**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**  **UPL Unit of Measure** 
All Kilns PM 225 31 12% Sources Industrywide 27 148 0.041 0.16 lb/ton Equivalent PM 225 31 12% Sources Industrywide 27 148 0.011 0.040 gr/dscf @ 7% O2 Large Kilns Mercury 122 19 12% Sources with Data 3 9 1.6E-05 2.2E-05 lb/ton Equivalent Mercury 122 19 12% Sources with Data 3 9 14 29 μg/dscm @ 7% O2 Equivalent Mercury 122 19 12% Sources with Data 3 9 1.7E-04 2.7E-04 lb/hr Small Kilns Mercury 103 2 12% Sources with Data 1 3 1.8E-04 2.0E-04 lb/ton Equivalent Mercury 103 2 12% Sources with Data 1 3 62 70 μg/dscm @ 7% O2 Equivalent Mercury 103 2 12% Sources with Data 1 3 0.0010 0.0011 lb/hr**Data**

**Distribution** lognormal lognormal normal normal skewed/unknown lognormal normal lognormalRuns 531 lb/ton 531 conc 531 lb/hr Example 4

1 1.84E-04 6.03E+01 1.04E-03 2 1.69E-04 6.21E+01 9.55E-04 3 1.76E-04 6.32E+01 9.96E-04 Average 1.76E-04 6.19E+01 9.97E-04 #DIV/0! Standard Deviation 7.42E-06 1.48E+00 4.20E-05 #DIV/0!

1 3.16E-21 6.01E+00 3.23E-18 #DIV/0! 2 2.91E-21 1.60E-03 2.98E-18 #DIV/0! 3 5.41E-28 3.48E+00 5.53E-25 #DIV/0! Testing Normality Raw Data Raw Data Raw Data Raw Data Sample Size 3.00 3.00 3.00 0.00 Kurtosis -2.00 -2.00 -2.00 #DIV/0! SE Kurtosis 2.83 2.83 2.83 #DIV/0! Result Kurtosis normal normal normal #DIV/0! Skewness 0.09 -0.60 0.09 #DIV/0! SE Skewness 1.22 1.22 1.22 0.00 Result Skewness normal normal normal #DIV/0! lb/ton 0.075456359 0.488415014 0.075456359 #DIV/0! Lognormal Normal Lognormal #DIV/0! 1 2 3 4Example 5 Example 6 Example 7 Example 8 Example 9

#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! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #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! 0.00 0.00 0.00 0.00 0.00 #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! 5 6 7 8 9Example 10 Example 11 Example 12 Example 13 Example 14

#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! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #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! 0.00 0.00 0.00 0.00 0.00 #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! 10 11 12 13 14Example 15 Example 16 Example 17 Example 18 Example 19

#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! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #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! 0.00 0.00 0.00 0.00 0.00 #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! 15 16 17 18 19Example 20 Example 21 Example 1 Example 2 -8.60E+00 4.10E+00 -8.69E+00 4.13E+00 -8.64E+00 4.15E+00 #DIV/0! #DIV/0! -8.64E+00 4.12E+00 #DIV/0! #DIV/0! 4.21E-02 2.39E-02 #DIV/0! #DIV/0! 3.19E-06 4.19E-07 #DIV/0! #DIV/0! 3.11E-06 1.36E-10 #DIV/0! #DIV/0! 5.67E-15 2.35E-07 Raw Data Raw Data Log Data Log Data 0.00 0.00 3.00 3.00 #DIV/0! #DIV/0! -2.00 -2.00 #DIV/0! #DIV/0! 2.83 2.83 #DIV/0! #DIV/0! normal normal #DIV/0! #DIV/0! 0.03 -0.63 0.00 0.00 1.22 1.22 #DIV/0! #DIV/0! normal normal #DIV/0! #DIV/0! 0.023935519 0.513997038 #DIV/0! #DIV/0! 20 21Example 3 Example 4 Example 5 Example 6 Example 7

-6.87E+00 #NUM! #NUM! #NUM! #NUM! -6.95E+00 #NUM! #NUM! #NUM! #NUM! -6.91E+00 #NUM! #NUM! #NUM! #NUM! -6.91E+00 #NUM! #NUM! #NUM! #NUM! 4.21E-02 #NUM! #NUM! #NUM! #NUM! 3.19E-06 #NUM! #NUM! #NUM! #NUM! 3.11E-06 #NUM! #NUM! #NUM! #NUM! 5.67E-15 #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 3.00 0.00 0.00 0.00 0.00 -2.00 #NUM! #NUM! #NUM! #NUM! 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! normal #NUM! #NUM! #NUM! #NUM! 0.03 #NUM! #NUM! #NUM! #NUM! 1.22 0.00 0.00 0.00 0.00 normal #NUM! #NUM! #NUM! #NUM! 0.023935519 #NUM! #NUM! #NUM! #NUM!Example 8 Example 9 Example 10 Example 11 Example 12

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Example 13 Example 14 Example 15 Example 16 Example 17

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Example 18 Example 19 Example 20 Example 21

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Raw Data Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567 1 2.99E-03 8.07E-03 0.008071368 1.10E-02 8.11E-03 0.00985 0.014691 0.029395 0.011364 0.026644 0.044467 0.0257190.063886 0.043787 0.027694 0.08631 0.003761 0.01295 0.03875 0.045215 0.08865 0.08865 0.0870390.113399 0.05275 0.040802 0.065536 2 9.53E-03 1.02E-02 0.010195412 1.26E-02 1.53E-02 0.013537 0.013642 0.020209 0.0234850.026077 0.0313680.049924 0.064942 0.043907 0.045821 0.004523 0.0055750.021133 0.05 0.044636 0.050124 0.050124 0.076493 0.0413790.0695850.035075 0.10717 3 5.61E-03 6.37E-03 0.006372133 6.74E-03 4.53E-03 0.009217 0.013642 0.0459296 0.0204550.014172 0.032403 0.022693 0.063886 0.047333 0.091641 0.02619 0.01585 0.033007 0.04625 0.052993 0.090307 0.090307 0.060864 0.023596 0.075196 0.037223 0.106399 4 0.008756567 0.014378 0.0041162 0.04589 0.0065850.028407 0.089596 0.052179 0.058712 0.042327 0.042327 0.0235850.157302 0.110354 0.024479 5 0.004108 0.0041298 0.011616 0.016438 0.0057680.028911 0.01413 0.076806 0.055327 0.055871 0.0410890.0410890.053774 0.107421 0.115804 0.013599 6 0.004300271 0.008901 0.0041838 0.0225590.018836 0.003212 0.0315480.010195 0.063728 0.036207 0.052083 0.0045050.0045050.034906 0.019087 0.091281 0.004932 7 0.027478 0.0728890.012142 0.024 0.17612 8 0.021659 0.048889 0.05862 0.212343 9 0.043956 0.043556 0.007607 0.029607 0.148328 10 0.02997 11 0.012615 12 0.116444 13 0.050074 14 0.115111 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 148.00 148.00 Kurtosis 3.67 -0.77 SE Kurtosis 0.40 0.40 Average of Raw Data 4.11E-02 Result Kurtosis non normal normal Skewness 1.68 -0.32 SE Skewness 0.20 0.20 Result Raw Data Result Log Data Result Skewness non normal normal non normal normal 1. Take LN( ) of Raw Data Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567 1 -5.81E+00 -4.82E+00 -4.82E+00 -4.51E+00 -4.81E+00 -4.62E+00-4.22E+00 -3.53E+00 -4.48E+00-3.63E+00-3.11E+00-3.66E+00-2.75E+00-3.13E+00-3.59E+00-2.45E+00-5.58E+00-4.35E+00-3.25E+00-3.10E+00-2.42E+00-2.42E+00-2.44E+00-2.18E+00-2.94E+00-3.20E+00-2.73E+00 2 -4.65E+00 -4.59E+00 -4.59E+00 -4.37E+00 -4.18E+00 -4.30E+00-4.29E+00 -3.90E+00 -3.75E+00-3.65E+00-3.46E+00-3.00E+00-2.73E+00-3.13E+00-3.08E+00-5.40E+00-5.19E+00-3.86E+00-3.00E+00-3.11E+00-2.99E+00-2.99E+00-2.57E+00-3.18E+00-2.67E+00-3.35E+00-2.23E+00 3 -5.18E+00 -5.06E+00 -5.06E+00 -5.00E+00 -5.40E+00 -4.69E+00-4.29E+00 -3.08E+00 -3.89E+00-4.26E+00-3.43E+00-3.79E+00-2.75E+00-3.05E+00-2.39E+00-3.64E+00-4.14E+00-3.41E+00-3.07E+00-2.94E+00-2.40E+00-2.40E+00-2.80E+00-3.75E+00-2.59E+00-3.29E+00-2.24E+00 4 -4.74E+00 -4.24E+00 -5.49E+00 -3.08E+00 -5.02E+00-3.56E+00 -2.41E+00-2.95E+00 -2.84E+00-3.16E+00-3.16E+00-3.75E+00-1.85E+00-2.20E+00-3.71E+00 5 -5.49E+00 -5.49E+00 -4.46E+00-4.11E+00 -5.16E+00-3.54E+00-4.26E+00 -2.57E+00-2.89E+00 -2.88E+00-3.19E+00-3.19E+00-2.92E+00-2.23E+00-2.16E+00-4.30E+00 6 -5.45E+00 -4.72E+00 -5.48E+00 -3.79E+00-3.97E+00 -5.74E+00-3.46E+00-4.59E+00 -2.75E+00-3.32E+00 -2.95E+00-5.40E+00-5.40E+00-3.36E+00-3.96E+00-2.39E+00-5.31E+00 7 -3.59E+00 -2.62E+00-4.41E+00-3.73E+00-1.74E+00 8 -3.83E+00 -3.02E+00 -2.84E+00-1.55E+00 9 -3.12E+00 -3.13E+00-4.88E+00-3.52E+00-1.91E+00 10 -3.51E+00 11 -4.37E+00 12 -2.15E+00 13 -2.99E+00 14 -2.16E+00 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 148

3. Calculate -3.63E+00

1.019643371 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
2 ˆˆˆ 4 3 2 2 2 ˆ 233 1 3 1 1 z e e e m

m e 

           



0900006481934216 PM-lbton 12/1/2014121.7260378
9.422141661

14.91914749
7. Calculate 3.66788443

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

3.484z value from "z-stat"

a- Calculate 0.044251848
b- Calculate 0.001958226

c- Calculate 1.772205938

d-calculate 0.010425784

e- Calculate 0.102931

f- Calculate UPL 0.16 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             

 

     1 m e 

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

0.101 14.91915 3.667884

-5=f(a) 171.1451 1 -4.899 154.1468 2 -4.798 138.3634 3 -4.697 123.7394 4 -4.596 110.2202 5 -4.495 97.75249 6 -4.394 86.28422 7 -4.293 75.76462 8 -4.192 66.14412 9 -4.091 57.3744 10 -3.99 49.40837 11 -3.889 42.20021 12 -3.788 35.7053 13 -3.687 29.8803 14 -3.586 24.68308 15 -3.485 20.07277 16 -3.384 16.00972 17 -3.283 12.45555 18 -3.182 9.37308 19 -3.081 6.726409 20 -2.98 4.480855 21 -2.879 2.602983 22 -2.778 1.060595 23 -2.677 -0.17726 24 -2.576 -1.14031 25 -2.475 -1.85701 26 -2.374 -2.35461 27 -2.273 -2.6591 28 -2.172 -2.79524 29 -2.071 -2.78654 30 -1.97 -2.65527 31 -1.869 -2.42246 32 -1.768 -2.10792 33 -1.667 -1.73018 34 -1.566 -1.30657 35 -1.465 -0.85316 36 -1.364 -0.38477 37 -1.263 0.084995 38 -1.162 0.543791 39 -1.061 0.980505 

#    1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 1.385267

41 -0.859 1.749446 42 -0.758 2.065654 43 -0.657 2.32774 44 -0.556 2.530794 45 -0.455 2.671148 46 -0.354 2.746374 47 -0.253 2.755283 48 -0.152 2.697926 49 -0.051 2.575596 50 0.05 2.390826 51 0.151 2.147389 52 0.252 1.850297 53 0.353 1.505805 54 0.454 1.121405 55 0.555 0.705833 56 0.656 0.269063 57 0.757 -0.17769 58 0.858 -0.62197 59 0.959 -1.05009 60 1.06 -1.4471 61 1.161 -1.79684 62 1.262 -2.08188 63 1.363 -2.28357 64 1.464 -2.38201 65 1.565 -2.35606 66 1.666 -2.18335 67 1.767 -1.84026 68 1.868 -1.30194 69 1.969 -0.54227 70 2.07 0.466074 71 2.171 1.751672 72 2.272 3.344346 73 2.373 5.275154 74 2.474 7.576397 75 2.575 10.28162 76 2.676 13.42559 77 2.777 17.04434 78 2.878 21.17513 79 2.979 25.85646 80 3.08 31.12807 81 3.181 37.03094 82 3.282 43.6073 83 3.383 50.90061 84 3.484 58.95557 85 3.585 67.81813 86 3.686 77.53547 87 3.787 88.15601 88 3.888 99.7294289 3.989 112.3066

90 4.09 125.9397 91 4.191 140.6821 92 4.292 156.5884 93 4.393 173.7146 94 4.494 192.1176 95 4.595 211.8559 96 4.696 232.989 97 4.797 255.5777 98 4.898 279.6842 99 4.999 305.3718distribution of Z Normal distribution

1.48672E-06 0.000254445 2.45094E-06 0.000377804 3.9995E-06 0.000553385 6.46025E-06 0.000799387 1.03291E-05 0.001138474 1.63472E-05 0.001597984 2.56093E-05 0.002209675 3.97118E-05 0.003008747 6.09553E-05 0.004031831 9.26132E-05 0.005313626 0.000139285 0.006881844 0.000207351 0.008750242 0.000305546 0.01090961 0.000445674 0.013316871 0.000643469 0.015882803 0.000919619 0.018459303 0.001300942 0.020827725 0.001821704 0.022690317 0.002525035 0.023667351 0.003464389 0.023302895 0.004704958 0.021082232 0.006324913 0.016463637 0.008416337 0.008926329 0.011085658 -0.001965069 0.014453386 -0.016481283 0.018652949 -0.034638733 0.023828414 -0.056106679 0.030130931 -0.080121292 0.03771375 -0.105419042 0.046725789 -0.130203175 0.057303789 -0.152156836 0.069563239 -0.168514263 0.083588399 -0.176197271 0.099421884 -0.172017858 0.117054396 -0.152939866 0.136415346 -0.11638371 0.157365126 -0.060549568 0.179689839 0.015272756 0.203099244 0.110443481 0.227228529 0.222798658 

# 

#  

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 0.348594543 0.275855243 0.482593977 0.299326439 0.618304831 0.321498297 0.748364303 0.341807853 0.865045289 0.359712192 0.960844678 0.37471238 1.029100385 0.386376486 1.064576417 0.394360216 1.063954656 0.398423793 1.026178861 0.398443914 0.95261022 0.394419966 0.84697308 0.386474058 0.715091895 0.374844865 0.564443205 0.359875719 0.403566586 0.34199778 0.241393398 0.32170943 0.086559994 0.299553264 -0.053228079 0.276092166 -0.171722226 0.25188591 -0.264502926 0.227469632 -0.329172129 0.203335281 -0.365360668 0.17991684 -0.374565027 0.157579839 -0.359844169 0.136615273 -0.325418619 0.117237788 -0.276219533 0.099587708 -0.217435206 0.083736272 -0.154096878 0.069693339 -0.090736244 0.05741676 -0.031135345 0.046822635 0.021822793 0.037795734 0.066205731 0.030199481 0.1009975 0.023885038 0.12599725 0.018699163 0.141672277 0.014490659 0.148987379 0.011115369 0.149230376 0.008439746 0.143849904 0.006343145 0.134316918 0.004718997 0.122016541 0.003475077 0.108172443 0.002533081 0.093802365 0.001827693 0.079700779 0.001305351 0.066443182 0.000922829 0.054405917 0.00064578 0.043795621 0.00044732 0.034683159 0.000306705 0.02703792 0.000208159 0.0207595310.000139842 0.015705153 9.29928E-05 0.011711485 6.12113E-05 0.008611332 3.98826E-05 0.006245153 2.5722E-05 0.00446829 1.64209E-05 0.003154741 1.03767E-05 0.002198357 6.49066E-06 0.001512252 4.01874E-06 0.0010271 2.46298E-06 0.000688855 1.49417E-06 0.000456278Absolute values

0.000254445 1.28495E-05 0.000377804 3.81582E-05 0.000553385 5.58919E-05 0.000799387 8.07381E-05 0.001138474 0.000114986 0.001597984 0.000161396 0.002209675 0.000223177 0.003008747 0.000303883 0.004031831 0.000407215 0.005313626 0.000536676 0.006881844 0.000695066 0.008750242 0.000883774 0.01090961 0.001101871 0.013316871 0.001345004 0.015882803 0.001604163 0.018459303 0.00186439 0.020827725 0.0021036 0.022690317 0.002291722 0.023667351 0.002390402 0.023302895 0.002353592 0.021082232 0.002129305 0.016463637 0.001662827 0.008926329 0.000901559 0.001965069 0.000198472 0.016481283 0.00166461 0.034638733 0.003498512 0.056106679 0.005666775 0.080121292 0.00809225 0.105419042 0.010647323 0.130203175 0.013150521 0.152156836 0.01536784 0.168514263 0.017019941 0.176197271 0.017795924 0.172017858 0.017373804 0.152939866 0.015446927 0.11638371 0.011754755 0.060549568 0.006115506 0.015272756 0.001542548 0.110443481 0.011154792 0.222798658 0.022502664 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 ( ) () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.348594543 0.035208049 0.482593977 0.048741992 0.618304831 0.062448788 0.748364303 0.075584795 0.865045289 0.087369574 0.960844678 0.097045312 1.029100385 0.103939139 1.064576417 0.107522218 1.063954656 0.10745942 1.026178861 0.103644065 0.95261022 0.096213632 0.84697308 0.085544281 0.715091895 0.072224281 0.564443205 0.057008764 0.403566586 0.040760225 0.241393398 0.024380733 0.086559994 0.008742559 0.053228079 0.005376036 0.171722226 0.017343945 0.264502926 0.026714796 0.329172129 0.033246385 0.365360668 0.036901428 0.374565027 0.037831068 0.359844169 0.036344261 0.325418619 0.032867281 0.276219533 0.027898173 0.217435206 0.021960956 0.154096878 0.015563785 0.090736244 0.009164361 0.031135345 0.00314467 0.021822793 0.002204102 0.066205731 0.006686779 0.1009975 0.010200748 0.12599725 0.012725722 0.141672277 0.0143089 0.148987379 0.015047725 0.149230376 0.015072268 0.143849904 0.01452884 0.134316918 0.013566009 0.122016541 0.012323671 0.108172443 0.010925417 0.093802365 0.009474039 0.079700779 0.008049779 0.066443182 0.006710761 0.054405917 0.005494998 0.043795621 0.004423358 0.034683159 0.003502999 0.02703792 0.00273083 0.020759531 0.0020967130.015705153 0.00158622 0.011711485 0.00118286 0.008611332 0.000869745 0.006245153 0.00063076 0.00446829 0.000451297 0.003154741 0.000318629 0.002198357 0.000222034 0.001512252 0.000152737 0.0010271 0.000103737 0.000688855 6.95744E-05 0.000456278 2.3042E-05Cumulative normalization
1.28495E-05 6.77666E-06 5.10077E-05 2.69009E-05 0.0001069 5.63776E-05 0.000187638 9.8958E-05 0.000302624 0.0001596 0.00046402 0.000244719 0.000687197 0.00036242 0.000991081 0.000522685 0.001398296 0.000737445 0.001934972 0.001020482 0.002630038 0.001387052 0.003513813 0.001853145 0.004615683 0.002434259 0.005960687 0.003143599 0.00756485 0.003989617 0.00942924 0.004972875 0.01153284 0.00608229 0.013824562 0.007290918 0.016214965 0.008551589 0.018568557 0.009792847 0.020697862 0.010915819 0.02236069 0.011792775 0.023262249 0.012268247 0.023460721 0.012372919 0.025125331 0.013250815 0.028623843 0.015095891 0.034290617 0.018084484 0.042382868 0.022352245 0.053030191 0.027967523 0.066180711 0.034902959 0.081548552 0.043007785 0.098568492 0.051983909 0.116364417 0.061369278 0.13373822 0.070532026 0.149185147 0.078678561 0.160939902 0.084877885 0.167055408 0.088103134 0.168597956 0.088916657 0.179752748 0.094799568 0.202255412 0.106667219 

## ( ) 2 ( ) () f a f x f b        0.237463461 0.125235546 0.286205453 0.150941522 0.348654241 0.183876307 0.424239036 0.223738873 0.51160861 0.269816599 0.608653922 0.3209972 0.712593061 0.375813528 0.820115279 0.43251953 0.927574699 0.489192414 1.031218764 0.543853123 1.127432397 0.594595105 1.212976678 0.639710192 1.285200959 0.677800462 1.342209723 0.70786624 1.382969948 0.729362722 1.407350681 0.742220845 1.41609324 0.746831572 1.421469276 0.749666833 1.438813221 0.758813833 1.465528017 0.772902915 1.498774402 0.790436682 1.535675829 0.809898078 1.573506897 0.829849756 1.609851158 0.849017308 1.642718439 0.866351141 1.670616612 0.881064322 1.692577567 0.892646282 1.708141352 0.900854446 1.717305713 0.905687626 1.720450382 0.907346089 1.722654485 0.908508508 1.729341263 0.91203504 1.739542011 0.917414799 1.752267733 0.924126201 1.766576633 0.931672553 1.781624358 0.939608553 1.796696626 0.947557497 1.811225467 0.955219843 1.824791475 0.962374403 1.837115146 0.968873767 1.848040563 0.974635709 1.857514602 0.97963221 1.86556438 0.983877572 1.872275142 0.987416752 1.877770139 0.990314751 1.882193497 0.99264758 1.885696496 0.994495022 1.888427326 0.995935231 1.890524038 0.9970410141.892110259 0.997877568 1.893293119 0.998501395 1.894162863 0.998960089 1.894793624 0.999292745 1.895244921 0.999530754 1.89556355 0.999698795 1.895785584 0.999815893 1.895938322 0.999896445 1.896042059 0.999951155 1.896111633 0.999987848 1.896134675 1 93.15042416Raw Data rank by lb/ton use gr/dscf @ 7% O2 values Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567 1 0.001024602 0.002138462 0.002138462 0.0278 0.002025429 0.0023490.003141 0.0085662 0.00417 0.007784 0.0150780.008084 0.035522 0.011024 0.004633 0.022757 0.000982 0.003353 0.0091190.0097850.015886 0.015886 0.021789 0.02445 0.0096780.008486 0.013187 2 0.003182773 0.002527273 0.002527273 0.0278 0.003772857 0.003132 0.002514 0.0054668 0.0073590.007784 0.0094680.013974 0.035522 0.011024 0.007841 0.0010080.001324 0.005523 0.0106150.009812 0.0114050.0114050.0195350.0087150.0168090.008096 0.021741 3 0.001782597 0.001782051 0.001782051 0.0139 0.001123347 0.00182 0.002547 0.0147517 0.00695 0.004544 0.0113080.0067580.035522 0.008554 0.0149690.006904 0.003734 0.008176 0.009856 0.012184 0.020672 0.020672 0.016154 0.0048080.015617 0.008034 0.022097 4 0.002494872 0.003282 0.0016079 0.021482 0.003742 0.023167 0.025608 0.012209 0.0182150.010692 0.010692 0.007174 0.024241 0.0238790.005673 5 0.000927 0.0014766 0.003375 0.00891 0.003451 0.0117050.003569 0.021898 0.014 0.016466 0.0102390.0102390.016776 0.0165550.022114 0.003159 6 0.00119472 0.001782 0.001456 0.0061550.009979 0.0020990.012437 0.002529 0.018266 0.009165 0.016113 0.00106 0.00106 0.009688 0.00339 0.023712 0.001137 7 0.005505 0.0163950.003061 0.004026 0.02889 8 0.004526 0.012311 0.0094490.036601 9 0.005272 0.013003 0.002033 0.00532 0.025076 10 0.003834 11 0.001554 12 0.027525 13 0.011851 14 0.02306 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 148.00 148.00 Kurtosis 0.52 -0.87 SE Kurtosis 0.40 0.40 Average of Raw Data 1.05E-02 Result Kurtosis normal non normal Note: Not ideal to use the Skewed/Unknown template, because the "current confidence level" is 100%, which is a higher confidence level than expected. Performed more statistical tests in R to determine the distribution, and results indicated a lognormal distribution was the best fit for the data. Skewness 1.04 -0.38 SE Skewness 0.20 0.20 Result Raw Data Result Log Data Result Skewness non normal normal non normal non normal 1. Take LN( ) of Raw Data Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567 1 -6.88E+00 -6.15E+00 -6.15E+00 -3.58E+00 -6.20E+00 -6.05E+00-5.76E+00 -4.76E+00 -5.48E+00-4.86E+00-4.19E+00-4.82E+00-3.34E+00-4.51E+00-5.37E+00-3.78E+00-6.93E+00-5.70E+00-4.70E+00-4.63E+00-4.14E+00-4.14E+00-3.83E+00-3.71E+00-4.64E+00-4.77E+00-4.33E+00 2 -5.75E+00 -5.98E+00 -5.98E+00 -3.58E+00 -5.58E+00 -5.77E+00-5.99E+00 -5.21E+00 -4.91E+00-4.86E+00-4.66E+00-4.27E+00-3.34E+00-4.51E+00-4.85E+00-6.90E+00-6.63E+00-5.20E+00-4.55E+00-4.62E+00-4.47E+00-4.47E+00-3.94E+00-4.74E+00-4.09E+00-4.82E+00-3.83E+00 3 -6.33E+00 -6.33E+00 -6.33E+00 -4.28E+00 -6.79E+00 -6.31E+00-5.97E+00 -4.22E+00 -4.97E+00-5.39E+00-4.48E+00-5.00E+00-3.34E+00-4.76E+00-4.20E+00-4.98E+00-5.59E+00-4.81E+00-4.62E+00-4.41E+00-3.88E+00-3.88E+00-4.13E+00-5.34E+00-4.16E+00-4.82E+00-3.81E+00 4 -5.99E+00 -5.72E+00 -6.43E+00 -3.84E+00 -5.59E+00-3.77E+00 -3.66E+00-4.41E+00 -4.01E+00-4.54E+00-4.54E+00-4.94E+00-3.72E+00-3.73E+00-5.17E+00 5 -6.98E+00 -6.52E+00 -5.69E+00-4.72E+00 -5.67E+00-4.45E+00-5.64E+00 -3.82E+00-4.27E+00 -4.11E+00-4.58E+00-4.58E+00-4.09E+00-4.10E+00-3.81E+00-5.76E+00 6 -6.73E+00 -6.33E+00 -6.53E+00 -5.09E+00-4.61E+00 -6.17E+00-4.39E+00-5.98E+00 -4.00E+00-4.69E+00 -4.13E+00-6.85E+00-6.85E+00-4.64E+00-5.69E+00-3.74E+00-6.78E+00 7 -5.20E+00 -4.11E+00-5.79E+00-5.51E+00-3.54E+00 8 -5.40E+00 -4.40E+00 -4.66E+00-3.31E+00 9 -5.25E+00 -4.34E+00-6.20E+00-5.24E+00-3.69E+00 10 -5.56E+00 11 -6.47E+00 12 -3.59E+00 13 -4.44E+00 14 -3.77E+00 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 148

3. Calculate -4.95E+00

0.946447098 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
2 ˆˆˆ 4 3 2 2 2 ˆ 233 1 3 1 1 z e e e m

m e 

           



0900006481934216 PM-conc 12/1/201495.19505418
7.456427425

14.76684513
7. Calculate 3.31763759

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

3.484z value from "z-stat"

a- Calculate 0.011379046
b- Calculate 0.000129483

c- Calculate 1.576539187

d-calculate 0.009441723

e- Calculate 0.024968

f- Calculate UPL 0.040 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 1 z m

m e    



0900006481934216 PM-conc 12/1/2014∆z z

0.101 14.76685 3.317638

-5=f(a) 174.533 1 -4.899 157.3918 2 -4.798 141.4653 3 -4.697 126.6982 4 -4.596 113.0362 5 -4.495 100.4266 6 -4.394 88.81752 7 -4.293 78.1586 8 -4.192 68.40058 9 -4.091 59.49545 10 -3.99 51.3964 11 -3.889 44.05788 12 -3.788 37.43554 13 -3.687 31.48627 14 -3.586 26.16816 15 -3.485 21.44056 16 -3.384 17.26402 17 -3.283 13.60032 18 -3.182 10.41246 19 -3.081 7.66468 20 -2.98 5.322424 21 -2.879 3.352374 22 -2.778 1.722431 23 -2.677 0.401725 24 -2.576 -0.6394 25 -2.475 -1.42935 26 -2.374 -1.99534 27 -2.273 -2.36335 28 -2.172 -2.55812 29 -2.071 -2.60317 30 -1.97 -2.52081 31 -1.869 -2.33212 32 -1.768 -2.05695 33 -1.667 -1.71393 34 -1.566 -1.32046 35 -1.465 -0.89272 36 -1.364 -0.44567 37 -1.263 0.006973 38 -1.162 0.45269 39 -1.061 0.880204 

#    1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 1.279457

41 -0.859 1.641618 42 -0.758 1.959078 43 -0.657 2.225453 44 -0.556 2.435585 45 -0.455 2.585539 46 -0.354 2.672604 47 -0.253 2.695295 48 -0.152 2.653351 49 -0.051 2.547734 50 0.05 2.380633 51 0.151 2.155459 52 0.252 1.876849 53 0.353 1.550664 54 0.454 1.18399 55 0.555 0.785136 56 0.656 0.363637 57 0.757 -0.06975 58 0.858 -0.50304 59 0.959 -0.92302 60 1.06 -1.31527 61 1.161 -1.66412 62 1.262 -1.95269 63 1.363 -2.16289 64 1.464 -2.27538 65 1.565 -2.2696 66 1.666 -2.12377 67 1.767 -1.8149 68 1.868 -1.31875 69 1.969 -0.60987 70 2.07 0.338411 71 2.171 1.554 72 2.272 3.066022 73 2.373 4.904827 74 2.474 7.101989 75 2.575 9.690307 76 2.676 12.70381 77 2.777 16.17773 78 2.878 20.14856 79 2.979 24.65399 80 3.08 29.73294 81 3.181 35.42555 82 3.282 41.7732 83 3.383 48.81848 84 3.484 56.60521 85 3.585 65.17843 86 3.686 74.58442 87 3.787 84.87066 88 3.888 96.0858789 3.989 108.28

90 4.09 121.5042 91 4.191 135.8109 92 4.292 151.2537 93 4.393 167.8873 94 4.494 185.768 95 4.595 204.953 96 4.696 225.5007 97 4.797 247.471 98 4.898 270.9248 99 4.999 295.9242distribution of Z Normal distribution

1.48672E-06 0.000259482 2.45094E-06 0.000385758 3.9995E-06 0.000565791 6.46025E-06 0.000818501 1.03291E-05 0.001167561 1.63472E-05 0.001641698 2.56093E-05 0.002274551 3.97118E-05 0.003103816 6.09553E-05 0.004169375 9.26132E-05 0.005510063 0.000139285 0.007158747 0.000207351 0.009135432 0.000305546 0.011438276 0.000445674 0.014032607 0.000643469 0.016838405 0.000919619 0.019717151 0.001300942 0.022459495 0.001821704 0.024775757 0.002525035 0.026291831 0.003464389 0.026553433 0.004704958 0.02504178 0.006324913 0.021203472 0.008416337 0.014496564 0.011085658 0.004453382 0.014453386 -0.009241426 0.018652949 -0.026661628 0.023828414 -0.047545907 0.030130931 -0.071209917 0.03771375 -0.096476136 0.046725789 -0.121635173 0.057303789 -0.144452185 0.069563239 -0.162230129 0.083588399 -0.171937529 0.099421884 -0.170402355 0.117054396 -0.154565776 0.136415346 -0.12178082 0.157365126 -0.070132299 0.179689839 0.001252934 0.203099244 0.091940952 0.227228529 0.200007356 

# 

#  

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 0.321968128 0.275855243 0.452848877 0.299326439 0.586403756 0.321498297 0.715479414 0.341807853 0.832502123 0.359712192 0.930049835 0.37471238 1.001457829 0.386376486 1.041398655 0.394360216 1.046375954 0.398423793 1.015077836 0.398443914 0.948548573 0.394419966 0.85015594 0.386474058 0.725353361 0.374844865 0.581258401 0.359875719 0.426089097 0.34199778 0.268514675 0.32170943 0.116985384 0.299553264 -0.020893351 0.276092166 -0.138884407 0.25188591 -0.232495603 0.227469632 -0.299182933 0.203335281 -0.338373769 0.17991684 -0.35132257 0.157579839 -0.340827901 0.136615273 -0.310851037 0.117237788 -0.266082314 0.099587708 -0.21150142 0.083736272 -0.151972751 0.069693339 -0.091907998 0.05741676 -0.035016841 0.046822635 0.015845302 0.037795734 0.058734586 0.030199481 0.092592277 0.023885038 0.117151971 0.018699163 0.132801241 0.014490659 0.140418941 0.011115369 0.141207497 0.008439746 0.136535971 0.006343145 0.127805262 0.004718997 0.1163421 0.003475077 0.103324255 0.002533081 0.089735775 0.001827693 0.076348598 0.001305351 0.063725265 0.000922829 0.05223693 0.00064578 0.042090954 0.00044732 0.033363095 0.000306705 0.026030285 0.000208159 0.0200010960.000139842 0.015142066 9.29928E-05 0.011299016 6.12113E-05 0.008313159 3.98826E-05 0.006032388 2.5722E-05 0.004318402 1.64209E-05 0.003050476 1.03767E-05 0.002126728 6.49066E-06 0.001463648 4.01874E-06 0.000994521 2.46298E-06 0.000667281 1.49417E-06 0.000442161Absolute values

0.000259482 1.31038E-05 0.000385758 3.89615E-05 0.000565791 5.71449E-05 0.000818501 8.26686E-05 0.001167561 0.000117924 0.001641698 0.000165811 0.002274551 0.00022973 0.003103816 0.000313485 0.004169375 0.000421107 0.005510063 0.000556516 0.007158747 0.000723033 0.009135432 0.000922679 0.011438276 0.001155266 0.014032607 0.001417293 0.016838405 0.001700679 0.019717151 0.001991432 0.022459495 0.002268409 0.024775757 0.002502351 0.026291831 0.002655475 0.026553433 0.002681897 0.02504178 0.00252922 0.021203472 0.002141551 0.014496564 0.001464153 0.004453382 0.000449792 0.009241426 0.000933384 0.026661628 0.002692824 0.047545907 0.004802137 0.071209917 0.007192202 0.096476136 0.00974409 0.121635173 0.012285152 0.144452185 0.014589671 0.162230129 0.016385243 0.171937529 0.01736569 0.170402355 0.017210638 0.154565776 0.015611143 0.12178082 0.012299863 0.070132299 0.007083362 0.001252934 0.000126546 0.091940952 0.009286036 0.200007356 0.020200743 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 ( ) () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.321968128 0.032518781 0.452848877 0.045737737 0.586403756 0.059226779 0.715479414 0.072263421 0.832502123 0.084082714 0.930049835 0.093935033 1.001457829 0.101147241 1.041398655 0.105181264 1.046375954 0.105683971 1.015077836 0.102522861 0.948548573 0.095803406 0.85015594 0.08586575 0.725353361 0.073260689 0.581258401 0.058707098 0.426089097 0.043034999 0.268514675 0.027119982 0.116985384 0.011815524 0.020893351 0.002110228 0.138884407 0.014027325 0.232495603 0.023482056 0.299182933 0.030217476 0.338373769 0.034175751 0.35132257 0.03548358 0.340827901 0.034423618 0.310851037 0.031395955 0.266082314 0.026874314 0.21150142 0.021361643 0.151972751 0.015349248 0.091907998 0.009282708 0.035016841 0.003536701 0.015845302 0.001600375 0.058734586 0.005932193 0.092592277 0.00935182 0.117151971 0.011832349 0.132801241 0.013412925 0.140418941 0.014182313 0.141207497 0.014261957 0.136535971 0.013790133 0.127805262 0.012908331 0.1163421 0.011750552 0.103324255 0.01043575 0.089735775 0.009063313 0.076348598 0.007711208 0.063725265 0.006436252 0.05223693 0.00527593 0.042090954 0.004251186 0.033363095 0.003369673 0.026030285 0.002629059 0.020001096 0.0020201110.015142066 0.001529349 0.011299016 0.001141201 0.008313159 0.000839629 0.006032388 0.000609271 0.004318402 0.000436159 0.003050476 0.000308098 0.002126728 0.0002148 0.001463648 0.000147828 0.000994521 0.000100447 0.000667281 6.73954E-05 0.000442161 2.23292E-05Cumulative normalization
1.31038E-05 7.12293E-06 5.20654E-05 2.83015E-05 0.00010921 5.93641E-05 0.000191879 0.000104301 0.000309803 0.000168401 0.000475614 0.000258533 0.000705344 0.000383408 0.001018829 0.000553811 0.001439936 0.000782715 0.001996452 0.001085224 0.002719486 0.001478248 0.003642164 0.001979794 0.00479743 0.002607769 0.006214724 0.003378176 0.007915402 0.004302625 0.009906835 0.00538512 0.012175244 0.006618174 0.014677595 0.007978393 0.01733307 0.009421846 0.020014967 0.010879661 0.022544187 0.012254485 0.024685737 0.013418581 0.02614989 0.01421446 0.026599682 0.014458956 0.027533066 0.014966322 0.03022589 0.016430077 0.035028027 0.019040405 0.042220228 0.022949915 0.051964318 0.028246571 0.064249471 0.034924488 0.078839141 0.042855087 0.095224384 0.051761717 0.112590075 0.061201294 0.129800713 0.070556589 0.145411856 0.079042436 0.157711719 0.08572835 0.164795081 0.089578697 0.164921627 0.089647485 0.174207663 0.094695154 0.194408406 0.105675799 

## ( ) 2 ( ) () f a f x f b        0.226927187 0.123352237 0.272664924 0.148214186 0.331891703 0.180408458 0.404155124 0.21968914 0.488237838 0.265394509 0.582172872 0.316455366 0.683320113 0.371436607 0.788501377 0.428610647 0.894185348 0.486057947 0.996708209 0.541786943 1.092511615 0.593863403 1.178377365 0.640537988 1.251638055 0.680360761 1.310345153 0.712272547 1.353380152 0.735665352 1.380500134 0.750407131 1.392315658 0.75682977 1.394425886 0.75797684 1.408453211 0.765601761 1.431935267 0.77836605 1.462152744 0.794791553 1.496328494 0.813368681 1.531812074 0.832656713 1.566235692 0.851368575 1.597631647 0.868434672 1.62450596 0.883042911 1.645867604 0.894654594 1.661216852 0.902998081 1.67049956 0.908043941 1.67403626 0.909966407 1.675636636 0.910836334 1.681568829 0.914060933 1.690920649 0.919144361 1.702752998 0.925576145 1.716165923 0.932867093 1.730348236 0.940576262 1.744610194 0.948328724 1.758400327 0.955824713 1.771308658 0.962841376 1.78305921 0.969228697 1.79349496 0.974901323 1.802558273 0.979827926 1.810269482 0.984019555 1.816705734 0.987518149 1.821981663 0.99038602 1.82623285 0.992696864 1.829602522 0.994528538 1.832231581 0.995957632 1.834251692 0.9970557161.835781041 0.997887034 1.836922241 0.998507363 1.83776187 0.998963766 1.838371141 0.999294951 1.8388073 0.999532037 1.839115398 0.999699511 1.839330198 0.999816271 1.839478026 0.999896627 1.839578473 0.999951228 1.839645868 0.999987862 1.839668197 1 90.51431457Raw Data rank by lb/ton use lb/ton Runs 568 558 655 1 1.37E-05 1.75E-05 2.04E-05 2 1.50E-05 1.57E-05 2.04E-05 3 1.56E-05 1.80E-05 1.08E-05 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 -0.10 SE Kurtosis 1.56 Result Kurtosis normal Skewness -0.29 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 1.48E-05 1.70E-05 1.72E-05 #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!

1.63E-05 Mean =
Pooled Variance = 9.70E-12

m= number future runs = 3 Term1 4.44E-01
Term2 4.31E-12

Squared root Term2 2.08E-06 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 = 2.2E-05 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 Hg-Large-lbton 12/1/2014Raw Data rank by lb/ton use conc Runs 568 558 655 1 1.17E+01 2.31E+01 9.79E+00 2 9.98E+00 2.19E+01 1.14E+01 3 9.71E+00 2.69E+01 5.67E+00 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 -1.09 SE Kurtosis 1.56 Result Kurtosis normal Skewness 0.76 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 1.05E+01 2.40E+01 8.94E+00 #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!

1.45E+01 Mean =
Pooled Variance = 5.55E+01

m= number future runs = 3 Term1 4.44E-01
Term2 2.47E+01

Squared root Term2 4.97E+00 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 = 29 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 Hg-Large-conc 12/1/2014Raw Data rank by lb/ton use lb/hr Runs 568 558 655 1 1.63E-04 1.99E-04 1.63E-04 2 1.78E-04 1.79E-04 1.62E-04 3 1.85E-04 2.05E-04 8.57E-05 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 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 Sample size 9 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY Testing Normality Raw Data Logtransformed Sample Size 9.00 9.00 Kurtosis 4.67 6.46 http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics SE Kurtosis 1.56 1.56 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 -1.90 -2.39 SE Skewness 0.72 0.72 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 1.69E-04 Standard Deviation 3.48E-05 http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt 568 558 655 1 6.78E-04 5.49E-01 8.97E-04 2 4.99E-03 6.74E-03 1.33E-03 3 4.77E-02 1.15E+00 3.27E+014 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 1.31 SE Kurtosis 1.63 Result Kurtosis normal Skewness -1.90 SE Skewness 0.72 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= 4.67 depending on sample size skewness= -1.90 same for n>3 and n=3 ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 3
n =Total # test runs = 9

1 N i df n       

###  1

N i in

  N

1 N i in

 8

This step is for QC only:
means 0.00018 0.00019 0.00014 #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 = 1.69E-04

Pooled Variance = 1.2091E-09

m= number future runs = 3
Term1 0.4444

Term2 5.3737E-10

Squared root Term2 2.31813E-05 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 = 2.896459 UPL based on t-statistic with df at 99% confidence level 0.000236

9 0.488118

98.07% 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 0.236 4.312459448 Adjusted UPL based on recalculated t-statistic 2.7E-04

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 **568 558**  **655** -8.72E+00 -8.52E+00-8.72E+00 -8.63E+00 -8.63E+00-8.73E+00 -8.59E+00 -8.49E+00-9.36E+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!Raw Data rank by lb/ton use lb/ton Runs 531 1 1.84E-04 2 1.69E-04 3 1.76E-04 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 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.76E-04 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.09 0.03 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 531 1 -8.60E+00 2 -8.69E+00 3 -8.64E+00 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 3

3. Calculate -8.64E+00

0.001773919 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 Hg-Small-lbton 12/1/20143.028455212
9.45713E-06

320231.8612
7. Calculate 0.07302593

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

3.484z value from "z-stat"

a- Calculate 0.000176319
b- Calculate 3.10885E-08

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 1.82E-05

f- Calculate UPL 2.0E-04 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 Hg-Small-lbton 12/1/2014∆z z

0.101 320231.9 0.073026

-5=f(a) 6377891 1 -4.899 5804260 2 -4.798 5268199 3 -4.697 4768108 4 -4.596 4302421 5 -4.495 3869604 6 -4.394 3468158 7 -4.293 3096616 8 -4.192 2753546 9 -4.091 2437548 10 -3.99 2147255 11 -3.889 1881334 12 -3.788 1638485 13 -3.687 1417443 14 -3.586 1216973 15 -3.485 1035876 16 -3.384 872986 17 -3.283 727169.3 18 -3.182 597326.2 19 -3.081 482390.2 20 -2.98 381328 21 -2.879 293140 22 -2.778 216859.4 23 -2.677 151553.1 24 -2.576 96321.29 25 -2.475 50297.27 26 -2.374 12647.86 27 -2.273 -17426.9 28 -2.172 -40693.5 29 -2.071 -57885.4 30 -1.97 -69702.4 31 -1.869 -76811.1 32 -1.768 -79845 33 -1.667 -79403.9 34 -1.566 -76054.5 35 -1.465 -70330.2 36 -1.364 -62730.9 37 -1.263 -53723.4 38 -1.162 -43741 39 -1.061 -33183.7 

#    1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -22418.4

41 -0.859 -11778.3 42 -0.758 -1563.57 43 -0.657 7959.066 44 -0.556 16556.17 45 -0.455 24027.64 46 -0.354 30206.71 47 -0.253 34959.9 48 -0.152 38187.09 49 -0.051 39821.47 50 0.05 39829.55 51 0.151 38211.15 52 0.252 34999.45 53 0.353 30260.93 54 0.454 24095.38 55 0.555 16635.93 56 0.656 8049.039 57 0.757 -1465.52 58 0.858 -11674.7 59 0.959 -22311.9 60 1.06 -33077.6 61 1.161 -43638.7 62 1.262 -53628.7 63 1.363 -62647.9 64 1.464 -70263.3 65 1.565 -76008.6 66 1.666 -79384.1 67 1.767 -79856.7 68 1.868 -76860.1 69 1.969 -69794.7 70 2.07 -58027.5 71 2.171 -40892.3 72 2.272 -17689.3 73 2.373 12314.41 74 2.474 49884.99 75 2.575 95822.1 76 2.676 150958.6 77 2.777 216160.8 78 2.878 292328.3 79 2.979 380393.8 80 3.08 481323.7 81 3.181 596117.4 82 3.282 725807.8 83 3.383 871461.1 84 3.484 1034177 85 3.585 1215088 86 3.686 1415360 87 3.787 1636193 88 3.888 187882089 3.989 2144506

90 4.09 2434551 91 4.191 2750289 92 4.292 3093084 93 4.393 3464337 94 4.494 3865480 95 4.595 4297979 96 4.696 4763334 97 4.797 5263077 98 4.898 5798774 99 4.999 6372025distribution of Z Normal distribution

1.48672E-06 9.482135231 2.45094E-06 14.22589281 3.9995E-06 21.07017796 6.46025E-06 30.80315723 1.03291E-05 44.44008603 1.63472E-05 63.25737464 2.56093E-05 88.8169625 3.97118E-05 122.9721103 6.09553E-05 167.8431131 9.26132E-05 225.7490785 0.000139285 299.0803286 0.000207351 390.0958602 0.000305546 500.6324415 0.000445674 631.7171745 0.000643469 783.0844331 0.000919619 952.6114436 0.001300942 1135.704304 0.001821704 1324.687107 0.002525035 1508.269244 0.003464389 1671.187086 0.004704958 1794.132213 0.006324913 1854.084705 0.008416337 1825.161903 0.011085658 1680.066356 0.014453386 1392.168772 0.018652949 938.1924676 0.023828414 301.3784363 0.030130931 -525.0880094 0.03771375 -1534.705674 0.046725789 -2704.740072 0.057303789 -3994.210046 0.069563239 -5343.231447 0.083588399 -6674.1139 0.099421884 -7894.483006 0.117054396 -8902.512616 0.136415346 -9594.112976 0.157365126 -9871.654295 0.179689839 -9653.544101 0.203099244 -8883.758627 0.227228529 -7540.291364 

# 

#  

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 -5641.457864 0.275855243 -3249.104849 0.299326439 -468.0164235 0.321498297 2558.8262 0.341807853 5659.029172 0.359712192 8643.036797 0.37471238 11318.82738 0.386376486 13507.68414 0.394360216 15059.47049 0.398423793 15865.82151 0.398443914 15869.84028 0.394419966 15071.24222 0.386474058 13526.38094 0.374844865 11343.15265 0.359875719 8671.340423 0.34199778 5689.45084 0.32170943 2589.451717 0.299553264 -439.0018051 0.276092166 -3223.280932 0.25188591 -5620.06389 0.227469632 -7524.158318 0.203335281 -8873.282863 0.17991684 -9648.701009 0.157579839 -9872.046658 0.136615273 -9599.045599 0.117237788 -8911.083046 0.099587708 -7905.677738 0.083736272 -6686.900326 0.069693339 -5356.637602 0.05741676 -4007.386645 0.046822635 -2717.001999 0.037795734 -1545.55264 0.030199481 -534.2062408 0.023885038 294.1300802 0.018699163 932.807623 0.014490659 1388.525396 0.011115369 1677.960975 0.008439746 1824.342699 0.006343145 1854.280771 0.004718997 1795.0773 0.003475077 1672.637104 0.002533081 1510.013548 0.001827693 1326.554202 0.001305351 1137.562967 0.000922829 954.3684071 0.00064578 784.6798087 0.00044732 633.1186513 0.000306705 501.829117 0.000208159 391.09233490.000139842 299.8914797 9.29928E-05 226.395735 6.12113E-05 168.3487008 3.98826E-05 123.3602238 2.5722E-05 89.10975263 1.64209E-05 63.47460261 1.03767E-05 44.59868829 6.49066E-06 30.91717381 4.01874E-06 21.15091801 2.46298E-06 14.28223616 1.49417E-06 9.520894664Absolute values

9.482135231 0.478847829 14.22589281 1.436815174 21.07017796 2.128087974 30.80315723 3.11111888 44.44008603 4.488448689 63.25737464 6.388994839 88.8169625 8.970513213 122.9721103 12.42018314 167.8431131 16.95215442 225.7490785 22.80065693 299.0803286 30.20711319 390.0958602 39.39968188 500.6324415 50.56387659 631.7171745 63.80343462 783.0844331 79.09152774 952.6114436 96.21375581 1135.704304 114.7061347 1324.687107 133.7933978 1508.269244 152.3351937 1671.187086 168.7898957 1794.132213 181.2073535 1854.084705 187.2625552 1825.161903 184.3413522 1680.066356 169.6867019 1392.168772 140.609046 938.1924676 94.75743923 301.3784363 30.43922207 525.0880094 53.03388895 1534.705674 155.005273 2704.740072 273.1787473 3994.210046 403.4152146 5343.231447 539.6663761 6674.1139 674.0855039 7894.483006 797.3427836 8902.512616 899.1537742 9594.112976 969.0054106 9871.654295 997.0370838 9653.544101 975.0079542 8883.758627 897.2596213 7540.291364 761.5694277 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 ( ) () 2 M i

i 

## xf a f x f b

 

##  

      

# 5641.457864 569.7872442 3249.104849 328.1595898 468.0164235 47.26965877 2558.8262 258.4414462 5659.029172 571.5619464 8643.036797 872.9467164 11318.82738 1143.201565 13507.68414 1364.276098 15059.47049 1521.006519 15865.82151 1602.447973 15869.84028 1602.853868 15071.24222 1522.195464 13526.38094 1366.164475 11343.15265 1145.658418 8671.340423 875.8053828 5689.45084 574.6345348 2589.451717 261.5346234 439.0018051 44.33918231 3223.280932 325.5513742 5620.06389 567.6264529 7524.158318 759.9399901 8873.282863 896.2015692 9648.701009 974.5188019 9872.046658 997.0767124 9599.045599 969.5036055 8911.083046 900.0193876 7905.677738 798.4734515 6686.900326 675.3769329 5356.637602 541.0203978 4007.386645 404.7460512 2717.001999 274.4172019 1545.55264 156.1008167 534.2062408 53.95483033 294.1300802 29.7071381 932.807623 94.21356992 1388.525396 140.241065 1677.960975 169.4740585 1824.342699 184.2586126 1854.280771 187.2823578 1795.0773 181.3028073 1672.637104 168.9363475 1510.013548 152.5113683 1326.554202 133.9819744 1137.562967 114.8938597 954.3684071 96.39120912 784.6798087 79.25266068 633.1186513 63.94498378 501.829117 50.68474082 391.0923349 39.50032582299.8914797 30.28903945 226.395735 22.86596924 168.3487008 17.00321878 123.3602238 12.4593826 89.10975263 9.000085016 63.47460261 6.410934864 44.59868829 4.504467517 30.91717381 3.122634555 21.15091801 2.136242719 14.28223616 1.442505852 9.520894664 0.480805181Cumulative normalization
0.478847829 1.28191E-05 1.915663003 5.12837E-05 4.043750977 0.000108254 7.154869857 0.000191541 11.64331855 0.0003117 18.03231338 0.000482738 27.0028266 0.000722885 39.42300974 0.001055382 56.37516416 0.001509204 79.17582108 0.002119593 109.3829343 0.002928259 148.7826162 0.003983017 199.3464927 0.005336648 263.1499274 0.007044712 342.2414551 0.009162049 438.4552109 0.01173776 553.1613456 0.014808525 686.9547434 0.01839027 839.2899371 0.022468393 1008.079833 0.026987019 1189.287186 0.03183807 1376.549741 0.036851222 1560.891094 0.041786172 1730.577796 0.046328807 1871.186842 0.050093012 1965.944281 0.052629736 1996.383503 0.053444616 2049.417392 0.054864371 2204.422665 0.059013973 2477.601412 0.066327164 2881.016627 0.077126878 3420.683003 0.091574133 4094.768507 0.109619885 4892.11129 0.130965322 5791.265065 0.155036313 6760.270475 0.180977282 7757.307559 0.207668678 8732.315513 0.233770339 9629.575134 0.257790622 10391.14456 0.278178381 

## ( ) 2 ( ) () f a f x f b        10960.93181 0.293431993 11289.0914 0.302217061 11336.36105 0.303482503 11594.8025 0.310401166 12166.36445 0.325702288 13039.31116 0.349071697 14182.51273 0.379676022 15546.78883 0.41619867 17067.79535 0.456917103 18670.24332 0.499815783 20273.09719 0.542725329 21795.29265 0.583475591 23161.45713 0.620048793 24307.11554 0.650718889 25182.92093 0.674164826 25757.55546 0.689548204 26019.09009 0.696549673 26063.42927 0.697736665 26388.98064 0.706451908 26956.6071 0.721647675 27716.54709 0.741991813 28612.74865 0.765983771 29587.26746 0.792072337 30584.34417 0.818764794 31553.84777 0.8447191 32453.86716 0.868813264 33252.34061 0.89018897 33927.71755 0.908269294 34468.73794 0.922752797 34873.484 0.933588139 35147.9012 0.940934484 35304.00201 0.945113414 35357.95684 0.946557824 35387.66398 0.947353105 35481.87755 0.94987527 35622.11862 0.953629624 35791.59268 0.958166566 35975.85129 0.963099301 36163.13365 0.968112983 36344.43645 0.972966589 36513.3728 0.977489136 36665.88417 0.981571975 36799.86614 0.985158768 36914.76 0.988234559 37011.15121 0.990815021 37090.40387 0.992936671 37154.34886 0.994648524 37205.0336 0.996005391 37244.53392 0.99706284337274.82296 0.997873703 37297.68893 0.998485841 37314.69215 0.998941029 37327.15153 0.999274576 37336.15162 0.999515515 37342.56255 0.99968714 37347.06702 0.999807728 37350.18966 0.999891323 37352.3259 0.999948512 37353.7684 0.999987129 37354.24921 1 1886224.019Raw Data rank by lb/ton using concentration data Runs 531 1 6.03E+01 2 6.21E+01 3 6.32E+01 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 3.00 Kurtosis #DIV/0! SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness -0.60 SE Skewness 1.22 Result Raw Data Result Skewness normal #DIV/0!
ni = number test runs = 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 0 0 Total Number of sources = 1 n =Total # test runs = 3

2
This step is for QC only: means 6.19E+01 #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!

6.19E+01 Mean =
Pooled Variance = 2.18E+00

m= number future runs = 3 Term1 6.67E-01
Term2 1.45E+00

Squared root Term2 1.21E+00 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 = 6.964556734 99 UPL POOLED VARIANCE = 70 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 Hg-Small-conc 12/1/2014Raw Data rank by lb/ton using lb/hr data Runs 531 1 1.04E-03 2 9.55E-04 3 9.96E-04 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 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 9.97E-04 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.09 0.03 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 531 1 -6.87E+00 2 -6.95E+00 3 -6.91E+00 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 3

3. Calculate -6.91E+00

0.001773919 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 Hg-Small-lbhr 12/1/20143.028455212
9.45713E-06

320231.8612
7. Calculate 0.07302593

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

3.484z value from "z-stat"

a- Calculate 0.000997086
b- Calculate 9.9418E-07

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 0.000103

f- Calculate UPL 0.0011 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 Hg-Small-lbhr 12/1/2014∆z z

0.101 320231.9 0.073026

-5=f(a) 6377891 1 -4.899 5804260 2 -4.798 5268199 3 -4.697 4768108 4 -4.596 4302421 5 -4.495 3869604 6 -4.394 3468158 7 -4.293 3096616 8 -4.192 2753546 9 -4.091 2437548 10 -3.99 2147255 11 -3.889 1881334 12 -3.788 1638485 13 -3.687 1417443 14 -3.586 1216973 15 -3.485 1035876 16 -3.384 872986 17 -3.283 727169.3 18 -3.182 597326.2 19 -3.081 482390.2 20 -2.98 381328 21 -2.879 293140 22 -2.778 216859.4 23 -2.677 151553.1 24 -2.576 96321.29 25 -2.475 50297.27 26 -2.374 12647.86 27 -2.273 -17426.9 28 -2.172 -40693.5 29 -2.071 -57885.4 30 -1.97 -69702.4 31 -1.869 -76811.1 32 -1.768 -79845 33 -1.667 -79403.9 34 -1.566 -76054.5 35 -1.465 -70330.2 36 -1.364 -62730.9 37 -1.263 -53723.4 38 -1.162 -43741 39 -1.061 -33183.7 

#    1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -22418.4

41 -0.859 -11778.3 42 -0.758 -1563.57 43 -0.657 7959.066 44 -0.556 16556.17 45 -0.455 24027.64 46 -0.354 30206.71 47 -0.253 34959.9 48 -0.152 38187.09 49 -0.051 39821.47 50 0.05 39829.55 51 0.151 38211.15 52 0.252 34999.45 53 0.353 30260.93 54 0.454 24095.38 55 0.555 16635.93 56 0.656 8049.039 57 0.757 -1465.52 58 0.858 -11674.7 59 0.959 -22311.9 60 1.06 -33077.6 61 1.161 -43638.7 62 1.262 -53628.7 63 1.363 -62647.9 64 1.464 -70263.3 65 1.565 -76008.6 66 1.666 -79384.1 67 1.767 -79856.7 68 1.868 -76860.1 69 1.969 -69794.7 70 2.07 -58027.5 71 2.171 -40892.3 72 2.272 -17689.3 73 2.373 12314.41 74 2.474 49884.99 75 2.575 95822.1 76 2.676 150958.6 77 2.777 216160.8 78 2.878 292328.3 79 2.979 380393.8 80 3.08 481323.7 81 3.181 596117.4 82 3.282 725807.8 83 3.383 871461.1 84 3.484 1034177 85 3.585 1215088 86 3.686 1415360 87 3.787 1636193 88 3.888 187882089 3.989 2144506

90 4.09 2434551 91 4.191 2750289 92 4.292 3093084 93 4.393 3464337 94 4.494 3865480 95 4.595 4297979 96 4.696 4763334 97 4.797 5263077 98 4.898 5798774 99 4.999 6372025distribution of Z Normal distribution

1.48672E-06 9.482135231 2.45094E-06 14.22589281 3.9995E-06 21.07017796 6.46025E-06 30.80315723 1.03291E-05 44.44008603 1.63472E-05 63.25737464 2.56093E-05 88.8169625 3.97118E-05 122.9721103 6.09553E-05 167.8431131 9.26132E-05 225.7490785 0.000139285 299.0803286 0.000207351 390.0958602 0.000305546 500.6324415 0.000445674 631.7171745 0.000643469 783.0844331 0.000919619 952.6114436 0.001300942 1135.704304 0.001821704 1324.687107 0.002525035 1508.269244 0.003464389 1671.187086 0.004704958 1794.132213 0.006324913 1854.084705 0.008416337 1825.161903 0.011085658 1680.066356 0.014453386 1392.168772 0.018652949 938.1924676 0.023828414 301.3784363 0.030130931 -525.0880094 0.03771375 -1534.705674 0.046725789 -2704.740072 0.057303789 -3994.210046 0.069563239 -5343.231447 0.083588399 -6674.1139 0.099421884 -7894.483006 0.117054396 -8902.512616 0.136415346 -9594.112976 0.157365126 -9871.654295 0.179689839 -9653.544101 0.203099244 -8883.758627 0.227228529 -7540.291364 

# 

#  

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 -5641.457864 0.275855243 -3249.104849 0.299326439 -468.0164235 0.321498297 2558.8262 0.341807853 5659.029172 0.359712192 8643.036797 0.37471238 11318.82738 0.386376486 13507.68414 0.394360216 15059.47049 0.398423793 15865.82151 0.398443914 15869.84028 0.394419966 15071.24222 0.386474058 13526.38094 0.374844865 11343.15265 0.359875719 8671.340423 0.34199778 5689.45084 0.32170943 2589.451717 0.299553264 -439.0018051 0.276092166 -3223.280932 0.25188591 -5620.06389 0.227469632 -7524.158318 0.203335281 -8873.282863 0.17991684 -9648.701009 0.157579839 -9872.046658 0.136615273 -9599.045599 0.117237788 -8911.083046 0.099587708 -7905.677738 0.083736272 -6686.900326 0.069693339 -5356.637602 0.05741676 -4007.386645 0.046822635 -2717.001999 0.037795734 -1545.55264 0.030199481 -534.2062408 0.023885038 294.1300802 0.018699163 932.807623 0.014490659 1388.525396 0.011115369 1677.960975 0.008439746 1824.342699 0.006343145 1854.280771 0.004718997 1795.0773 0.003475077 1672.637104 0.002533081 1510.013548 0.001827693 1326.554202 0.001305351 1137.562967 0.000922829 954.3684071 0.00064578 784.6798087 0.00044732 633.1186513 0.000306705 501.829117 0.000208159 391.09233490.000139842 299.8914797 9.29928E-05 226.395735 6.12113E-05 168.3487008 3.98826E-05 123.3602238 2.5722E-05 89.10975263 1.64209E-05 63.47460261 1.03767E-05 44.59868829 6.49066E-06 30.91717381 4.01874E-06 21.15091801 2.46298E-06 14.28223616 1.49417E-06 9.520894664Absolute values

9.482135231 0.478847829 14.22589281 1.436815174 21.07017796 2.128087974 30.80315723 3.11111888 44.44008603 4.488448689 63.25737464 6.388994839 88.8169625 8.970513213 122.9721103 12.42018314 167.8431131 16.95215442 225.7490785 22.80065693 299.0803286 30.20711319 390.0958602 39.39968188 500.6324415 50.56387659 631.7171745 63.80343462 783.0844331 79.09152774 952.6114436 96.21375581 1135.704304 114.7061347 1324.687107 133.7933978 1508.269244 152.3351937 1671.187086 168.7898957 1794.132213 181.2073535 1854.084705 187.2625552 1825.161903 184.3413522 1680.066356 169.6867019 1392.168772 140.609046 938.1924676 94.75743923 301.3784363 30.43922207 525.0880094 53.03388895 1534.705674 155.005273 2704.740072 273.1787473 3994.210046 403.4152146 5343.231447 539.6663761 6674.1139 674.0855039 7894.483006 797.3427836 8902.512616 899.1537742 9594.112976 969.0054106 9871.654295 997.0370838 9653.544101 975.0079542 8883.758627 897.2596213 7540.291364 761.5694277 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 ( ) () 2 M i

i 

## xf a f x f b

 

##  

      

# 5641.457864 569.7872442 3249.104849 328.1595898 468.0164235 47.26965877 2558.8262 258.4414462 5659.029172 571.5619464 8643.036797 872.9467164 11318.82738 1143.201565 13507.68414 1364.276098 15059.47049 1521.006519 15865.82151 1602.447973 15869.84028 1602.853868 15071.24222 1522.195464 13526.38094 1366.164475 11343.15265 1145.658418 8671.340423 875.8053828 5689.45084 574.6345348 2589.451717 261.5346234 439.0018051 44.33918231 3223.280932 325.5513742 5620.06389 567.6264529 7524.158318 759.9399901 8873.282863 896.2015692 9648.701009 974.5188019 9872.046658 997.0767124 9599.045599 969.5036055 8911.083046 900.0193876 7905.677738 798.4734515 6686.900326 675.3769329 5356.637602 541.0203978 4007.386645 404.7460512 2717.001999 274.4172019 1545.55264 156.1008167 534.2062408 53.95483033 294.1300802 29.7071381 932.807623 94.21356992 1388.525396 140.241065 1677.960975 169.4740585 1824.342699 184.2586126 1854.280771 187.2823578 1795.0773 181.3028073 1672.637104 168.9363475 1510.013548 152.5113683 1326.554202 133.9819744 1137.562967 114.8938597 954.3684071 96.39120912 784.6798087 79.25266068 633.1186513 63.94498378 501.829117 50.68474082 391.0923349 39.50032582299.8914797 30.28903945 226.395735 22.86596924 168.3487008 17.00321878 123.3602238 12.4593826 89.10975263 9.000085016 63.47460261 6.410934864 44.59868829 4.504467517 30.91717381 3.122634555 21.15091801 2.136242719 14.28223616 1.442505852 9.520894664 0.480805181Cumulative normalization
0.478847829 1.28191E-05 1.915663003 5.12837E-05 4.043750977 0.000108254 7.154869857 0.000191541 11.64331855 0.0003117 18.03231338 0.000482738 27.0028266 0.000722885 39.42300974 0.001055382 56.37516416 0.001509204 79.17582108 0.002119593 109.3829343 0.002928259 148.7826162 0.003983017 199.3464927 0.005336648 263.1499274 0.007044712 342.2414551 0.009162049 438.4552109 0.01173776 553.1613456 0.014808525 686.9547434 0.01839027 839.2899371 0.022468393 1008.079833 0.026987019 1189.287186 0.03183807 1376.549741 0.036851222 1560.891094 0.041786172 1730.577796 0.046328807 1871.186842 0.050093012 1965.944281 0.052629736 1996.383503 0.053444616 2049.417392 0.054864371 2204.422665 0.059013973 2477.601412 0.066327164 2881.016627 0.077126878 3420.683003 0.091574133 4094.768507 0.109619885 4892.11129 0.130965322 5791.265065 0.155036313 6760.270475 0.180977282 7757.307559 0.207668678 8732.315513 0.233770339 9629.575134 0.257790622 10391.14456 0.278178381 

## ( ) 2 ( ) () f a f x f b        10960.93181 0.293431993 11289.0914 0.302217061 11336.36105 0.303482503 11594.8025 0.310401166 12166.36445 0.325702288 13039.31116 0.349071697 14182.51273 0.379676022 15546.78883 0.41619867 17067.79535 0.456917103 18670.24332 0.499815783 20273.09719 0.542725329 21795.29265 0.583475591 23161.45713 0.620048793 24307.11554 0.650718889 25182.92093 0.674164826 25757.55546 0.689548204 26019.09009 0.696549673 26063.42927 0.697736665 26388.98064 0.706451908 26956.6071 0.721647675 27716.54709 0.741991813 28612.74865 0.765983771 29587.26746 0.792072337 30584.34417 0.818764794 31553.84777 0.8447191 32453.86716 0.868813264 33252.34061 0.89018897 33927.71755 0.908269294 34468.73794 0.922752797 34873.484 0.933588139 35147.9012 0.940934484 35304.00201 0.945113414 35357.95684 0.946557824 35387.66398 0.947353105 35481.87755 0.94987527 35622.11862 0.953629624 35791.59268 0.958166566 35975.85129 0.963099301 36163.13365 0.968112983 36344.43645 0.972966589 36513.3728 0.977489136 36665.88417 0.981571975 36799.86614 0.985158768 36914.76 0.988234559 37011.15121 0.990815021 37090.40387 0.992936671 37154.34886 0.994648524 37205.0336 0.996005391 37244.53392 0.99706284337274.82296 0.997873703 37297.68893 0.998485841 37314.69215 0.998941029 37327.15153 0.999274576 37336.15162 0.999515515 37342.56255 0.99968714 37347.06702 0.999807728 37350.18966 0.999891323 37352.3259 0.999948512 37353.7684 0.999987129 37354.24921 1 1886224.019Raw 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 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 Lognormal Template 12/1/2014∆z z

0.101 14.78551 3.387135

-5=f(a) 173.6307 1 -4.899 156.5385 2 -4.798 140.6598 3 -4.697 125.9391 4 -4.596 112.3223 5 -4.495 99.75638 6 -4.394 88.18975 7 -4.293 77.57188 8 -4.192 67.85352 9 -4.091 58.98665 10 -3.99 50.92447 11 -3.889 43.62141 12 -3.788 37.03312 13 -3.687 31.11647 14 -3.586 25.82959 15 -3.485 21.13179 16 -3.384 16.98365 17 -3.283 13.34694 18 -3.182 10.18468 19 -3.081 7.461103 20 -2.98 5.141681 21 -2.879 3.193102 22 -2.778 1.583284 23 -2.677 0.281372 24 -2.576 -0.74226 25 -2.475 -1.51602 26 -2.374 -2.06709 27 -2.273 -2.42141 28 -2.172 -2.6037 29 -2.071 -2.63747 30 -1.97 -2.54498 31 -1.869 -2.34728 32 -1.768 -2.06419 33 -1.667 -1.71429 34 -1.566 -1.31496 35 -1.465 -0.88233 36 -1.364 -0.43132 37 -1.263 0.024392 38 -1.162 0.472344 39 -1.061 0.901303 

#    1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 1.301261

41 -0.859 1.663439 42 -0.758 1.980282 43 -0.657 2.245462 44 -0.556 2.45388 45 -0.455 2.601659 46 -0.354 2.686152 47 -0.253 2.705937 48 -0.152 2.660818 49 -0.051 2.551826 50 0.05 2.381219 51 0.151 2.152479 52 0.252 1.870318 53 0.353 1.540671 54 0.454 1.170702 55 0.555 0.7688 56 0.656 0.344581 57 0.757 -0.09111 58 0.858 -0.52622 59 0.959 -0.94743 60 1.06 -1.34023 61 1.161 -1.68888 62 1.262 -1.97639 63 1.363 -2.18458 64 1.464 -2.294 65 1.565 -2.28401 66 1.666 -2.13273 67 1.767 -1.81705 68 1.868 -1.31264 69 1.969 -0.59394 70 2.07 0.365836 71 2.171 1.594699 72 2.272 3.121887 73 2.373 4.977865 74 2.474 7.194324 75 2.575 9.804181 76 2.676 12.84158 77 2.777 16.34189 78 2.878 20.34171 79 2.979 24.87887 80 3.08 29.9924 81 3.181 35.7226 82 3.282 42.11095 83 3.383 49.20019 84 3.484 57.03427 85 3.585 65.65838 86 3.686 75.1189387 3.787 85.46354

88 3.888 96.74107 89 3.989 109.0016 90 4.09 122.2965 91 4.191 136.6783 92 4.292 152.2006 93 4.393 168.9187 94 4.494 186.8885 95 4.595 206.1677 96 4.696 226.8148 97 4.797 248.8898 98 4.898 272.4537 99 4.999 297.569distribution of Z Normal distribution

1.48672E-06 0.00025814 2.45094E-06 0.000383666 3.9995E-06 0.000562569 6.46025E-06 0.000813598 1.03291E-05 0.001160187 1.63472E-05 0.001630742 2.56093E-05 0.002258474 3.97118E-05 0.003080516 6.09553E-05 0.004136028 9.26132E-05 0.005462942 0.000139285 0.007093014 0.000207351 0.009044929 0.000305546 0.011315317 0.000445674 0.0138678 0.000643469 0.016620542 0.000919619 0.0194332 0.001300942 0.022094744 0.001821704 0.02431417 0.002525035 0.025716663 0.003464389 0.025848163 0.004704958 0.024191389 0.006324913 0.02019609 0.008416337 0.013325453 0.011085658 0.003119197 0.014453386 -0.010728213 0.018652949 -0.028278328 0.023828414 -0.049255451 0.030130931 -0.072959218 0.03771375 -0.098195275 0.046725789 -0.123237716 0.057303789 -0.145836905 0.069563239 -0.163284321 0.083588399 -0.172542023 0.099421884 -0.170438144 0.117054396 -0.153922026 0.136415346 -0.120363821 0.157365126 -0.067874755 0.179689839 0.004383042 0.203099244 0.095932777 0.227228529 0.204801723 

# 

#  

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 0.327454986 0.275855243 0.458868267 0.299326439 0.592750629 0.321498297 0.72191232 0.341807853 0.838755395 0.359712192 0.93584856 0.37471238 1.006534541 0.386376486 1.045510485 0.394360216 1.049320814 0.398423793 1.016708275 0.398443914 0.948782077 0.394419966 0.848980757 0.386474058 0.722829327 0.374844865 0.577512693 0.359875719 0.421307289 0.34199778 0.262927915 0.32170943 0.110854823 0.299553264 -0.027293533 0.276092166 -0.145283984 0.25188591 -0.238643935 0.227469632 -0.304862135 0.203335281 -0.343408575 0.17991684 -0.355586415 0.157579839 -0.344245108 0.136615273 -0.31339536 0.117237788 -0.267772357 0.099587708 -0.212393663 0.083736272 -0.152153024 0.069693339 -0.091482269 0.05741676 -0.034102101 0.046822635 0.017129395 0.037795734 0.060272801 0.030199481 0.094279357 0.023885038 0.118896489 0.018699163 0.134527832 0.014490659 0.142069051 0.011115369 0.142738913 0.008439746 0.137921435 0.006343145 0.129030453 0.004718997 0.117403303 0.003475077 0.104225924 0.002533081 0.090488222 0.001827693 0.076965905 0.001305351 0.064223535 0.000922829 0.052632887 0.00064578 0.042400898 0.00044732 0.0336021920.000306705 0.026212124 0.000208159 0.020137481 0.000139842 0.01524298 9.29928E-05 0.011372694 6.12113E-05 0.008366251 3.98826E-05 0.006070157 2.5722E-05 0.004344929 1.64209E-05 0.003068875 1.03767E-05 0.002139333 6.49066E-06 0.001472177 4.01874E-06 0.001000222 2.46298E-06 0.000671047 1.49417E-06 0.000444619Absolute values

0.00025814 1.30361E-05 0.000383666 3.87503E-05 0.000562569 5.68195E-05 0.000813598 8.21734E-05 0.001160187 0.000117179 0.001630742 0.000164705 0.002258474 0.000228106 0.003080516 0.000311132 0.004136028 0.000417739 0.005462942 0.000551757 0.007093014 0.000716394 0.009044929 0.000913538 0.011315317 0.001142847 0.0138678 0.001400648 0.016620542 0.001678675 0.0194332 0.001962753 0.022094744 0.002231569 0.02431417 0.002455731 0.025716663 0.002597383 0.025848163 0.002610664 0.024191389 0.00244333 0.02019609 0.002039805 0.013325453 0.001345871 0.003119197 0.000315039 0.010728213 0.00108355 0.028278328 0.002856111 0.049255451 0.004974801 0.072959218 0.007368881 0.098195275 0.009917723 0.123237716 0.012447009 0.145836905 0.014729527 0.163284321 0.016491716 0.172542023 0.017426744 0.170438144 0.017214253 0.153922026 0.015546125 0.120363821 0.012156746 0.067874755 0.00685535 0.004383042 0.000442687 0.095932777 0.009689211 0.204801723 0.020684974 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 ( ) () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.327454986 0.033072954 0.458868267 0.046345695 0.592750629 0.059867814 0.72191232 0.072913144 0.838755395 0.084714295 0.93584856 0.094520705 1.006534541 0.101659989 1.045510485 0.105596559 1.049320814 0.105981402 1.016708275 0.102687536 0.948782077 0.09582699 0.848980757 0.085747056 0.722829327 0.073005762 0.577512693 0.058328782 0.421307289 0.042552036 0.262927915 0.026555719 0.110854823 0.011196337 0.027293533 0.002756647 0.145283984 0.014673682 0.238643935 0.024103037 0.304862135 0.030791076 0.343408575 0.034684266 0.355586415 0.035914228 0.344245108 0.034768756 0.31339536 0.031652931 0.267772357 0.027045008 0.212393663 0.02145176 0.152153024 0.015367455 0.091482269 0.009239709 0.034102101 0.003444312 0.017129395 0.001730069 0.060272801 0.006087553 0.094279357 0.009522215 0.118896489 0.012008545 0.134527832 0.013587311 0.142069051 0.014348974 0.142738913 0.01441663 0.137921435 0.013930065 0.129030453 0.013032076 0.117403303 0.011857734 0.104225924 0.010526818 0.090488222 0.00913931 0.076965905 0.007773556 0.064223535 0.006486577 0.052632887 0.005315922 0.042400898 0.004282491 0.033602192 0.0033938210.026212124 0.002647425 0.020137481 0.002033886 0.01524298 0.001539541 0.011372694 0.001148642 0.008366251 0.000844991 0.006070157 0.000613086 0.004344929 0.000438838 0.003068875 0.000309956 0.002139333 0.000216073 0.001472177 0.00014869 0.001000222 0.000101022 0.000671047 6.77757E-05 0.000444619 2.24533E-05Cumulative normalization
1.30361E-05 7.04745E-06 5.17864E-05 2.79963E-05 0.000108606 5.87135E-05 0.000190779 0.000103137 0.000307958 0.000166486 0.000472663 0.000255527 0.000700769 0.000378843 0.001011901 0.000547045 0.00142964 0.000772879 0.001981397 0.001071165 0.002697791 0.001458456 0.003611329 0.001952325 0.004754176 0.00257016 0.006154824 0.003327366 0.007833499 0.004234876 0.009796252 0.005295962 0.012027821 0.006502373 0.014483552 0.007829968 0.017080935 0.009234142 0.0196916 0.010645496 0.02213493 0.011966388 0.024174735 0.013069129 0.025520606 0.013796721 0.025835645 0.013967035 0.026919195 0.014552814 0.029775306 0.016096859 0.034750106 0.018786291 0.042118987 0.02276999 0.05203671 0.02813162 0.064483719 0.034860611 0.079213247 0.042823556 0.095704963 0.05173916 0.113131707 0.06116025 0.13034596 0.070466465 0.145892085 0.078870872 0.15804883 0.085442943 0.164904181 0.089149021 0.165346868 0.089388343 0.175036079 0.094626437 0.195721052 0.105808962 

## ( ) 2 ( ) () f a f x f b        0.228794006 0.123688566 0.275139701 0.14874356 0.335007515 0.181108761 0.407920659 0.220526412 0.492634954 0.266323895 0.587155658 0.317422832 0.688815647 0.372381344 0.794412206 0.42946801 0.900393608 0.486762726 1.003081144 0.542276742 1.098908134 0.594081871 1.18465519 0.640437676 1.257660952 0.679905397 1.315989734 0.711438581 1.35854177 0.734442682 1.38509749 0.748798997 1.396293827 0.75485186 1.399050474 0.756342134 1.413724156 0.764274889 1.437827193 0.777305257 1.468618269 0.793951252 1.503302535 0.812701949 1.539216763 0.832117578 1.573985519 0.850913951 1.60563845 0.868025875 1.632683458 0.882646705 1.654135218 0.894243763 1.669502674 0.90255158 1.678742383 0.907546668 1.682186695 0.909408702 1.683916764 0.910343996 1.690004317 0.913634994 1.699526532 0.918782809 1.711535077 0.925274761 1.725122388 0.932620211 1.739471362 0.940377425 1.753887993 0.948171215 1.767818058 0.955701962 1.780850133 0.962747246 1.792707867 0.969157667 1.803234685 0.974848582 1.812373996 0.979789394 1.820147552 0.983991864 1.826634129 0.987498579 1.831950051 0.990372425 1.836232541 0.992687587 1.839626363 0.9945223241.842273787 0.995953551 1.844307673 0.997053092 1.845847214 0.997885385 1.846995856 0.998506354 1.847840847 0.998963165 1.848453933 0.999294606 1.848892771 0.999531847 1.849202727 0.999699413 1.8494188 0.999816224 1.84956749 0.999896607 1.849668512 0.999951221 1.849736288 0.999987862 1.849758741 1 90.98161566**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! 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

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

new t-statistic from tab Recalculate t-statistic #NUM!Logtransformed

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! 0.236 #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  

    