010

40.98200.90800.76200.56700.37100.21500.11100.05100.02100.0080

50.99300.96000.87500.73500.56000.38400.23800.13300.06800.0320

60.99800.98300.93800.84900.71500.55400.39400.25

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