Key primary_f uel Hg_Min_lb_ per_MWh_ Rank Reason_for_Removal ORIS code Plant Name 52071_5B coal 1 52071 Sandow Station 6180_1 coal 2 6180 Oak Grove 52071_5A coal 3 52071 Sandow Station 55076_AA-001 coal 4 55076 Red Hills Generating Facility 55076_AA-002 coal 5 55076 Red Hills Generating Facility 7030_U2 coal 6 7030Twin Oaks Power One 6147_2 coal 7 6147 Monticello 6030_1 coal 8 6030 Coal Creek 6183_SM-1 coal 9 6183 San Miguel 2817_1 coal 10 2817 Leland Olds 6469_B1 coal 11 6469Antelope Valley Stationphysical_

state NonContI ndicator Unit

Number boiler_i

d Unit Type boiler_type MWe_cap acity TX 5B 5B Fluidized bed firing Fluidized bed firing 282.4 TX OG1 1 Conventional Boiler Tangential firing 817.0 TX 5A 5A Fluidized bed firing Fluidized bed firing 282.4 MS 001 AA-001 Fluidized bed firing Fluidized bed firing 250.0 MS 002 AA-002 Fluidized bed firing Fluidized bed firing 250.0 TX U2 U2 Fluidized bed firing Fluidized bed firing 172.0 TX 2 2 Conventional Boiler Tangential firing 583.0 ND Unit 1b 1 Conventional Boiler Tangential firing 619.0 TX SMEC1 SM-1 Conventional Boiler Wall firing - opposed firing 440.0 ND 1 1 Conventional Boiler Wall firing - opposed firing 230.0 ND AVS Unit 1 B1 Conventional Boiler Tangential firing 450.0max_heat_i

nput Unit_heat_

rate Fuel_types fuel Fuel_Conc_l b/MMBtu Mercury -

ERT Mercury MW - ERT Mercury -

ETTS Mercury

MW - ETTS 2,960.0 10.5 lignite lignite 3.1E-05 5.6E-06 5.9E-05 8,970.0 11.0 lignite lignite 2.1E-05 1.1E-06 1.2E-05 2,960.0 10.5 lignite lignite 3.0E-05 3.7E-06 3.9E-05 2,401.0 9.6 lignite lignite 2.4E-05 1.1E-05 1.0E-04 2,382.3 9.5 lignite lignite 2.4E-05 1.1E-05 1.0E-04 1,717.0 10.0 lignite lignite 2.8E-05 6.9E-06 6.9E-05 7,000.0 12.0 lignite lignite 6.7E-06 8.0E-05 6,015.0 9.7 coal - lignite lignite 4,980.0 11.3 coal - lignite lignite 2,622.0 11.4 coal - lignite lignite 6,275.0 13.9 coal - lignite ligniteMercury -

CEMS Mercury -

Part II Mercury MW - Part II Mercury -

min Mercury MW - min control_group_1 control_type_1
1.0E-06 1.1E-05 1.0E-06 1.1E-05 NOx control Selective Noncatalytic Reduction 3.9E-06 4.3E-05 1.1E-06 1.22E-05 NOx control Selective Catalytic Reduction 1.4E-06 1.5E-05 1.4E-06 1.52E-05 NOx control Selective Noncatalytic Reduction 3.9E-06 3.7E-05 3.9E-06 3.72E-05 PM control Fabric Filter, unspecified 3.9E-06 3.7E-05 3.9E-06 3.72E-05 PM control Fabric Filter, unspecified

6.9E-06 6.86E-05 PM control Fabric Filter, reverse air 6.7E-06 8.04E-05 NOx control Selective Noncatalytic Reduction 1.1E-05 1.1E-04 1.1E-05 0.000106 PM control Electrostatic precipitator, cold side, w/o flue gas conditioning 1.1E-05 1.2E-04 1.1E-05 0.000123 PM control Electrostatic precipitator, cold side, w/o flue gas conditioning 5.3E-03 6.0E-02 5.3E-03 0.06 PM control Electrostatic precipitator, cold side, w/o flue gas conditioning 5.1E-03 7.1E-02 5.1E-03 0.071 SO2 control Dry FGD - Spray Dryerinstall_date_1 control_group_2 control_type_2 install_date_ 2 control_group_3 01-Jul-09 Other control Activated carbon injection 01-Aug-09 SO2 control 01-Aug-09 Other control Activated carbon injection 01-Aug-09 PM control 01-Aug-09 Other control Activated carbon injection 01-Aug-09 SO2 control 01-May-02 01-May-02 01-Aug-90 01-Nov-08 Other control Activated carbon injection 01-Feb-09 PM control 01-Aug-79 SO2 control Wet FGD - Spray Type 01-Aug-79 01-Jan-82 SO2 control Wet FGD - Tray Type 01-Jan-82 01-Nov-74 01-Jul-84 PM control Fabric Filter, reverse air 01-Jul-84control_type_3 install_date_ 3 control_group_4 control_type_4 install_date_4 Dry FGD - Circulating Dry Scrubber 01-Jul-09 PM control Fabric Filter, pulse 01-Jul-09 Fabric Filter, pulse 01-Aug-09 SO2 control Wet FGD - Spray Type 01-Aug-09 Dry FGD - Circulating Dry Scrubber 01-Aug-09 PM control Fabric Filter, pulse 01-Aug-09
Fabric Filter, shake and deflate 01-Dec-75 PM control Electrostatic precipitator, cold side, w/ flue gas conditioning 01-Dec-75control_grou

p_5 control_type_5 install_date_5 control_group_6 control_type_6 install_date_6control_group_7 control_type_7 install_date_7 control_group_8 control_type_8 install_date_8control_g

roup_9 control_t

ype_9 install_d

ate_9 control_gr oup_10 control_t

ype_10 install_d

ate_10 control_ group_1 1 control_t ype_11 install_d

ate_11 control_ group_1 2control_t

ype_12 install_d

ate_12facility_name facility_ id boiler_i d configuration_i

d compound Final_EF Final_EF_ Units Final_Outp

ut_EF Sandow Station 52071 5B 5B Mercury 1.0E-06 lb/MMBtu 1.1E-05 Sandow Station 52071 5B 5B Mercury 1.2E-06 lb/MMBtu 1.3E-05 Sandow Station 52071 5B 5B Mercury Invalid Test Avglb/MMBtu

Invalid Test Avg Oak Grove 6180 1 OG1 Mercury 1.1E-06 lb/MMBtu 1.2E-05 Oak Grove 6180 1 OG1 Mercury 3.9E-06 lb/MMBtu 4.3E-05 Sandow Station 52071 5A 5A Mercury 1.4E-06 lb/MMBtu 1.5E-05 Sandow Station 52071 5A 5A Mercury 1.9E-06 lb/MMBtu 2.0E-05 Sandow Station 52071 5A 5A Mercury 3.7E-06 lb/MMBtu 3.9E-05 Red Hills Generating Facility

55076AA-001 001 Mercury 3.9E-06 lb/MMBtu 3.7E-05 Red Hills Generating Facility

55076AA-001 001 Mercury 1.1E-05 lb/MMBtu 1.0E-04 Red Hills Generating Facility

55076AA-002 002 Mercury 3.9E-06 lb/MMBtu 3.7E-05 Red Hills Generating Facility

55076AA-002 002 Mercury 1.1E-05 lb/MMBtu 1.0E-04 Twin Oaks Power One

7030 U2 U2 Mercury 6.9E-06 lb/MMBtu 6.9E-05 Monticello 6147 2 2 Mercury 6.7E-06 lb/MMBtu 8.0E-05 Coal Creek 6030 1 Unit 1b Mercury 1.1E-05 lb/MMBtu 1.1E-04 San Miguel 6183 SM-1 SMEC1 Mercury 1.1E-05 lb/MMBtu 1.2E-04 San Miguel 6183 SM-1 SMEC1 Mercury 1.9E-05 lb/MMBtu 2.2E-04 Leland Olds 2817 1 1 Mercury 5.3E-03 lb/MMBtu 6.0E-02 Antelope Valley Station

6469 B1 AVS Unit 1 Mercury 5.1E-03 lb/MMBtu 7.1E-02Final_Output_

EF_Units From Report_id Hg_Min_lb_per_ MWh_Rank New Souce

MACT Floor Test_ID New Source MACT Floor

QA Calc (lb/MWh) lb/MW Part II 2-Source Emissions Survey, Stacks 1 2_Hg_T2 1.1E-05 lb/MW Part II 3-Source Emissions Survey, Inlet Ducts and Stacks 1 2_Hg_T3 1.3E-05 lb/MW ETTS 1 Invalid Test Avg 5.8E-05 (One Valid Run) lb/MW ETTS 2 lb/MW Part II 1-Compliance Test Report 2 lb/MW Part II 2-Source Emissions Survey, Stacks 3 lb/MW Part II 3-Source Emissions Survey, Inlet Ducts and Stacks 3 lb/MW ETTS 3 lb/MW Part II 06-05 4 lb/MW ERT 4 lb/MW Part II 06-05 5 lb/MW ERT 5 lb/MW ERT 6 lb/MW ETTS 7 lb/MW Part II DryFining Baseline\01 8 lb/MW Part II Demonstration of Sorbent Enhancement 9 lb/MW Part II Assessment of Mercury Control Options 9 lb/MW Part II LOS Mercury Alstom Demo 10 lb/MW Part II AVS Mercury Demo 11New Source Floor

Agreement of QA Calc Run 1

(lb/MWh) Run 2

(lb/MWh) Run 3

(lb/MWh) 100.00% 7.0E-06 9.9E-06 1.5E-05 100.00% 9.8E-06 1.3E-05 1.6E-05 100.30% 5.8E-05 Failed Spike Recovery Failed Spike Recovery

Run 1 is a Far Outlier based on Tukey's tests for near and far outliers.Run 1 is a Far Outlier based on Tukey's tests for near and far outliers.Runs 52071 Sandow 5B 1 7.05E-06 2 9.85E-06 3 1.53E-05 4 9.84E-06 5 1.32E-05 6 1.59E-05 7 5.84E-05 Outlier 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Emissions

0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05 7.00E-05

25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 7.00 Kurtosis 6.30 SE Kurtosis 1.00 Result Kurtosis **non normal** Skewness 2.47 SE Skewness 0.79 RESULT Result Skewness **non normal** non normal ### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectivelyni = number test runs = 7 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1 n =Total # test runs = 7

6
This step is for QC only: means 1.85E-05

1.85E-05 Mean = 

##  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 

2 1 1 s n m        1 1 N i i df n

         

## 

2 1 1

s n m        1 1 i n i ik

k i x x n   

##  1

N i i n

 

###  N

Pooled Variance = 3.20E-10

m= number future runs = 3 Term1 0.48
Term2 1.52E-10

Squared root Term2 1.23E-05



##  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 

2 1 1 s n m        1 1 N i i df n

         

## 

2 1 1

s n m        1 1 i n i ik

k i x x n   

##  1

N i i n

 

###  NNOTE: 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 = 3.14

7 0.38

99.13% Conclusion interval of differences for t-values used 0.453 5.86 Adjusted UPL for skewness and kurtosis for all sample size 9.08E-05 lb/MWh 9.08E-02 lb/GWhr Confidence Level is not 99%, Go to recalculate tab to fix the Confidence Level
new t-statistic from tab Recalculate t-statistic 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  

 df t 2 1 1 s n m       

2
, 1 1 df p X t s n m

   

    9.08E-02 lb/GWhr df t 2 1 1 s n m       

2
, 1 1 df p X t s n m

   

   (When Outlier is included, data distribution is non-normal)

0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7

### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectivelyPart II Data Part II Data Part II Data Part II Data Part II Data

Comments Use Run in UPL Calc? Grade Key Facility_Nam e BDL Yes A 2_Hg_T2_R1 BDL Yes A 2_Hg_T2_R2 BDL Yes B 2_Hg_T2_R3 Yes A 2_Hg_T3_R1 Yes A 2_Hg_T3_R2 Yes A 2_Hg_T3_R3 Part III Part III Part III Part III Part III

Used only a 3-point traverse. Ask EPA to determine if Run 1 should be included in UPL run analysis (yields non-normal data distribution and requires beyond the floor standard); or should run be excluded as an outlier. in UPL run analyses only. Yes C 2_Hg_T1_R1 Used only a 3-point traverse. Failed spike recovery. Do not use in any UPL analyses. No Exclude 2_Hg_T1_R2 Used only a 3-point traverse. Failed spike recovery. Do not use in any UPL analyses. No Exclude 2_Hg_T1_R3 RTIConversions 1 ppm SO2= 1.66E-07 lb/scf 1 ppm Hg= 5.20E-07 lb/scf 1000 mg= 1 g 1000000 ug= 1 g 454g= 1 lb 1 m^3= 35.3145ft^3 200.59 Hg molecular weight 64.06SO2 molecular weight Table 19-2 Fd Fw Fc Fuel Type dscf/MMBtu wscf/MMBtu scf/MMBtu Coal Anthracite 10100 10540 1970 Bituminous 9780 10640 1800 Lignite 9860 11950 1910 Oil 9190 10320 1420 Subbituminous 9820 1840 Petroleum Coke 9830 1850 QA CalcsColumn 1 Column 2 Column 3 Column 4 Column 5 Column Source==> 43 43 43 Test Avgs for UPL R1 R2 R3 Test_IDs lb/MW lb/MW lb/MW lb/MW 2_Hg_T2 1.1E-05 7.0E-06 9.9E-06 1.5E-05 2_Hg_T3 1.3E-05 9.8E-06 1.3E-05 1.6E-05 2_Hg_T1 Invalid Test Avg 5.8E-05 Invalid Run Invalid RunAnthracite Anthracite Bituminous Bituminous Lignite Lignite Oil Oil Pet Coke Pet Coke Subbituminous Sub Bituminous Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data
Facility_ID Boiler_ID Fuel1 Used Fuel1 Fraction Fuel2 Used Fuel2 Fractio n "Fuel Used/Fraction" Data Source 52071 Lignite 100% Part II Submittal 52071 Lignite 100% Part II Submittal 52071 Lignite 100% Part II Submittal 52071 Lignite 100% Part II Submittal 52071 Lignite 100% Part II Submittal 52071 Lignite 100% Part II Submittal Part III Part III Part III Part III Part III Part III Part III

52071 Lignite ETTS 52071 Lignite ETTS 52071 Lignite ETTS As SubmittedColumn 6 Column 7 Column 8 Column 9 ####### ##### Column 12

43 43 43 43 43 43 R4 R5 R6 R7 R8 R9 R10 lb/MW lb/MW lb/MW lb/MW lb/MW lb/MWfPM 30B μg/dscm No μg/dscm 2.205E-09

Se M5 lb/dscf Yes ppm 2.205E-12 Pb M5B ppm Hg M29 ppm @ 3% O2 HCl OTM27 ppm @ 7% O2 SO2 26A

M6C Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data

Pollutant Reference

Method Concentrati

on Concentr

ation UOM NonDetect? MDL MDL

UOM HAP Conversion Value to lbs Hg M29 Yes 0.879 μg/dscm 2.20E-09 Hg M29 Yes 1.253 μg/dscm 2.20E-09 Hg M29 Yes 2.033 μg/dscm 2.20E-09 Hg 30B 1.238 μg/dscm No 2.20E-09 Hg 30B 1.657 μg/dscm No 2.20E-09 Hg 30B 2.018 μg/dscm No 2.20E-09 Part III Part III Part III Part III Part III Part III Part III Part III

Hg 30B 7.62 μg/dscm No 2.20E-09 Hg 30B 8.63 μg/dscm No 2.20E-09 Hg 30B 6.69 μg/dscm No 2.20E-09Column 13 Column 14 Column 15 ####### ######## ####### ######## Column 20

Test_IDs 2_Hg_T2 2_Hg_T3 2_Hg_T1lbs/μg 0.0283 cm/cf

lb/mg lb/scf

Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data HAP Conversion UOM Volume

Conversion Value to cubic feet Reserve Reserve Reserve Volume Conversion UOM Bws

(percent) %O2d lbs/μg 0.0283 cm/cf 17.46 4.6 lbs/μg 0.0283 cm/cf 17.53 4.6 lbs/μg 0.0283 cm/cf 17.85 4.8 lbs/μg 0.0283 cm/cf 18.13 4.2 lbs/μg 0.0283 cm/cf 4.55 lbs/μg 0.0283 cm/cf 4.55 Part III Part III Part III Part III Part III Part III Part III Part III

lbs/μg 0.0283 cm/cf 19.67 3.27 lbs/μg 0.0283 cm/cf 20.47 3.31 lbs/μg 0.0283 cm/cf 19.76 3.39 RTI-QAColumn 21 Column 22 Column 23 Column 24 Column 25 Column 26 Column 27 ########lb/hr lb/MMBtu lb/MWh

Part II Data Part II DataPart II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II DataPart II Data

%CO2d Emissio

n Rate Value Emission Rate UOM Emission

Rate MDL Emission

Factor Value Emission

Factor UOM Emission Factor MDL Fd Fw Fc 15.2 lb/hr 0.0021 7.05E-06 lb/MWh 9860 11950 1910 15.6 lb/hr 0.003 9.85E-06 lb/MWh 9860 11950 1910 15.3 lb/hr 0.0048 1.53E-05 lb/MWh 9860 11950 1910 15.6 3.0E-03 lb/hr 9.84E-06 lb/MWh 9860 11950 1910 15.425 4.0E-03 lb/hr 1.32E-05 lb/MWh 9860 11950 1910 15.425 5.0E-03 lb/hr 1.59E-05 lb/MWh 9860 11950 1910 Part III Part III Part III Part III Part III Part III Part III Part III Part III Part III

16.22 5.57E-06 lb/MMBtu 9860 11950 1910 15.94 6.31E-06 lb/MMBtu 9860 11950 1910 16.08 4.92E-06 lb/MMBtu 9860 11950 1910

As Submitted####### ###### ######## ######## ######## ######## Column 35 ######## ####### #####From Min Test Sheet

Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Final for UPL Calculated conc lb/dscf 19-1

lb/MW %diff

19-1 19-6

lb/MW %diff

19-6 lb/MMBtu Value Used for UPL Calcs Unit Heat

Rate (MMBtu/MW h) 5.48E-11 7.26E-06 3.0% 7.22E-06 2.4% 6.93E-07 10.5 7.82E-11 1.04E-05 4.9% 1.00E-05 1.8% 9.88E-07 10.5 1.27E-10 1.70E-05 10.2% 1.66E-05 7.9% 1.62E-06 10.5 7.72E-11 9.99E-06 1.5% 9.91E-06 0.8% 9.53E-07 10.5 1.03E-10 1.37E-05 3.0% 1.34E-05 1.3% 1.30E-06 10.5 1.26E-10 1.66E-05 4.6% 1.63E-05 2.9% 1.59E-06 10.5 Part III Part III Part III Part III Part III Part III Part III

lb/MMBtu lb/MMBtu lb/MMBtu 4.75E-10 5.56E-06 -0.26% 5.60E-06 0.48% 5.57E-06 10.5 5.38E-10 6.31E-06 -0.06% 6.45E-06 2.22% 6.31E-06 10.5 4.17E-10 4.91E-06 -0.19% 4.96E-06 0.74% 4.92E-06 10.5

RTI-QA######## Column 40 Column 41 Column 42 Column 43 Column 44 Column 45original database export===>

original database export===> original database export===> original database export===> original database export===> original database export===> original database export===> original database export===> Final for UPL original database export===> lb/MWh Value Used in UPL Calcs original database export===> Key RTI Calculated QA lb/MMBtu Run Grade Submitted

Value 7.05E-06 2_Hg_T2_R1 6.89E-07A 0.0021 9.85E-06 2_Hg_T2_R2 9.57E-07A 0.003 1.53E-05 2_Hg_T2_R3 1.58E-06 B 0.0048 9.84E-06 2_Hg_T3_R1 9.53E-07A 0.003 1.32E-05 2_Hg_T3_R2 1.28E-06A 0.004 1.59E-05 2_Hg_T3_R3 1.55E-06A 0.005 Part III Part III Part III Part III Part III Part III RTI Calculated QA lb/MMBtu Run Grade Submitted Value lb/MWh Value Used in UPL Calcs Key

5.8E-05 100.3% 2_Hg_T1_R1 5.56E-06A 5.57E-06 6.6E-05original database export===>

2_Hg_T1_R2 6.31E-06 Exclude 6.31E-06 5.2E-05original database export===>

2_Hg_T1_R3 4.91E-06 Exclude 4.92E-06 original database export===> original database export===>Column 46 Column 47Submitted

Value UOM Use ? Comment facility_id configurat ion_id report_id sampling_

port_id order pollutant_ id pound per hour (lb/hr)

Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130 pound per hour (lb/hr)

Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130 pound per hour (lb/hr)

Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130 pound per hour (lb/hr)

Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130 pound per hour (lb/hr)

Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130 pound per hour (lb/hr)

Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130 Part III Part III Part III Part III Part III Part III Part III Part III Part III Submitted Value UOM Use ? Comment facility_id configurat ion_id submittal_

id Hg_type run_numb

er start_date lb/MMBtu Yes Used only a 3-point traverse. 52071 5B 832 30B 1 40246.72 lb/MMBtu No Used only a 3-point traverse. Failed spike recovery. 52071 5B 832 30B 2 40247.55 lb/MMBtu No Used only a 3-point traverse. Failed spike recovery. 52071 5B 832 30B 3 40248.517.05E-06 9.85E-06 1.53E-05

pollutant_ name run_id start_datetim e end_datetime unit_load fuel_flow_ rate fuel_heati ng_value fg_moistu

re_conten t fg_oxygen _content mercury 10 02-Nov-09 02-Nov-09 298.08 341180 7830 17.46 4.6 mercury 2 31-Oct-09 01-Nov-09 304.55 357154 8232 17.53 4.6 mercury 6 01-Nov-09 01-Nov-09 314.07 356244 7905 17.85 4.8 mercury 19 11/1/2009 11/1/2009 305 18.13 4.2 mercury 20 11/1/2009 11/1/2009 302 mercury 21 11/2/2009 11/2/2009 315 Part III Part III Part III Part III Part III Part III Part III Part III Part III
end_date measured

_value_Hg measured_val

ue_Hg0 measured_val

ue_Hg+2 measured

_value_Hg P basis test_meth

od data_flag method_c

omments μg/dscm μg/dscm μg/dscm μg/dscm 40246.8 7.62 Dry U.S. EPA Method 30B ADL Based on values in previous test, only a 3 point traverse was required.Spike recovery for Runs 2 & 3 were 168 & 136%, respectively. According to the method, these runs are invalid because the recovery was >115%. 40247.63 8.63 Dry U.S. EPA Method 30B ADL 40248.59 6.69 Dry U.S. EPA Method 30B ADLfg_CO2_c

ontent fg_temper

ature test_meth

od conc conc_unit

s conc_ND conc_MDL conc2 conc2_uni ts 15.2 217 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-037 microgram per dry standard cubic meter (μg/dscm)

TRUE 0.879 15.6 213 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-029 microgram per dry standard cubic meter (μg/dscm)

TRUE 1.253 15.3 215 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-033 microgram per dry standard cubic meter (μg/dscm)

TRUE 2.033 15.6 224 EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B 1.238 microgram per dry standard cubic meter (μg/dscm)

FALSE EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B

1.657 microgram per dry standard cubic meter (μg/dscm)

FALSE EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B

2.018 microgram per dry standard cubic meter (μg/dscm)

FALSE Part III Part III Part III Part III Part III Part III Part III Part III Part III emission_ rate steam_ou

tput heat_inpu

t fg_moistu

re_conten t fg_CO2_c

ontent 19-6 %Diff

19-6 Ffactor

calculated 19-9 lb/MMBtu 5.57E-06 1977 2484 19.67 16.22 5.6E-06 -0.00479 4.5E-06 6.31E-06 1993 2487 20.47 15.94 6.45E-06 -0.02216 5.13E-06 4.92E-06 1990 2480 19.76 16.08 4.96E-06 -0.00739 3.98E-06emission_

rate emission_ rate_units emission_

rate_ND emission_

rate_MDL Calculated

lb/MWh at Prom Boiler

Heat Rate (lb/MMBt u)/MWh Calculated

lb/MMBtu at Prom Calculated

conc lb/dscf 19-1 lb/MMBtu pound per hour (lb/hr)

TRUE 0.0021 pound per hour (lb/hr)

TRUE 0.003 pound per hour (lb/hr)

TRUE 0.0048 0.003 pound per hour (lb/hr)

FALSE 0.004 pound per hour (lb/hr)

FALSE 0.005 pound per hour (lb/hr)

FALSE Part III Part III Part III Part III Part III Part III Part III Part III Part III
%Diff 19-9 fg_oxygen

_content 19-1 %Diff

19-1 Ffactor

calculated 19-5 %Diff

19-5 fg_flow_r

ate fg_temper

ature 0.192856 3.27 5.56E-06 0.002617 4.46E-06 0.198803 579345 192 0.187079 3.31 6.31E-06 0.000622 5.02E-06 0.205194563279 194 0.191667 3.39 4.91E-06 0.001865 3.94E-06 0.199097 580823 193if use Run

1 O2% for all runs % Diff

19-1 19-6 lb/MMBtu if use Run

1 O2% for all runs % Diff

19-6 emission_

factor emission_ factor_uni ts emission_ factor_ND emission_

factor_M DL FALSE FALSE FALSE FALSE FALSE FALSE Part III Part III Part III Part III Part III Part III Part III Part III Part III fuel1_inp ut_rate fuel1_typ

e fuel1_rate

_units fuel1_met

hod fuel2_inp

ut_rate fuel2_typ

e fuel2_rate

_units fuel2_met

hod fuel3_inp

ut_rate 5544 lignite ton/day gas fuel meter 5616 lignite ton/day gas fuel meter 5592 lignite ton/day gas fuel meterrun_pollut

ant_com ment Fuel1 Fuel1%Btu Fuel2 Fuel2%Btu QA Comment

Part III Part III Part III Part III Part III Part III Part III Part III Part III fuel3_typ e fuel3_rate

_units fuel3_met

hod fuel4_inp

ut_rate fuel4_typ

e fuel4_rate

_units fuel4_met

hod other_info

rmation exclude_d

ata FALSE FALSE FALSEPart III Part III Part III Part III Part III

Paired Trap RD % (<10%) Spike

Recovery % (bet. 85 and 115%) QA

Comment Questions

for EPA 0.797937 108 Used only a 3-point traverse. 6.427881 168 Used only a 3-point traverse. Failed spike recovery. Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one unit. 3.847204 136 Used only a 3-point traverse. Failed spike recovery. Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one unit.average of test runs 1.07E-05 1.30E-05

Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one unit. Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one unit.Hg Runs 52071 Sandow 5B 1 7.05E-06 2 9.84E-06 3 9.85E-06 4 1.32E-05 5 1.53E-05 6 1.59E-05 **Outlier Removed (This was a single run test**  **because 2 of 3 runs failed spike**  **recovery).==>**

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 6.00 Kurtosis -1.69 SE Kurtosis 1.03 Result Kurtosis normal Skewness -0.14 SE Skewness 0.85 Result Skewness normal ni = number test Runs = 6 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test Runs = 6 Emissions (lb/MW)

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

1 N i i n

  N

090000648116d066.xlsx UPL Pooled Template_noOutlier 11/19/20125

This step is for QC only: means 1.2E-05

1.2E-05 Mean =
Pooled Variance = 1.2E-11

m= number future Runs = 3 Term1 0.50
Term2 6.1E-12
Squared root Term2 2.5E-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 = 3.36 UPL POOLED VARIANCE = 2.02E-05 lb/MWh 2.02E-02 lb/GWh 

###  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 

2 1 1 s n m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i ik

k i x x n   

### 

1 i i n

 

2
, 1 1 df p X t s n m    

   

090000648116d066.xlsx UPL Pooled Template_noOutlier 11/19/2012Runs Parameters for Tukey's Test==> 25th Percentile75th Percentile IQ Dist. 1.5 x IQD 3 x IQD Subcategory 2_Hg 52071 Sandow 5B Outliers Removed 9.8E-06 1.6E-05 5.7E-06 8.6E-06 1.7E-05 2_Hg_T2_R1 7.05E-06 2_Hg_T3_R1 9.84E-06 2_Hg_T2_R2 9.85E-06 2_Hg_T3_R2 1.32E-05 2_Hg_T2_R3 1.53E-05 2_Hg_T3_R3 1.59E-05 2_Hg_T1_R1 5.8E-05 When the outlier was removed the curve became normal and lognormal. Far Outlier High (By Tukey's Test) 8 With the outlier in the set. The data are neither normal nor lognormal. 9 This was a single run test because 2 of 3 runs failed spike recovery 10 (See "Lignite_NonNormal_UPL" for the nonNormal UPL Analysis") 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 Log Data Sample Size 6.00 6.00 Kurtosis -1.69 -0.88 SE Kurtosis 1.03 1.03 Result Kurtosis normal normal Skewness -0.14 -0.51 SE Skewness 0.85 0.85 Result Skewness normal normal

Runs 1 -11.9 2 -11.5 3 -11.5 4 -11.2 5 -11.1 6 -11.1 7 8 9 Emissions (lb/MW)

Use for small sample sets that are Lognormal.

1. Take LN( ) of Raw Data y = 5E-06ln(x) + 6E-06 R2 = 0.909

0.00E+00 2.00E-06 4.00E-06 6.00E-06 8.00E-06 1.00E-05 1.20E-05 1.40E-05 1.60E-05 1.80E-05

0 1 2 3 4 5 6 7 Series1 Log. (Series1) 0.00E+00 1.00E-05 2.00E-05 3.00E-05 4.00E-05 5.00E-05 6.00E-05 7.00E-05

0 1 2 3 4 5 6 7 Series1 Series2

Outlier Removed

090000648116d066.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012

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 6 Average of Raw Data 1.19E-05 3. Calculate -11.38 2. Calculate sample size, n=

1 ˆ n i

i x n      2 2 1 ˆ 1

n i i x x n    

 

### 

090000648116d066.xlsx Lignite_Logormal UPL_noOutlier 11/19/20129.85E-02 4. Calculate

5. m 3 = number of future runs to be used to assess compliance with the standard 6. Calculate
4.82 3.22E-02
151.9 7. Calculate 0.58

9. substitue the values of in cell D7 and F7 respectively 93 is the Row number in Column AE with the first value > 0.99. 3.48 z value from "lognormal distribution 99%"

a- Calculate 1.20E-05 b- Calculate 1.43E-10
c- Calculate 1.04E-01 b- denominator first term a- Numerator first term

8. Go to tab distribution for 99 percentile "lognormal distribution 99%"
10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 11. In column B, go down to the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula  2 2 1 ˆ 1

n i i x x n    

 

### 

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

m e 

           



090000648116d066.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012 c- Calculate 1.04E-01
d-calculate 1.74E-02
e- Calculate 8.19E-06
f- Calculate UPL 2.15E-05 lb/MW 2.15E-02 lb/GW  2 2 1 ˆ 1

n i i x x n    

 

### 

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

m e 

           



090000648116d066.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012∆z z

0.101 151.9333 0.576654

-5 =f(a) 2956.683 1 -4.899 2690.581 2 -4.798 2441.923 3 -4.697 2209.966 4 -4.596 1993.982 5 -4.495 1793.258 6 -4.394 1607.098 7 -4.293 1434.82 8 -4.192 1275.759 9 -4.091 1129.264 10 -3.99 994.6991 11 -3.889 871.4463 12 -3.788 758.9013 13 -3.687 656.4756 14 -3.586 563.5963 15 -3.485 479.706 16 -3.384 404.2628 17 -3.283 336.7403 18 -3.182 276.6275 19 -3.081 223.4291 20 -2.98 176.665 21 -2.879 135.871 22 -2.778 100.598 23 -2.677 70.41261 24 -2.576 44.89694 25 -2.475 23.64854 26 -2.374 6.280479 27 -2.273 -7.57869 28 -2.172 -18.2849 29 -2.071 -26.1786 30 -1.97 -31.5847 31 -1.869 -34.8127 32 -1.768 -36.1565 33 -1.667 -35.8945 34 -1.566 -34.2898 35 -1.465 -31.5897 36 -1.364 -28.0262 37 -1.263 -23.8158 

#     2 1 3 

## 336 1 3 6 24 z z 

## z z    

    

     2z 

###  1z



## 38 -1.162 -19.1595

39 -1.061 -14.2426 40 -0.96 -9.23514 41 -0.859 -4.29162 42 -0.758 0.44903 43 -0.657 4.863335 44 -0.556 8.843327 45 -0.455 12.29654 46 -0.354 15.14599 47 -0.253 17.33021 48 -0.152 18.80323 49 -0.051 19.53455 50 0.05 19.50922 51 0.151 18.72773 52 0.252 17.2061 53 0.353 14.97586 54 0.454 12.08401 55 0.555 8.593056 56 0.656 4.581013 57 0.757 0.141383 58 0.858 -4.61683 59 0.959 -9.56912 60 1.06 -14.5755 61 1.161 -19.4804 62 1.262 -24.113 63 1.363 -28.2866 64 1.464 -31.7994 65 1.565 -34.4337 66 1.666 -35.9567 67 1.767 -36.1198 68 1.868 -34.659 69 1.969 -31.2949 70 2.07 -25.7325 71 2.171 -17.6613 72 2.272 -6.75539 73 2.373 7.326806 74 2.474 24.94222 75 2.575 46.46332 76 2.676 72.27807 77 2.777 102.7899 78 2.878 138.4179 79 2.979 179.5964 80 3.08 226.7754 81 3.181 280.4204 82 3.282 341.0124 83 3.383 409.0477 84 3.484 485.038385 3.585 569.5117

86 3.686 663.0109 87 3.787 766.0943 88 3.888 879.3358 89 3.989 1003.325 90 4.09 1138.666 91 4.191 1285.981 92 4.292 1445.904 93 4.393 1619.088 94 4.494 1806.199 95 4.595 2007.92 96 4.696 2224.948 97 4.797 2457.998 98 4.898 2707.797 99 4.999 2975.091distribution of Z Normal distribution

1.48672E-06 0.004395759 2.45094E-06 0.006594452 3.9995E-06 0.009766478 6.46025E-06 0.014276926 1.03291E-05 0.020596014 1.63472E-05 0.029314834 2.56093E-05 0.041156595 3.97118E-05 0.056979252 6.09553E-05 0.07776421 9.26132E-05 0.104584692 0.000139285 0.138546642 0.000207351 0.180695003 0.000305546 0.231879187 0.000445674 0.292574045 0.000643469 0.36265682 0.000919619 0.441146806 0.001300942 0.525922553 0.001821704 0.613441086 0.002525035 0.698494025 0.003464389 0.774045163 0.004704958 0.831201439 0.006324913 0.859372041 0.008416337 0.846666498 0.011085658 0.780570136 0.014453386 0.648912863 0.018652949 0.441115078 0.023828414 0.149653847 0.030130931 -0.22835296 0.03771375 -0.689592056 0.046725789 -1.223215123 0.057303789 -1.809922867 0.069563239 -2.421682456 0.083588399 -3.022261044 0.099421884 -3.568700224 0.117054396 -4.013768631 0.136415346 -4.309318783 0.157365126 -4.410350549 0.179689839 -4.279462529 

#  

#   2 4 2 

## 336 6 24 z 

## z z   

    

     

#    2 1 3 

### ( ) 1 3 ( )

6 24 z z G 

### f z z z z             

## ( ) z 0.203099244 -3.891271841 0.227228529 -3.236319862 0.251644341 -2.323971263 0.275855243 -1.183865354 0.299326439 0.134406501 0.321498297 1.563553778 0.341807853 3.022718637 0.359712192 4.42321406 0.37471238 5.675389821 0.386376486 6.69598638 0.394360216 7.415244478 0.398423793 7.783031387 0.398443914 7.773328313 0.394419966 7.386589552 0.386474058 6.649713024 0.374844865 5.613624598 0.359875719 4.348741234 0.34199778 2.938806238 0.32170943 1.473754994 0.299553264 0.042351745 0.276092166 -1.274670276 0.25188591 -2.410326736 0.227469632 -3.315482119 0.203335281 -3.961062377 0.17991684 -4.33833279 0.157579839 -4.457401349 0.136615273 -4.344277538 0.117237788 -4.036930511 0.099587708 -3.580841496 0.083736272 -3.024534073 0.069693339 -2.415502093 0.05741676 -1.796853298 0.046822635 -1.20486494 0.037795734 -0.667523509 0.030199481 -0.204009302 0.023885038 0.175001038 0.018699163 0.466398593 0.014490659 0.673284083 0.011115369 0.80339742 0.008439746 0.867521063 0.006343145 0.878004905 0.004718997 0.847514951 0.003475077 0.788062225 0.002533081 0.710327622 0.001827693 0.623266057 0.001305351 0.533950936 0.000922829 0.4476074570.00064578 0.36777953 0.00044732 0.296577985 0.000306705 0.234965214 0.000208159 0.183041252 0.000139842 0.140306712 9.29928E-05 0.105887771 6.12113E-05 0.078716542 3.98826E-05 0.057666418 2.5722E-05 0.041646221 1.64209E-05 0.029659389 1.03767E-05 0.020835511 6.49066E-06 0.014441379 4.01874E-06 0.009878044 2.46298E-06 0.006669237 1.49417E-06 0.004445294Absolute values

0.004395759 0.000221986 0.006594452 0.00066604 0.009766478 0.000986414 0.014276926 0.00144197 0.020596014 0.002080197 0.029314834 0.002960798 0.041156595 0.004156816 0.056979252 0.005754904 0.07776421 0.007854185 0.104584692 0.010563054 0.138546642 0.013993211 0.180695003 0.018250195 0.231879187 0.023419798 0.292574045 0.029549979 0.36265682 0.036628339 0.441146806 0.044555827 0.525922553 0.053118178 0.613441086 0.06195755 0.698494025 0.070547897 0.774045163 0.078178561 0.831201439 0.083951345 0.859372041 0.086796576 0.846666498 0.085513316 0.780570136 0.078837584 0.648912863 0.065540199 0.441115078 0.044552623 0.149653847 0.015115039 0.22835296 0.023063649 0.689592056 0.069648798 1.223215123 0.123544727 1.809922867 0.18280221 2.421682456 0.244589928 3.022261044 0.305248365 3.568700224 0.360438723 4.013768631 0.405390632 4.309318783 0.435241197 4.410350549 0.445445405 4.279462529 0.432225715 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

           1 

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

i 

## x f a f x f b

 

##   

      

# 3.891271841 0.393018456 3.236319862 0.326868306 2.323971263 0.234721098 1.183865354 0.119570401 0.134406501 0.013575057 1.563553778 0.157918932 3.022718637 0.305294582 4.42321406 0.44674462 5.675389821 0.573214372 6.69598638 0.676294624 7.415244478 0.748939692 7.783031387 0.78608617 7.773328313 0.78510616 7.386589552 0.746045545 6.649713024 0.671621015 5.613624598 0.566976084 4.348741234 0.439222865 2.938806238 0.29681943 1.473754994 0.148849254 0.042351745 0.004277526 1.274670276 0.128741698 2.410326736 0.243443 3.315482119 0.334863694 3.961062377 0.4000673 4.33833279 0.438171612 4.457401349 0.450197536 4.344277538 0.438772031 4.036930511 0.407729982 3.580841496 0.361664991 3.024534073 0.305477941 2.415502093 0.243965711 1.796853298 0.181482183 1.20486494 0.121691359 0.667523509 0.067419874 0.204009302 0.02060494 0.175001038 0.017675105 0.466398593 0.047106258 0.673284083 0.068001692 0.80339742 0.081143139 0.867521063 0.087619627 0.878004905 0.088678495 0.847514951 0.08559901 0.788062225 0.079594285 0.710327622 0.07174309 0.623266057 0.062949872 0.533950936 0.053929044 0.447607457 0.0452083530.36777953 0.037145733 0.296577985 0.029954376 0.234965214 0.023731487 0.183041252 0.018487166 0.140306712 0.014170978 0.105887771 0.010694665 0.078716542 0.007950371 0.057666418 0.005824308 0.041646221 0.004206268 0.029659389 0.002995598 0.020835511 0.002104387 0.014441379 0.001458579 0.009878044 0.000997682 0.006669237 0.000673593 0.004445294 0.000224487Cumulative normalization 0.000221986 1.2692E-05 9 0.000888025 5.07729E-05 10 0.00187444 0.000107171 11 0.003316409 0.000189616 12 0.005396607 0.000308551 13 0.008357405 0.000477835 14 0.012514221 0.000715501 15 0.018269125 0.001044537 16 0.026123311 0.001493601 17 0.036686365 0.002097543 18 0.050679575 0.002897605 19 0.068929771 0.003941061 20 0.092349569 0.005280088 21 0.121899547 0.006969608 22 0.158527886 0.009063834 23 0.203083713 0.011611314 24 0.256201891 0.014648347 25 0.318159441 0.01819077 26 0.388707337 0.022224347 27 0.466885899 0.026694208 28 0.550837244 0.031494127 29 0.63763382 0.036456722 30 0.723147137 0.041345947 31 0.80198472 0.045853487 32 0.86752492 0.049600749 33 0.912077543 0.052148046 34 0.927192581 0.053012248 35 0.95025623 0.054330913 36 1.019905028 0.058313084 37 1.143449755 0.065376755 38 1.326251965 0.075828474 39 1.570841893 0.089812907 40 1.876090258 0.107265486 41 2.236528981 0.127873575 42 2.641919613 0.15105179 43 3.07716081 0.175936711 44 3.522606215 0.201405058 45 3.95483193 0.226117569 46 

## ( ) 2 ( ) ( ) f a f x f b        4.347850386 0.248588405 47 4.674718692 0.267277105 48 4.90943979 0.280697287 49 5.029010191 0.287533726 50 5.042585247 0.28830988 51 5.200504179 0.297338897 52 5.505798761 0.314794119 53 5.952543381 0.340336749 54 6.525757753 0.373110288 55 7.202052377 0.411777443 56 7.95099207 0.45459808 57 8.73707824 0.499542567 58 9.522184399 0.544431023 59 10.26822994 0.587086187 60 10.93985096 0.625486127 61 11.50682704 0.657902993 62 11.94604991 0.683015566 63 12.24286934 0.699986221 64 12.39171859 0.708496679 65 12.39599612 0.708741247 66 12.52473782 0.716102055 67 12.76818082 0.730020912 68 13.10304451 0.749166748 69 13.50311181 0.772040602 70 13.94128342 0.797093069 71 14.39148096 0.82283312 72 14.83025299 0.847919916 73 15.23798297 0.871231884 74 15.59964796 0.891910085 75 15.9051259 0.90937579 76 16.14909162 0.923324533 77 16.3305738 0.933700779 78 16.45226516 0.940658484 79 16.51968503 0.944513216 80 16.54028997 0.945691304 81 16.55796508 0.946701879 82 16.60507133 0.949395179 83 16.67307303 0.953283177 84 16.75421617 0.957922537 85 16.84183579 0.962932191 86 16.93051429 0.968002385 87 17.0161133 0.972896509 88 17.09570758 0.977447314 89 17.16745067 0.981549226 90 17.23040055 0.985148386 91 17.28432959 0.988231779 92 17.32953794 0.990816568x 9317.36668368 0.992940375x 94 17.39663805 0.994653017x 95 17.42036954 0.996009865x 96 17.43885671 0.997066869x 97 17.45302768 0.997877095x 98 17.46372235 0.998488563 x 99 17.47167272 0.998943125x 100 17.47749703 0.99927613 x 101 17.4817033 0.999516624x 102 17.48469889 0.999687897x 103 17.48680328 0.999808215x 104 17.48826186 0.99989161x 105 17.48925954 0.999948652x 106 17.48993314 0.999987165x 107 17.49015762 1x 108 882.453626The UPL analysis for non-normal data distributions is included because inclusion of an outlier results in a non-normal data distribution.The UPL analysis for non-normal data distributions is included because inclusion of an outlier results in a non-normal data distribution.**Step 1. Identify size of sample set.**

Large: When N (total number of sources)≥20. For a large sample set can assume normal distribution based on the Central Limit Theorem; use the "**UPL Pooled Template**" . Small: When N (total number of sources) <20. For a small sample set the Central Limit Theorem cannot be used to support the normality assumption; move on to**Step 2**. **Step 2. Identify Normality of sample set.** Statistical tests of the kurtosis and skewness are used to evaluate the normality assumption. If you have three or less runs in your entire sample set, skip down to**Step 6**. If you have more than three runs in your entire sample set, start in the "**UPL Pooled Template**" and test for normality in cells B45:B51. If the Kurtosis and Skewness tests show that raw data is Normal, you may continue to If the Kurtosis and Skewness tests show that raw data is Not Normal, move on to**Step 3**. **Step 3. Transform raw data to lognormal data.** In the "**Lognormal Template**", take the LN() of the raw data in cells B55:L89 and test the normalityagain of the LNdata transformed in cells C45:C51. If Kurtosis and Skewness tests show that the LN-data transformed is Lognormal, continue to use the "**Lognormal** If Kurtosis and Skewness tests show that the LN-data transformed is Not Lognormal, move on to**Step 4**. **Step 4. A pply Skewness A** **djustment.** For data that is Not Normal and Not Lognormal, use the "**Skewness Adjustment Template**"**.** This is essentially the UPL Pooled Template with a Skewness Adjustment. Note: DO NOT not use this template if n<13 (n is number For data that is Not Normal and Not Lognormal with n<13, move on to**Step 5**. **Step 5.** See "**Step 5. Intructions**" tab **Step 6.** See "**Step 6. Intructions**"tab **Things to keep in mind:** Type of data affects m (number of future runs) that will be used in calculation, for example: If the average of three test runs is used to determine compliance; m=3 If 30-day averages are used to determine compliance; m=30 N = total number of sources n = number of individual runs df = degrees of freedom = n - 1 X = mean of data s2= pooled vaiance of data m = number of future runs tdf = t-statistic: t-distribution with df degrees of freedom at .99 confidence level**Step 5. Instructions**

For data sets that are Non Normal & Non Lognormal with n<13 1) Go to the "**Template_nonnormal**" tab. The adjustment starts in Row 150. Copy the kurtosis value (B41) and skewness value (B44) in the "**Template_nonnormal**" tab into cells H14 and H12 respectively of the "**Calculations for Template** " tab. 2) Cell E114 of the "**Template_nonnormal**" tab shows the actual confidence level (1 - 'Calculations for Template'!J139) after accounting for the skewness and kurtosis. If the confidence level is larger than 99.00% or lower than 99.00% then we need to adjust the t-statistic. 3) Row 115 of the "**Template_nonnormal**" tab uses the IF function in EXCEL to conclude about the need to adjust or not. If you don’t need to adjust (because the difference between the actual level and 99.00% is not that large) then you can use the t-statistic in cell H102 of the "**Template_nonnormal**" tab. If the difference is significant, then you need to adjust the t-statistic. 4) The “**Calculations_for_Template**” tab has the calculations for the confidence level for the first t-statistic in cell H102 from the“**Template_nonnormal**” tab. Input values for this tab are obtained from the “**Template_nonnormal**” tab, so you don’t need to worry about it. 5) To adjust the t-statistic go to the “**Recalculate t-stat**” tab. This tab starts with the t-statistic in cell H102 from the “**Template_nonnormal**” tab. If the confidence level from cell E114 of the "**Template_nonnormal**" tab was larger or lower THAN 99.00% you need to decrease or increase the t-statistic so you get closer to 99.00%. Cell G3 has the value of the increase also called "interval", the candidate t-values will be calculated as the t-value in cell H102 plus multiples of the interval, for example if the t-value is 2, and the interval is 0.3, then the candidate t-values are: 2.03, 2.06, 2.09, 3.12, etc. These are calculated automatically for you in row 5 after a value is placed in cell G3. The corresponding p-values and confidence levels are also calculated automatically for you. Cell G3, the “interval” can be manipulated. For example, start with 0.1 and notice what the values in row 9 are. If they are not 99.00% keep increaseing or decreasing the value in G3 until one of the values in row 9 is 99.00% and you get a "YES" value in row 11 in the same column. You may have to go out to several significant figures. Calculations are in the tabs: Recalculations1 to Recalculations6. You don’t need to worry about them, they

are basicallyduplicates of tab “Calculations_for_Template” and each one use a different value of the candidate tvalues in row 5. 6) Sometimes the actual significance level in cell E114 of tab “**Template_nonnormal**” is larger than 99%, then you have to use a negative interval (meaning that your value in cell G3, tab “**Recalculate t-stat**” has to be negative), for example instead of 0.389, it has to be -0.389 or another negative number. 7) Once you get the adjusted t-value from tab “**Recalculate t-stat**”, you copy the t-value and paste special the value into cell E119 in tab “**Template_nonnormal**”. 8) The template will calculate the UPL for you in cell E122.Runs 52071 Sandow 5B 1 7.0E-06 2 9.9E-06 3 1.5E-05 4 9.8E-06 5 1.3E-05 6 1.6E-05 7 5.8E-05 Outlier is included. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Emissions

26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 7.00 Kurtosis 6.30 SE Kurtosis 1.00 Result Kurtosis non normal Skewness 2.47 SE Skewness 0.79 RESULT Result Skewness non normal non normal ### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectivelyni = number test runs = 7 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1 n =Total # test runs = 7

6
This step is for QC only: means 1.85E-05

1.85E-05 Mean = 

##  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 

2 1 1 s n m        1 1 N i i df n

         

## 

2 1 1

s n m        1 1 i n i ik

k i x x n   

##  1

N i i n

 

###  N

Pooled Variance = 3.20E-10

m= number future runs = 3 Term1 0.4762
Term2 1.52E-10

Squared root Term2 1.23E-05



##  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 

2 1 1 s n m        1 1 N i i df n

         

## 

2 1 1

s n m        1 1 i n i ik

k i x x n   

##  1

N i i n

 

###  NNOTE: 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 = 3.142668

7 0.37792

99.13% Conclusion interval of differences for t-values used 0.453 5.860668403 Adjusted UPL for skewness and kurtosis for all sample size 9.08E-05 lb/MWh 9.08E-02 lb/GWhr Confidence Level is not 99%, Go to recalculate tab to fix the Confidence Level
new t-statistic from tab Recalculate t-statistic 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  

 df t 2 1 1 s n m       

2
, 1 1 df p X t s n m

   

    9.08E-02 lb/GWhr df t 2 1 1 s n m       

2
, 1 1 df p X t s n m

   

   Outlier is included.

### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectivelyIncomplete Beta Function approximation

where Inputs skewness= 2.47 x=u0= 0.377920268 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= b= 0.5 b= w= 0.303755 w= Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##  , x 

### I a b a a w a a

0 

### 1 1 , 2 2 u 

### n I       

w= 0.303755 w=
c.1.1. 0.003726024 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.010342442 d.1.3 c.1.4. 1.69624451 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 2.531296 d.1.6 c.1.7 4.00E-01 d.1.7 Calculations Calculations



### ( , ) ( ) 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   1 2 a w b  Result1: 2.48E-02 Result2:

1- Calculate 1.24E-02
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115
2.3. Calculate 49.02293 0 0 

### 1 1 1 ( ) , 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 n n  

### 2 1 6 2 nn  

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

##                              Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703
Result 6.09E-03 0 0 

### 1 1 1 ( ) , 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 n n  

### 2 1 6 2 nn  

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

##                             4-Calculate

4.1. Calculate 6.20E-03

4.2. Calculate 6.13E-03
4.3. Calculate 7.39E-04
Result: 8.12E-04 5- Calculate

3.93E-02 5.1. Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

       5.1. Calculate 4.80E-02 5.2. Calculate
5.3. Calculate 9.74E-03
5.4. Calculate 3.25E-03

Result -2.24E-03 Corrected probability is:: 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      4 a= 5

0.5 b= 0.5 0.303755 w= 0.303755 d.1.) Calculate e.1.) Calculate 

##    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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I       

0.303755 w= 0.303755

0.000278544 e.1.1. 1.6746E-05 5.999999999 e.1.2. 24 0.001047191 e.1.3 7.95225E-05 2.69624451 e.1.4. 3.69624451 11 e.1.5.
5.163843 e.1.6 5.38E-01 e.1.7 Calculations Calculations 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  

    9.53E-03 Result3:



#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

1 1 1 1 ( ) , ,

72 2 2 24 2 2

3 1 5 1 , , 24 2 2 72 2 2

n n P t I I n n         

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

0.00868 

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

1 1 1 1 ( ) , ,

72 2 2 24 2 2

3 1 5 1 , , 24 2 2 72 2 2

n n P t I I n n         

                     a= 5 b= 0.5 w= 0.303755 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        
2 

### 3 1 2 6 w a  

     w= 0.303755
f.1.1. 1.6746E-05 f.1.2. 24 f.1.3 7.95225E-05 f.1.4. 3.69624451 28 f.1.5. 28
8.707657 f.1.6 8.707657 6.77E-01 f.1.7 6.77E-01

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  

    1.88E-03 Result4: 1.88E-03a= 3 a= 4

b= 1 b= 1 w= 0.607511 w= 0.607511 h.1.) Calculate 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  

     w= 0.607511 w= 0.607511
g.1.1. 0.023864 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.061065 h.1.3 g.1.4. 0.696245 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 5.062592 h.1.6 g.1.7 1.60E+00 h.1.7 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        
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: 4.06E-02 Result6:0 1,1 2 u n I       

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  

     0.003509 6 0.012366 1.196245 11

10.32769 2.15E+00 0 1,1 2 u n I       

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  

    2.38E-023.142668 0.453 3.595668 4.048668 4.501668 4.954668 5.407668 5.860668 0.043638 0.03994 0.029781 0.020908 0.014444 0.009991 95.64% 96.01% 97.02% 97.91% 98.56% 99.00% NO NO NO NO NO YES 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 E119**Template_nonnormal** 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 skewness= 2.47 x=u0= 0.316976781 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 

### 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.23204 w=
c.1.1. 0.001751454 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.004953161 d.1.3 c.1.4. 1.767960465 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 1.933663 d.1.6 Calculations Calculations



### ( , ) ( ) 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 2.33E-01 d.1.7

Result1: 1.44E-02 Result2: 1- Calculate 7.19E-03
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 5.49E-03

4-Calculate 4.1. Calculate 3.60E-03

4.2. Calculate 2.51E-03
4.3. Calculate 2.23E-04
Result: 1.31E-03 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      2.28E-02 5.1. Calculate 1.96E-02 5.2. Calculate
5.3. Calculate 2.94E-03
5.4. Calculate 9.79E-04

Result 5.11E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      3.595668 u0= 0.316977

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.23204 w= 0.23204

0.0001004 e.1.1. 4.62259E-06 5.999999999 e.1.2. 24 0.00038311 e.1.3 2.22242E-05 2.767960465 e.1.4. 3.767960465 11 e.1.5.
3.944672 e.1.6 Calculations 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        3.14E-01 e.1.7
3.90E-03 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.04364a= 5 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         a= 5 b= 0.5 w= 0.23204
f.1.1. 4.62259E-06 f.1.2. 24 f.1.3 2.22242E-05 f.1.4. 3.767960465 28 f.1.5. 28
6.6518 f.1.6 6.6518

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        3.95E-01 f.1.7 3.95E-01
5.67E-04 Result4: 5.67E-04 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.464079 w= 0.464079
g.1.1. 0.01181 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.031419 h.1.3 g.1.4. 0.76796 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 3.867326 h.1.6 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 9.33E-01 h.1.7

Result5: 2.75E-02 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

0.001337 6 0.00486 1.26796 11

7.889344 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        1.26E+00
1.16E-02 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation
where Inputs skewness= 2.47 x=u0= 0.267956241 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.183019 w=
c.1.1. 0.000891254 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.002552562 d.1.3 c.1.4. 1.816980722 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 1.525161 d.1.6 Calculations Calculations



### ( , ) ( ) 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 1.45E-01 d.1.7

Result1: 8.36E-03 Result2: 1- Calculate 4.18E-03
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 4.08E-03

4-Calculate 4.1. Calculate 2.09E-03

4.2. Calculate 1.10E-03
4.3. Calculate 7.51E-05
Result: 1.07E-03 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      1.32E-02 5.1. Calculate 8.58E-03 5.2. Calculate
5.3. Calculate 9.90E-04
5.4. Calculate 3.30E-04

Result 5.32E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      4.048668 u0= 0.267956

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.183019 w= 0.183019

4.04003E-05 e.1.1. 1.46964E-06 5.999999999 e.1.2. 24 0.000155723 e.1.3 7.12506E-06 2.816980722 e.1.4. 3.816980722 11 e.1.5.
3.111328 e.1.6 Calculations 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        1.95E-01 e.1.7
1.70E-03 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.03994a= 5 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         a= 5 b= 0.5 w= 0.183019
f.1.1. 1.46964E-06 f.1.2. 24 f.1.3 7.12506E-06 f.1.4. 3.816980722 28 f.1.5. 28
5.246553 f.1.6 5.246553

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        2.46E-01 f.1.7 2.46E-01
1.91E-04 Result4: 1.91E-04 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.366039 w= 0.366039
g.1.1. 0.006228 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.017005 h.1.3 g.1.4. 0.816981 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 3.050321 h.1.6 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 5.81E-01 h.1.7

Result5: 1.77E-02 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

0.000559 6 0.002075 1.316981 11

6.222655 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        7.82E-01
5.70E-03 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation
where Inputs skewness= 2.47 x=u0= 0.228440731 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.148038 w=
c.1.1. 0.000484097 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.001398945 d.1.3 c.1.4. 1.851961645 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 1.233653 d.1.6 Calculations Calculations



### ( , ) ( ) 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 9.50E-02 d.1.7

Result1: 4.97E-03 Result2: 1- Calculate 2.48E-03
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 2.83E-03

4-Calculate 4.1. Calculate 1.24E-03

4.2. Calculate 5.11E-04
4.3. Calculate 2.79E-05
Result: 7.59E-04 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      7.87E-03 5.1. Calculate 4.00E-03 5.2. Calculate
5.3. Calculate 3.67E-04
5.4. Calculate 1.22E-04

Result 4.11E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      4.501668 u0= 0.228441

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.148038 w= 0.148038

1.7782E-05 e.1.1. 5.23852E-07 5.999999999 e.1.2. 24 6.90325E-05 e.1.3 2.55486E-06 2.851961645 e.1.4. 3.851961645 11 e.1.5.
2.516652 e.1.6 Calculations 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        1.28E-01 e.1.7
7.94E-04 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.02978a= 5 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         a= 5 b= 0.5 w= 0.148038
f.1.1. 5.23852E-07 f.1.2. 24 f.1.3 2.55486E-06 f.1.4. 3.851961645 28 f.1.5. 28
4.243766 f.1.6 4.243766

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        1.61E-01 f.1.7 1.61E-01
7.10E-05 Result4: 7.10E-05 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.296077 w= 0.296077
g.1.1. 0.00347 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.009652 h.1.3 g.1.4. 0.851962 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 2.467306 h.1.6 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 3.80E-01 h.1.7

Result5: 1.13E-02 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

0.000253 6 0.000953 1.351962 11

5.033304 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        5.11E-01
2.87E-03 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation
where Inputs skewness= 2.47 x=u0= 0.196407452 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.122206 w=
c.1.1. 0.000277615 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.000807556 d.1.3 c.1.4. 1.87779413 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 1.018382 d.1.6 Calculations Calculations



### ( , ) ( ) 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 6.47E-02 d.1.7

Result1: 3.04E-03 Result2: 1- Calculate 1.52E-03
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 1.93E-03

4-Calculate 4.1. Calculate 7.59E-04

4.2. Calculate 2.52E-04
4.3. Calculate 1.12E-05
Result: 5.18E-04 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      4.81E-03 5.1. Calculate 1.98E-03 5.2. Calculate
5.3. Calculate 1.48E-04
5.4. Calculate 4.94E-05

Result 2.93E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      4.954668 u0= 0.196407

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.122206 w= 0.122206

8.42918E-06 e.1.1. 2.05171E-07 5.999999999 e.1.2. 24 3.2896E-05 e.1.3 1.00502E-06 2.87779413 e.1.4. 3.87779413 11 e.1.5.
2.0775 e.1.6 Calculations 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        8.71E-02 e.1.7
3.92E-04 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.02091a= 5 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         a= 5 b= 0.5 w= 0.122206
f.1.1. 2.05171E-07 f.1.2. 24 f.1.3 1.00502E-06 f.1.4. 3.87779413 28 f.1.5. 28
3.503235 f.1.6 3.503235

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        1.10E-01 f.1.7 1.10E-01
2.86E-05 Result4: 2.86E-05 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.244412 w= 0.244412
g.1.1. 0.002028 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.005717 h.1.3 g.1.4. 0.877794 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 2.036764 h.1.6 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 2.59E-01 h.1.7

Result5: 7.29E-03 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

0.000122 6 0.000466 1.377794 11
4.155 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        3.48E-01
1.50E-03 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation
where Inputs skewness= 2.47 x=u0= 0.170247165 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.102589 w=
c.1.1. 0.000166657 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.000487214 d.1.3 c.1.4. 1.897410917 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 0.854909 d.1.6 Calculations Calculations



### ( , ) ( ) 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 4.56E-02 d.1.7

Result1: 1.91E-03 Result2: 1- Calculate 9.56E-04
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 1.31E-03

4-Calculate 4.1. Calculate 4.78E-04

4.2. Calculate 1.31E-04
4.3. Calculate 4.88E-06
Result: 3.52E-04 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      3.03E-03 5.1. Calculate 1.03E-03 5.2. Calculate
5.3. Calculate 6.43E-05
5.4. Calculate 2.14E-05

Result 2.04E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      5.407668 u0= 0.170247

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.102589 w= 0.102589

4.25218E-06 e.1.1. 8.69445E-08 5.999999999 e.1.2. 24 1.66609E-05 e.1.3 4.27308E-07 2.897410917 e.1.4. 3.897410917 11 e.1.5.
1.744014 e.1.6 Calculations 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        6.14E-02 e.1.7
2.04E-04 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.01444a= 5 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         a= 5 b= 0.5 w= 0.102589
f.1.1. 8.69445E-08 f.1.2. 24 f.1.3 4.27308E-07 f.1.4. 3.897410917 28 f.1.5. 28
2.940887 f.1.6 2.940887

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        7.72E-02 f.1.7 7.72E-02
1.24E-05 Result4: 1.24E-05 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.205178 w= 0.205178
g.1.1. 0.001235 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.003518 h.1.3 g.1.4. 0.897411 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 1.709818 h.1.6 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 1.82E-01 h.1.7

Result5: 4.77E-03 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

6.27E-05 6 0.000241 1.397411 11

3.488029 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        2.46E-01
8.13E-04 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    t-statistic=

Incomplete Beta Function approximation
where Inputs skewness= 2.47 x=u0= 0.148708341 kurtosis= 6.30 n= 7 t0=t= 3.142668403

a= 3 a= Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 1x w b x     

   

##   , x 

### I a b a a w a a         

0 

### 1 1 , 2 2 u 

### n I        2 1 0 1 1 u t n  



a= 3 a= b= 0.5 b= w= 0.087343 w=
c.1.1. 0.000104026 d.1.1. c.1.2. 2 d.1.2. c.1.3 0.000305294 d.1.3 c.1.4. 1.912657231 d.1.4. c.1.5. 2.666667 d.1.5.
c.1.6 0.727856 d.1.6 Calculations Calculations



### ( , ) ( ) 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 3.31E-02 d.1.7

Result1: 1.24E-03 Result2: 1- Calculate 6.19E-04
2- Calculate

2.1. Calculate 0.025131
2.2. Calculate 7.809115 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn  2.3. Calculate 49.02293
Result: 0.004003
3-Calculate

3.1. Calculate 0.301572
3.2. Calculate 0.326703 2 

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

### 3 1 2 6 w a  

    

0 0 

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

### n P t I 

      

3 0 0 

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

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 n n  

### 2 1 6 2 nn Result 9.02E-04

4-Calculate 4.1. Calculate 3.09E-04

4.2. Calculate 7.15E-05
4.3. Calculate 2.25E-06
Result: 2.40E-04 5- Calculate 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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    

      1.96E-03 5.1. Calculate 5.60E-04 5.2. Calculate
5.3. Calculate 2.96E-05
5.4. Calculate 9.87E-06

Result 1.42E-03 

#       4 0 0 0 

## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n  

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

0 

### 1 1 1

, 24 2 2 u 

### n n I         

##    0 

### 1 2 1 1

, 12 2 2

u 

### n n n

I n   

      

##    0 

### 4 1 3 1

, 24 2 2

u 

### n n n

I n   

      

##    

#    

#    

#   2 3/2 0 0

2 2

0 0



### 1 2 5 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   

      5.860668 u0= 0.148708

4 a= 5 e.1.) Calculate 

##   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 a b a a w a a

a a b b

w a w  

###    


           

           

            

0 1 1 , 2 2 u n I        4 a= 5 0.5 b= 0.5 0.087343 w= 0.087343

2.26149E-06 e.1.1. 3.93891E-08 5.999999999 e.1.2. 24 8.88841E-06 e.1.3 1.94085E-07 2.912657231 e.1.4. 3.912657231 11 e.1.5.
1.484827 e.1.6 Calculations 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        4.45E-02 e.1.7
1.11E-04 Result3: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



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

#   0 0 0



## 1 2 4 1 1 1 1 1 1 3 1 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u



## n n n n n n n n P t I I I n n                          0.00999a= 5 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         a= 5 b= 0.5 w= 0.087343
f.1.1. 3.93891E-08 f.1.2. 24 f.1.3 1.94085E-07 f.1.4. 3.912657231 28 f.1.5. 28
2.503826 f.1.6 2.503826

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        5.59E-02 f.1.7 5.59E-02
5.72E-06 Result4: 5.72E-06 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    a= 3 a= 4

g.1.) Calculate h.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         ( , ) ( ) 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= 3 a= 4 b= 1 b= 1 w= 0.174686 w= 0.174686
g.1.1. 0.00078 h.1.1. g.1.2. 2 h.1.2. g.1.3 0.002238 h.1.3 g.1.4. 0.912657 h.1.4. g.1.5. 2.666667 h.1.5.
g.1.6 1.455713 h.1.6 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 1.32E-01 h.1.7

Result5: 3.18E-03 Result6: 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

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

3.38E-05 6 0.00013 1.412657 11

2.969654 0 1,1 2 u n I       

3 2 5 3 1 2 3 2 3 a a a   
2 

### 3 11 1 2 6 3 w a a        1.78E-01
4.57E-04 2 

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

### 3 1 2 6 w a  

     2 

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

### 3 1 2 6 w a  

    **Emissions (lb/MW) Hg**  52071 **ORIS/Facility/Unit** Sandow **Runs** 5B

1 7.2E-06 2 1.0E-05 3 1.0E-05 4 1.3E-05 5 1.7E-05 6 1.6E-05 **Outlier Removed (This was a single run test because 2 of 3 runs failed**  **spike recovery).==>** 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 ni = number test Runs = 6
Total Number of sources = 1 1 N i i 

### n

 

#  Nn =Total # test Runs = 6

5
This step is for QC only: means 1.2E-05

1.2E-05 UPL Mean =
Pooled Variance = 1.4E-11 

# m= number future Runs = 3
Term1 0.5000
Term2 7.2E-12
Squared root Term2 2.7E-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 = 3.36 1 

### 1 N i i 

### df n

 

###         

# 

d f

t 1 i

i 

### n

 

# 



#  2 2 1 

## 1 1 n i i 

## s x X n  

##  

 

#  1 1 

## 1 i

n N ij i j 

## X x n   

##  

# 

1 1 n m



2 

### 1 1 s n m

 

    

d f 

### t 2 

### 1 1

s n m  

     1 

## 1 i n i ik

k i 

## x x n  

##  

# 

1 1

 UPL POOLED VARIANCE = 2.13E-05

Normal UPL 2.02E-05 Lognormal UPL 2.15E-05 Final UPL (lognormal distribution) 2.15E-05 2
, 

## 1 1 dfp 

## X t s n m        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)lb/MW 2.13E-02 lb/GWhr

lb/MW lb/MW 2.15E-02 lb/GWhr2
, 

## 1 1 df p 

## X t s n m

   

   2
, 

## 1 1 df p 

## X t s n m

   

   