**Subcategory Pollutant**  **No. of Sources**  **Industryw ide**  **No. of Sources**  **with Data Determination** 

**of No. of Sources in**  **MA CT Floor**  **No. of**

**Sources in MA** **CT Floor (N)**  **No. Data**

**Points in MA** **CT Floor (n)**  **Av erage** 

**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**  **Me asure**  **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 normalRaw 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 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 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 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 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 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-Large-lbton 12/1/201414.60789904 0.671102327
23.7670219 7. Calculate 1.378977872

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

3.484z 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 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e


2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  
2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e


2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me 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 3.888 133.669589 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 4.999 439.494distribution 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 6 24 z 

## z z   

    

     

#    2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z 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 0.0278244430.000139842 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 0.000656679Absolute 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 ( ) 1 3 ( )

6 24 z 

### z z f 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 0.0028102690.021243364 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-05Cumulative 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 

## ( ) 2 ( ) () f a 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 0.9970972372.652034149 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 component is zero). 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+004 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 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 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 in

  N

1 N i in

 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 11n i i s xX 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 21 1

s n m        1 1 i n i ik

k i 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 LevelLogtransformed **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

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 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+00Raw 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 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 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 11 n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 s n m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2
, 1 1 df p X t s n m    

   

0900006481934216 PM-Small-lbton 12/1/2014Raw 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 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 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 11 n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 s n m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2
, 1 1 df p X t s n m    

   

0900006481934216 PM-Small-conc 12/1/2014Raw 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 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 11 n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 s n m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2
, 1 1 df p X t s n m    

   

0900006481934216 UPL Pooled Template 12/1/2014Raw 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 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 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 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 ˆˆˆ 4 3 2 233 1 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

3.484z 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 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e


2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  
2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e


2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me 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 3.888 135.623489 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 4.999 446.2528distribution 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 6 24 z 

## z z   

    

     

#    2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z 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 0.0282311680.000139842 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 0.000666778Absolute 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 ( ) 1 3 ( )

6 24 z 

### z z f 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 0.0028513480.021556082 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-05Cumulative 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 

## ( ) 2 ( ) () f a 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 0.9970973182.691624922 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 component is zero). 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 34 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 in

  N

1 N i in

 -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!

#NUM! Current Confidence Level is 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 11n i i s xX 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 21 1

s n m        1 1 i n i ik

k i 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-statisticLogtransformed

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

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 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 

### 1x w b x     

   

## , x 

### I ab a a w a a

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn  

#      

1 2 4 1 1 11 11 31 ( ) , , ,

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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        
2 

### 3 1 2 6 w a  

    #NUM! Result3:



# 1 2 4 1 1 11 11 31 ( ) , , , n n n n P t I I I                    #NUM! 0 0 0



# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I                          

1 1 1 1 72 2 2 24 2 2

3 1 5 1 24 2 2 72 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2

5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 8 1 (2 5) 1 1 1 1 ( ) , ,

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a         

## ( , ) ( ) a  ( ) w a e w a  1 2 a w b  
3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 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   
2 

### 3 11 1 2 6 3 w a a        #NUM!
#NUM! 

### 2 6 3     3 1 2 6 w a  

    