Subcategory Pollutant

No. of 

Sources 

Industryw

ide

No. of 

Sources 

with Data

Determination 

of No. of 

Sources in 

MACT Floor

No. of 

Sources 

in MACT 

Floor (N)

No. Data 

Points in 

MACT 

Floor (n)

Average 

of Top 

Sources

Large Kilns PM 122 71 12% Sources with 

Data 9 34 0.012

Equivalent PM 122 71 12% Sources with 

Data 9 34 0.0045

Small Kilns PM 103 21 12% Sources with 

Data 3 9 0.026

Equivalent PM 103 21 12% Sources with 

Data 3 9 0.0029
UPL Unit of 

Measure

Data 

Distribution

0.028 lb/ton lognormal

0.016 gr/dscf @ 7% O2 skewed/ unknown

0.063 lb/ton normal

0.0050 gr/dscf @ 7% O2 normal
Raw Data

Runs 514 686 687 538 508 695 503 696 600 1 2.99E-03 8.07E-03 8.07E-03 1.10E-02 0.00985023 0.009312 0.014691 0.0045537 0.029395 2 9.53E-03 1.02E-02 1.02E-02 1.26E-02 0.013536866 0.013578 0.013642 0.0209472 0.020209 3 5.61E-03 6.37E-03 6.37E-03 6.74E-03 0.00921659 0.011992 0.013642 0.0273224 0.04593 4 0.008756567 0.014377947 0.004116 5 0.004107985 0.00413 6 0.004300271 0.008900634 0.004184 789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 35.00 35.00

Kurtosis 6.83 0.08

SE Kurtosis 0.80 0.80 Average of Raw Data 1.17E-02

Result Kurtosis non normal normal

Skewness 2.31 0.32

SE Skewness 0.40 0.40 Result Raw Data Result Log Data

Result Skewness non normal normal non normal normal

1. Take LN( ) of Raw Data

Runs 514 686 687 538 508 695 503 696 600 1 -5.81E+00 -4.82E+00 -4.82E+00 -4.51E+00 -4.62E+00 -4.68E+00 -4.22E+00 -5.39E+00 -3.53E+00 2 -4.65E+00 -4.59E+00 -4.59E+00 -4.37E+00 -4.30E+00 -4.30E+00 -4.29E+00 -3.87E+00 -3.90E+00 3 -5.18E+00 -5.06E+00 -5.06E+00 -5.00E+00 -4.69E+00 -4.42E+00 -4.29E+00 -3.60E+00 -3.08E+00 4 -4.74E+00 -4.24E+00 -5.49E+00 5 -5.49E+00 -5.49E+00 6 -5.45E+00 -4.72E+00 -5.48E+00 789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

35

3. Calculate -4.65E+00

0.387280882

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 PM-Large-lbton 12/1/2014
14.60789904

0.671102327

23.7670219

7. Calculate

1.378977872

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat "

a- Calculate 0.011605907

b- Calculate 0.000134697

c- Calculate 0.472970163

d-calculate 0.013270852

e- Calculate 0.014395

f- Calculate UPL 2.83E-02

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat "

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 PM-Large-lbton 12/1/2014
∆z z

0.101 23.76702 1.378978

-5 =f(a) 389.3286

1 -4.899 353.7647

2 -4.798 320.5683

3 -4.697 289.6371

4 -4.596 260.8709

5 -4.495 234.1718

6 -4.394 209.4438

7 -4.293 186.5933

8 -4.192 165.5288

9 -4.091 146.161

10 -3.99 128.4027

11 -3.889 112.1687

12 -3.788 97.37633

13 -3.687 83.94478

14 -3.586 71.79553

15 -3.485 60.85219

16 -3.384 51.04053

17 -3.283 42.28849

18 -3.182 34.52616

19 -3.081 27.68581

20 -2.98 21.70185

21 -2.879 16.51085

22 -2.778 12.05157

23 -2.677 8.2649

24 -2.576 5.093903

25 -2.475 2.483804

26 -2.374 0.381986

27 -2.273 -1.262

28 -2.172 -2.49646

29 -2.071 -3.36752

30 -1.97 -3.91914

31 -1.869 -4.19315

32 -1.768 -4.22918

33 -1.667 -4.06474

34 -1.566 -3.73514

35 -1.465 -3.27355

36 -1.364 -2.71098

37 -1.263 -2.07627

38 -1.162 -1.39611

39 -1.061 -0.69501

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 0.00466

41 -0.859 0.6827

42 -0.758 1.321073

43 -0.657 1.9039

44 -0.556 2.417464

45 -0.455 2.850209

46 -0.354 3.19274

47 -0.253 3.437823

48 -0.152 3.580385

49 -0.051 3.617513

50 0.05 3.548458

51 0.151 3.374629

52 0.252 3.099597

53 0.353 2.729094

54 0.454 2.271014

55 0.555 1.73541

56 0.656 1.134497

57 0.757 0.482653

58 0.858 -0.20359

59 0.959 -0.90552

60 1.06 -1.60229

61 1.161 -2.27088

62 1.262 -2.88611

63 1.363 -3.42063

64 1.464 -3.84494

65 1.565 -4.12737

66 1.666 -4.23411

67 1.767 -4.12915

68 1.868 -3.77437

69 1.969 -3.12944

70 2.07 -2.1519

71 2.171 -0.79711

72 2.272 0.981703

73 2.373 3.233507

74 2.474 6.009411

75 2.575 9.36269

76 2.676 13.34878

77 2.777 18.02528

78 2.878 23.45194

79 2.979 29.69069

80 3.08 36.80559

81 3.181 44.86291

82 3.282 53.93103

83 3.383 64.08052

84 3.484 75.3841

85 3.585 87.91666

86 3.686 101.7552

87 3.787 116.9791

88 3.888 133.6695
89 3.989 151.91

90 4.09 171.7864

91 4.191 193.3864

92 4.292 216.8001

93 4.393 242.1197

94 4.494 269.4394

95 4.595 298.8559

96 4.696 330.4676

97 4.797 364.3756

98 4.898 400.6826

99 4.999 439.494
distribution of Z

Normal 

distribution

1.48672E-06 0.000578822

2.45094E-06 0.000867056

3.9995E-06 0.001282114

6.46025E-06 0.001871127

1.03291E-05 0.002694559

1.63472E-05 0.003828064

2.56093E-05 0.0053637

3.97118E-05 0.00740995

6.09553E-05 0.010089853

9.26132E-05 0.013536438

0.000139285 0.017884562

0.000207351 0.023258261

0.000305546 0.029752938

0.000445674 0.037411997

0.000643469 0.046198204

0.000919619 0.05596083

0.001300942 0.066400773

0.001821704 0.077037098

0.002525035 0.08717975

0.003464389 0.095914406

0.004704958 0.102106263

0.006324913 0.104429707

0.008416337 0.101430086

0.011085658 0.091621859

0.014453386 0.07362415

0.018652949 0.046330261

0.023828414 0.009102118

0.030130931 -0.03802537

0.03771375 -0.094150909

0.046725789 -0.157349859

0.057303789 -0.224581671

0.069563239 -0.291688933

0.083588399 -0.353510689

0.099421884 -0.404123896

0.117054396 -0.437214302

0.136415346 -0.446562432

0.157365126 -0.426613575

0.179689839 -0.373084637

0.203099244 -0.283548052

0.227228529 -0.157925827

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.001172659

0.275855243 0.188326452

0.299326439 0.395432055

0.321498297 0.612100542

0.341807853 0.826308103

0.359712192 1.02525485

0.37471238 1.196359116

0.386376486 1.328293843

0.394360216 1.411961209

0.398423793 1.441303364

0.398443914 1.41386151

0.394419966 1.331020992

0.386474058 1.197913741

0.374844865 1.022986849

0.359875719 0.817282616

0.34199778 0.593506233

0.32170943 0.364978526

0.299553264 0.144580351

0.276092166 -0.056208446

0.25188591 -0.22808808

0.227469632 -0.364473286

0.203335281 -0.461750852

0.17991684 -0.519259707

0.157579839 -0.539022049

0.136615273 -0.525277094

0.117237788 -0.483883834

0.099587708 -0.421664815

0.083736272 -0.345759856

0.069693339 -0.263048204

0.05741676 -0.179682183

0.046822635 -0.100757503

0.037795734 -0.030127519

0.030199481 0.029646918

0.023885038 0.077232431

0.018699163 0.112370951

0.014490659 0.135671549

0.011115369 0.148376612

0.008439746 0.152128763

0.006343145 0.148759056

0.004718997 0.140110249

0.003475077 0.12790229

0.002533081 0.11364137

0.001827693 0.098569387

0.001305351 0.083647592

0.000922829 0.06956664

0.00064578 0.056774858

0.00044732 0.045517147

0.000306705 0.035878104

0.000208159 0.027824443
0.000139842 0.021243364

9.29928E-05 0.015974895

6.12113E-05 0.01183743

3.98826E-05 0.008646551

2.5722E-05 0.006227808

1.64209E-05 0.004424434

1.03767E-05 0.003101127

6.49066E-06 0.002144953

4.01874E-06 0.001464329

2.46298E-06 0.000986871

1.49417E-06 0.000656679
Absolute values

0.000578822 2.92305E-05

0.000867056 8.75727E-05

0.001282114 0.000129493

0.001871127 0.000188984

0.002694559 0.00027215

0.003828064 0.000386634

0.0053637 0.000541734

0.00740995 0.000748405

0.010089853 0.001019075

0.013536438 0.00136718

0.017884562 0.001806341

0.023258261 0.002349084

0.029752938 0.003005047

0.037411997 0.003778612

0.046198204 0.004666019

0.05596083 0.005652044

0.066400773 0.006706478

0.077037098 0.007780747

0.08717975 0.008805155

0.095914406 0.009687355

0.102106263 0.010312733

0.104429707 0.0105474

0.101430086 0.010244439

0.091621859 0.009253808

0.07362415 0.007436039

0.046330261 0.004679356

0.009102118 0.000919314

0.03802537 0.003840562

0.094150909 0.009509242

0.157349859 0.015892336

0.224581671 0.022682749

0.291688933 0.029460582

0.353510689 0.03570458

0.404123896 0.040816514

0.437214302 0.044158645

0.446562432 0.045102806

0.426613575 0.043087971

0.373084637 0.037681548

0.283548052 0.028638353

0.157925827 0.015950509

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.001172659 0.000118439

0.188326452 0.019020972

0.395432055 0.039938638

0.612100542 0.061822155

0.826308103 0.083457118

1.02525485 0.10355074

1.196359116 0.120832271

1.328293843 0.134157678

1.411961209 0.142608082

1.441303364 0.14557164

1.41386151 0.142800013

1.331020992 0.13443312

1.197913741 0.120989288

1.022986849 0.103321672

0.817282616 0.082545544

0.593506233 0.05994413

0.364978526 0.036862831

0.144580351 0.014602615

0.056208446 0.005677053

0.22808808 0.023036896

0.364473286 0.036811802

0.461750852 0.046636836

0.519259707 0.05244523

0.539022049 0.054441227

0.525277094 0.053052987

0.483883834 0.048872267

0.421664815 0.042588146

0.345759856 0.034921745

0.263048204 0.026567869

0.179682183 0.018147901

0.100757503 0.010176508

0.030127519 0.003042879

0.029646918 0.002994339

0.077232431 0.007800476

0.112370951 0.011349466

0.135671549 0.013702826

0.148376612 0.014986038

0.152128763 0.015365005

0.148759056 0.015024665

0.140110249 0.014151135

0.12790229 0.012918131

0.11364137 0.011477778

0.098569387 0.009955508

0.083647592 0.008448407

0.06956664 0.007026231

0.056774858 0.005734261

0.045517147 0.004597232

0.035878104 0.003623688

0.027824443 0.002810269
0.021243364 0.00214558

0.015974895 0.001613464

0.01183743 0.00119558

0.008646551 0.000873302

0.006227808 0.000629009

0.004424434 0.000446868

0.003101127 0.000313214

0.002144953 0.00021664

0.001464329 0.000147897

0.000986871 9.9674E-05

0.000656679 3.31623E-05
Cumulative normalization

2.92305E-05 1.09988E-05

0.000116803 4.39506E-05

0.000246297 9.26763E-05

0.000435281 0.000163787

0.000707431 0.000266191

0.001094065 0.000411674

0.001635799 0.000615517

0.002384204 0.000897126

0.003403279 0.001280582

0.00477046 0.001795023

0.0065768 0.002474711

0.008925885 0.003358622

0.011930931 0.004489358

0.015709543 0.00591117

0.020375562 0.007666895

0.026027605 0.00979364

0.032734084 0.012317146

0.04051483 0.015244877

0.049319985 0.018558071

0.05900734 0.022203219

0.069320073 0.026083683

0.079867473 0.030052447

0.090111912 0.033907214

0.09936572 0.037389227

0.106801759 0.040187252

0.111481115 0.041947995

0.112400429 0.042293913

0.116240991 0.043739036

0.125750233 0.047317163

0.141642569 0.053297114

0.164325318 0.061832155

0.1937859 0.072917551

0.229490479 0.086352432

0.270306993 0.101710826

0.314465637 0.118326794

0.359568443 0.135298029

0.402656414 0.151511125

0.440337963 0.165689898

0.468976316 0.176465907

0.484926824 0.182467747

( f x ) ( f b)



 




0.485045263 0.182512313

0.504066234 0.189669504

0.544004872 0.204697572

0.605827027 0.227959946

0.689284145 0.259363101

0.792834885 0.298327063

0.913667156 0.3437937

1.047824834 0.394274408

1.190432916 0.447934825

1.336004556 0.502710366

1.478804568 0.556443002

1.613237688 0.607027353

1.734226976 0.652553072

1.837548648 0.691430841

1.920094192 0.72249099

1.980038322 0.745046703

2.016901153 0.758917409

2.031503768 0.764412066

2.037180821 0.766548221

2.060217718 0.77521652

2.097029519 0.789068025

2.143666355 0.806616484

2.196111586 0.826350519

2.250552813 0.846835606

2.303605799 0.866798327

2.352478067 0.885187932

2.395066213 0.901212954

2.429987958 0.914353271

2.456555827 0.9243502

2.474703727 0.931178873

2.484880235 0.935008078

2.487923115 0.93615305

2.490917453 0.937279756

2.498717929 0.940214911

2.510067395 0.944485476

2.523770221 0.94964156

2.538756259 0.955280491

2.554121264 0.961062018

2.569145929 0.966715483

2.583297064 0.972040257

2.596215195 0.976901078

2.607692974 0.981219924

2.617648482 0.984965972

2.626096889 0.988144929

2.633123119 0.990788752

2.63885738 0.992946433

2.643454612 0.994676275

2.6470783 0.996039793

2.649888569 0.997097237
2.652034149 0.997904574

2.653647613 0.998511687

2.654843194 0.998961558

2.655716495 0.999290163

2.656345504 0.999526846

2.656792372 0.999694993

2.657105585 0.999812849

2.657322226 0.999894366

2.657470123 0.999950016

2.657569797 0.999987522

2.657602959 1

132.704374
Raw Data

Runs 514 686 687 538 508 695 503 696 600

1 0.001024602 0.002138462 0.002138462 0.0278 0.002349 0.002027 0.003141 0.0003658 0.008566

2 0.003182773 0.002527273 0.002527273 0.0278 0.003132 0.002823 0.002514 0.0018289 0.005467

3 0.001782597 0.001782051 0.001782051 0.0139 0.00182 0.001793 0.002547 0.0023677 0.014752

4 0.002494872 0.003282 0.001608

5 0.000927 0.001477

6 0.00119472 0.001782 0.001456

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

n for each source 3 3 3 5 6 3 3 3 6 0 0 0 0 0 0 0

Sample size 35

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 35.00 35.00

Kurtosis 7.93 1.87 http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

SE Kurtosis 0.80 0.80 note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second compone

Result Kurtosis non normal non normal

Skewness 2.89 1.08

SE Skewness 0.40 0.40 Result Raw Data Result Log Data

Result Skewness non normal non normal non normal non normal

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average 4.52E-03

Standard Deviation 6.60E-03 http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt

514 686 687 538 508 695 503 696 600

1 7.82E-02 1.68E-02 1.68E-02 1.54E+02 1.16E-02 2.02E-02 1.88E-03 1.56E-01 1.41E-01

2 1.67E-03 8.24E-03 8.24E-03 1.54E+02 1.93E-03 4.32E-03 8.46E-03 2.74E-02 4.27E-04

3 2.94E-02 2.94E-02 2.94E-02 4.07E+00 2.78E-02 2.89E-02 7.91E-03 1.12E-02 5.76E+00
4 8.79E-03 1.22E-03 3.76E-02

5 8.73E-02 4.49E-02

6 6.40E-02 2.94E-02 4.61E-02

7

8

9

10

11

12

13

14

15

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17

18

19

20

21

22

23

24

25

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27

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29

30

31

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33

34

35

Testing Normality Raw Data

Sample Size 35.00

Kurtosis 6.40

SE Kurtosis 0.83

Result Kurtosis non normal

Skewness 2.89

SE Skewness 0.40 Result Raw Data

Result Skewness non normal non normal

### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively

kurtosis= 7.93 depending on sample size

skewness= 2.89 same for n>3 and n=3

ni = number test runs = 3 3 3 5 6 3 3 3 6 0 0 0 0 0 0 0

Total Number of sources = 9

n =Total # test runs = 35

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




34

This step is for QC only:

means 0.00200 0.00215 0.00215 0.01464 0.00222 0.00221 0.00273 0.00152 0.00555 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = 4.52E-03

Pooled Variance = 4.3626E-05

m= number future runs = 3

Term1 0.3619

Term2 1.5788E-05

Squared root Term2 0.003973453

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.44115

UPL based on t-statistic with df at 99% confidence level 0.014217

35

0.850868

81.96%

Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples 

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used 0.0636

2.822749628

Adjusted UPL based on recalculated t-statistic 1.57E-02

new t-statistic from tab Recalculate t-statistic

Confidence Level is not 99%, Go to Recalculate t-stat n=3 tab to fix the Confidence Level
Logtransformed

514 686 687 538 508

-6.88E+00 -6.15E+00 -6.15E+00 -3.58E+00 -6.05E+00

-5.75E+00 -5.98E+00 -5.98E+00 -3.58E+00 -5.77E+00

-6.33E+00 -6.33E+00 -6.33E+00 -4.28E+00 -6.31E+00

-5.99E+00 -5.72E+00

-6.98E+00

-6.73E+00 -6.33E+00

0 0 0 0 0 0 0 0 0 0 0 0 0 0

e term n-3 in denominator of second component is zero).
0 0 0 0 0 0 0 0 0 0 0 0 0 0
#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

695 503 696 600

-6.20E+00 -5.76E+00 -7.91E+00 -4.76E+00

-5.87E+00 -5.99E+00 -6.30E+00 -5.21E+00

-6.32E+00 -5.97E+00 -6.05E+00 -4.22E+00

-6.43E+00

-6.52E+00

-6.53E+00




Raw Data

Runs 599 549 665

1 0.008109483 0.018474576 0.044545455

2 0.015262515 0.023728814 0.045454545

3 0.00452788 0.016610169 0.060113636

4

5

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Testing Normality Raw Data

Sample Size 9.00

Kurtosis -0.76

SE Kurtosis 1.56

Result Kurtosis normal

Skewness 0.75

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 3

n =Total # test runs = 9

8

This step is for QC only:

means 9.30E-03 1.96E-02 5.00E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

2.63E-02

Mean =

Pooled Variance = 3.66E-04

m= number future runs = 3

Term1 4.44E-01

Term2 1.63E-04

Squared root Term2 1.28E-02

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448

UPL POOLED VARIANCE = 6.33E-02

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 PM-Small-lbton 12/1/2014
Raw Data

Runs 599 549 665

1 0.002025429 0.003003086 0.002394258

2 0.003772857 0.004854779 0.002527273

3 0.001123347 0.003321691 0.003258852

4

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7

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35

Testing Normality Raw Data

Sample Size 9.00

Kurtosis 0.63

SE Kurtosis 1.56

Result Kurtosis normal

Skewness 0.16

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 3

n =Total # test runs = 9

8

This step is for QC only:

means 2.31E-03 3.73E-03 2.73E-03 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

2.92E-03

Mean =

Pooled Variance = 1.15E-06

m= number future runs = 3

Term1 4.44E-01

Term2 5.12E-07

Squared root Term2 7.15E-04

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448

UPL POOLED VARIANCE = 4.99E-03

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 PM-Small-conc 12/1/2014
Raw Data

Runs

1

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Testing Normality Raw Data

Sample Size 0.00

Kurtosis #DIV/0!

SE Kurtosis 0.00

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

Result Skewness #DIV/0! #DIV/0!

ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 0

n =Total # test runs = 0

-1

This step is for QC only:

means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0!

Mean =

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM!

UPL POOLED VARIANCE = #DIV/0!

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 UPL Pooled Template 12/1/2014
Raw Data

Runs 123456789

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Testing Normality Raw Data Logtransformed Data

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis 0.00 0.00 Average of Raw Data #DIV/0!

Result Kurtosis #DIV/0! #DIV/0!

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 123456789

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0

3. Calculate #DIV/0!

#DIV/0!

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 Lognormal Template 12/1/2014
#DIV/0!

#DIV/0!

#DIV/0!

7. Calculate

#DIV/0!

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat"

a- Calculate #DIV/0!

b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0!

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

b- denominator first term

a- Numerator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Lognormal Template 12/1/2014
∆z z

0.101 24.12133 1.363077

-5 =f(a) 396.6768

1 -4.899 360.4593

2 -4.798 326.6518

3 -4.697 295.1499

4 -4.596 265.852

5 -4.495 238.6581

6 -4.394 213.4709

7 -4.293 190.195

8 -4.192 168.7373

9 -4.091 149.0069

10 -3.99 130.9151

11 -3.889 114.3752

12 -3.788 99.30312

13 -3.687 85.61659

14 -3.586 73.23572

15 -3.485 62.08277

16 -3.384 52.08221

17 -3.283 43.16072

18 -3.182 35.24714

19 -3.081 28.27255

20 -2.98 22.1702

21 -2.879 16.87554

22 -2.778 12.32624

23 -2.677 8.46214

24 -2.576 5.225296

25 -2.475 2.559955

26 -2.374 0.412563

27 -2.273 -1.26824

28 -2.172 -2.5316

29 -2.071 -3.42449

30 -1.97 -3.99166

31 -1.869 -4.27569

32 -1.768 -4.31694

33 -1.667 -4.15357

34 -1.566 -3.82156

35 -1.465 -3.35468

36 -1.364 -2.78451

37 -1.263 -2.14041

38 -1.162 -1.44958

39 -1.061 -0.737

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.02543

41 -0.859 0.664518

42 -0.758 1.314469

43 -0.657 1.908233

44 -0.556 2.431816

45 -0.455 2.873425

46 -0.354 3.223463

47 -0.253 3.474533

48 -0.152 3.621435

49 -0.051 3.661166

50 0.05 3.592923

51 0.151 3.418097

52 0.252 3.140281

53 0.353 2.765264

54 0.454 2.301034

55 0.555 1.757774

56 0.656 1.147869

57 0.757 0.4859

58 0.858 -0.21136

59 0.959 -0.92492

60 1.06 -1.63362

61 1.161 -2.31408

62 1.262 -2.94074

63 1.363 -3.48582

64 1.464 -3.91936

65 1.565 -4.20919

66 1.666 -4.32095

67 1.767 -4.21808

68 1.868 -3.86183

69 1.969 -3.21124

70 2.07 -2.22315

71 2.171 -0.85222

72 2.272 0.949112

73 2.373 3.230583

74 2.474 6.044142

75 2.575 9.443932

76 2.676 13.48629

77 2.777 18.22976

78 2.878 23.73508

79 2.979 30.06518

80 3.08 37.2852

81 3.181 45.46245

82 3.282 54.66648

83 3.383 64.96901

84 3.484 76.44396

85 3.585 89.16746

86 3.686 103.2178

87 3.787 118.6756

88 3.888 135.6234
89 3.989 154.1462

90 4.09 174.3312

91 4.191 196.2676

92 4.292 220.047

93 4.393 245.7629

94 4.494 273.5115

95 4.595 303.3907

96 4.696 335.5008

97 4.797 369.9444

98 4.898 406.8261

99 4.999 446.2528
distribution of Z

Normal 

distribution

1.48672E-06 0.000589747

2.45094E-06 0.000883464

3.9995E-06 0.001306445

6.46025E-06 0.001906741

1.03291E-05 0.002746009

1.63472E-05 0.003901404

2.56093E-05 0.005466833

3.97118E-05 0.007552981

6.09553E-05 0.010285427

9.26132E-05 0.013800006

0.000139285 0.018234501

0.000207351 0.023715784

0.000305546 0.030341661

0.000445674 0.038157079

0.000643469 0.047124921

0.000919619 0.057092498

0.001300942 0.067755949

0.001821704 0.078626046

0.002525035 0.089000254

0.003464389 0.097947106

0.004704958 0.104309835

0.006324913 0.10673633

0.008416337 0.103741783

0.011085658 0.093808392

0.014453386 0.075523218

0.018652949 0.047750703

0.023828414 0.009830716

0.030130931 -0.03821313

0.03771375 -0.095476112

0.046725789 -0.160011912

0.057303789 -0.228737477

0.069563239 -0.297430965

0.083588399 -0.360845903

0.099421884 -0.412955775

0.117054396 -0.447330438

0.136415346 -0.457629943

0.157365126 -0.43818421

0.179689839 -0.384610712

0.203099244 -0.29440934

0.227228529 -0.167466401

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.006400143

0.275855243 0.183310648

0.299326439 0.393455457

0.321498297 0.613493584

0.341807853 0.831213719

0.359712192 1.033605832

0.37471238 1.207871534

0.386376486 1.342478021

0.394360216 1.42815002

0.398423793 1.45869584

0.398443914 1.431578111

0.394419966 1.348165671

0.386474058 1.213637172

0.374844865 1.036545118

0.359875719 0.828086167

0.34199778 0.601154969

0.32170943 0.369280442

0.299553264 0.145552829

0.276092166 -0.058353815

0.25188591 -0.23297474

0.227469632 -0.371599393

0.203335281 -0.470535074

0.17991684 -0.529088588

0.157579839 -0.549294713

0.136615273 -0.535443759

0.117237788 -0.493475585

0.099587708 -0.43031327

0.083736272 -0.353206452

0.069693339 -0.26914386

0.05741676 -0.184378871

0.046822635 -0.104093763

0.037795734 -0.032210166

0.030199481 0.028662687

0.023885038 0.077162607

0.018699163 0.113020402

0.014490659 0.1368488

0.011115369 0.149905125

0.008439746 0.153854586

0.006343145 0.150555076

0.004718997 0.141877499

0.003475077 0.129568938

0.002533081 0.115160063

0.001827693 0.09991357

0.001305351 0.084807386

0.000922829 0.070544709

0.00064578 0.057582596

0.00044732 0.046171384

0.000306705 0.036398428

0.000208159 0.028231168
0.000139842 0.021556082

9.29928E-05 0.016211547

6.12113E-05 0.012013793

3.98826E-05 0.008776044

2.5722E-05 0.00632152

1.64209E-05 0.004491301

1.03767E-05 0.003148183

6.49066E-06 0.002177621

4.01874E-06 0.001486709

2.46298E-06 0.001002002

1.49417E-06 0.000666778
Absolute values

0.000589747 2.97822E-05

0.000883464 8.92299E-05

0.001306445 0.000131951

0.001906741 0.000192581

0.002746009 0.000277347

0.003901404 0.000394042

0.005466833 0.00055215

0.007552981 0.000762851

0.010285427 0.001038828

0.013800006 0.001393801

0.018234501 0.001841685

0.023715784 0.002395294

0.030341661 0.003064508

0.038157079 0.003853865

0.047124921 0.004759617

0.057092498 0.005766342

0.067755949 0.006843351

0.078626046 0.007941231

0.089000254 0.008989026

0.097947106 0.009892658

0.104309835 0.010535293

0.10673633 0.010780369

0.103741783 0.01047792

0.093808392 0.009474648

0.075523218 0.007627845

0.047750703 0.004822821

0.009830716 0.000992902

0.03821313 0.003859526

0.095476112 0.009643087

0.160011912 0.016161203

0.228737477 0.023102485

0.297430965 0.030040527

0.360845903 0.036445436

0.412955775 0.041708533

0.447330438 0.045180374

0.457629943 0.046220624

0.43818421 0.044256605

0.384610712 0.038845682

0.29440934 0.029735343

0.167466401 0.016914107

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.006400143 0.000646414

0.183310648 0.018514375

0.393455457 0.039739001

0.613493584 0.061962852

0.831213719 0.083952586

1.033605832 0.104394189

1.207871534 0.121995025

1.342478021 0.13559028

1.42815002 0.144243152

1.45869584 0.14732828

1.431578111 0.144589389

1.348165671 0.136164733

1.213637172 0.122577354

1.036545118 0.104691057

0.828086167 0.083636703

0.601154969 0.060716652

0.369280442 0.037297325

0.145552829 0.014700836

0.058353815 0.005893735

0.23297474 0.023530449

0.371599393 0.037531539

0.470535074 0.047524042

0.529088588 0.053437947

0.549294713 0.055478766

0.535443759 0.05407982

0.493475585 0.049841034

0.43031327 0.04346164

0.353206452 0.035673852

0.26914386 0.02718353

0.184378871 0.018622266

0.104093763 0.01051347

0.032210166 0.003253227

0.028662687 0.002894931

0.077162607 0.007793423

0.113020402 0.011415061

0.1368488 0.013821729

0.149905125 0.015140418

0.153854586 0.015539313

0.150555076 0.015206063

0.141877499 0.014329627

0.129568938 0.013086463

0.115160063 0.011631166

0.09991357 0.010091271

0.084807386 0.008565546

0.070544709 0.007125016

0.057582596 0.005815842

0.046171384 0.00466331

0.036398428 0.003676241

0.028231168 0.002851348
0.021556082 0.002177164

0.016211547 0.001637366

0.012013793 0.001213393

0.008776044 0.00088638

0.00632152 0.000638474

0.004491301 0.000453621

0.003148183 0.000317966

0.002177621 0.00021994

0.001486709 0.000150158

0.001002002 0.000101202

0.000666778 3.36723E-05
Cumulative normalization

2.97822E-05 1.10416E-05

0.000119012 4.41231E-05

0.000250963 9.30431E-05

0.000443544 0.000164441

0.000720891 0.000267266

0.001114933 0.000413355

0.001667083 0.000618061

0.002429934 0.000900884

0.003468762 0.001286024

0.004862562 0.001802767

0.006704247 0.002485561

0.009099541 0.003373603

0.012164049 0.004509751

0.016017914 0.00593855

0.020777531 0.00770315

0.026543873 0.009840989

0.033387224 0.012378122

0.041328455 0.015322287

0.05031748 0.018654917

0.060210138 0.022322563

0.070745431 0.026228463

0.081525801 0.030225223

0.092003721 0.034109851

0.101478368 0.037622523

0.109106214 0.040450502

0.113929034 0.042238536

0.114921937 0.042606648

0.118781463 0.044037546

0.12842455 0.047612665

0.144585753 0.053604338

0.167688239 0.062169452

0.197728766 0.073306805

0.234174202 0.086818741

0.275882735 0.102281941

0.32106311 0.119032305

0.367283734 0.136168336

0.411540339 0.152576218

0.450386021 0.166978032

0.480121364 0.17800224

0.497035471 0.184273048

( f x ) ( f b)



 




0.497681885 0.184512702

0.516196261 0.191376801

0.555935262 0.206109807

0.617898114 0.229082179

0.701850699 0.260207118

0.806244889 0.298910664

0.928239914 0.34413962

1.063830194 0.394408938

1.208073346 0.447886258

1.355401625 0.502507372

1.499991015 0.556113058

1.636155747 0.606595351

1.758733102 0.652040202

1.863424159 0.69085381

1.947060862 0.721861638

2.007777513 0.744371988

2.045074838 0.758199757

2.059775674 0.763650007

2.065669409 0.765835076

2.089199858 0.774558855

2.126731397 0.788473457

2.174255439 0.806092723

2.227693386 0.825904535

2.283172152 0.846472969

2.337251972 0.866522752

2.387093006 0.885001029

2.430554646 0.901114183

2.466228498 0.914340059

2.493412028 0.924418196

2.512034294 0.931322295

2.522547764 0.935220103

2.525800991 0.936426219

2.528695922 0.937499498

2.536489345 0.940388865

2.547904406 0.944620933

2.561726135 0.949745259

2.576866552 0.955358482

2.592405866 0.961119594

2.607611928 0.966757154

2.621941556 0.972069781

2.635028018 0.976921512

2.646659185 0.9812337

2.656750455 0.984974981

2.665316001 0.988150608

2.672441017 0.990792166

2.678256859 0.992948357

2.682920169 0.994677252

2.68659641 0.996040197

2.689447758 0.997097318
2.691624922 0.997904489

2.693262289 0.998511533

2.694475682 0.998961392

2.695362062 0.999290012

2.696000536 0.999526723

2.696454157 0.9996949

2.696772123 0.999812785

2.696992063 0.999894326

2.697142221 0.999949996

2.697243423 0.999987516

2.697277095 1

134.7207183
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

n for each source 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Sample size 0

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0! http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

SE Kurtosis 0.00 0.00 note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second compone

Result Kurtosis #DIV/0! #DIV/0!

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0!

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average #DIV/0!

Standard Deviation #DIV/0! http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt

1

2

3
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis -3.00

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

Result Skewness #DIV/0! #DIV/0!

### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively

kurtosis= #DIV/0! depending on sample size

skewness= #DIV/0! same for n>3 and n=3

ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 0

n =Total # test runs = 0

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




-1

This step is for QC only:

means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM!

UPL based on t-statistic with df at 99% confidence level #DIV/0!

0

#NUM!

Current Confidence Level is #NUM!

Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples 

a) n = number of test-runs =

b) Calculate u0

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used

Adjusted UPL based on recalculated t-statistic #DIV/0!

#NUM!

new t-statistic from tab Recalculate t-statistic
Logtransformed

0 0 0 0 0 0 0 0 0 0 0 0 0 0

e term n-3 in denominator of second component is zero).
0 0 0 0 0 0 0 0 0 0 0 0 0 0
#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!






Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

c.1.7 #NUM! d.1.7

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )



( )

w a e w 



a w  
Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0!

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



1 1 1 1  1 2   1 3  4 1   1

( ) , , ,

n n n n n n n n

P t I I I

             

        
4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

#NUM! e.1.7

d.1.) Calculate e.1.) Calculate 

Calculations Calculations



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result3:

3 4 1 1

,

n

I

 

   

 
#NUM!

0

3 4 1 1

,

24 2 2

u

n

I

 

   

  

 

1 1

2 2

5 1

2 2



 








a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

#NUM! f.1.7 #NUM!

f.1.) Calculate 

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result4: #NUM!

a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.7 #NUM! h.1.7

h.1.) Calculate 

Calculations Calculations

g.1.) Calculate 0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: #NUM! Result6:

#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3 1

2 6

w a

    

 
#NUM!
#NUM!

0.0636

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01

5) Once you get a YES, use the t-statistic in the calculations of the UPL.

Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11.

Copy the new t-value in cell E164 of Template_nonnormal n<3

1) Start with initial t-statistic from Template

2) Determine interval of differences for t-values

2) Candidate values for t-statistic

3) Corrected p-values

Acutal significance level with t-value

4) Is p-value closed enough to 0.01?
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 