# Y-1 Y.0 EXAMPLE CALCULATIONS—INITIAL CORRELATION TEST

This section presents several examples of initial correlation test calculations. For each of the

examples, a brief description of the correlation test scenario is provided, followed by a summary

and discussion of the results of the calculations. For the first example, all of the equations used

to evaluate the data and some sample calculations are presented. The calculations for all of the

examples were performed using the Correlation Te st Calculation Spreadsheet, which is described

in Section X. Appendix A includes the entire spreadsheet output for each of the examples.

Y.1 EXAMPLE 1: HOW DO I DEVELOP AN INITIAL CORRELATION?

Y.1.1 Correlation Test Scenario

A correlation test is performed on a boiler at Facility A. The boiler is equipped with an in situ

PM CEMS that responds to PM concentrations at actual conditions. The emission limit, when

converted to units that are consistent with the PM CEMS sampling conditions, is 20.0 mg/acm.

Twenty-four test runs are performed using a single Method 5 sampling train. The facility

determines that four of the test runs (Runs 7, 8, 9, and 18) were suspect because of problems

with the sampling train. Table Y-1 summarizes the correlation test data. Figure Y-1 shows a

plot of the data.

Table Y-1. Summary of Correlation Test Data for Facility A

Run no. PM CEMS response,

mA PM concentration,

mg/acm

1 4.90 2.1

2 7.60 2.9

3 7.80 5.1

4 9.20 4.8

5 10.10 2.9

6 11.06 6.3

7 13.40 0.2

8 16.10 0.6

9 15.80 0.3

10 12.00 8.4

(continued)

# Table Y-1. (continued)

Run no. PM CEMS response,

mA PM concentration,

mg/acm



# Y-2 0 2 4 6 8 10 12 14 16 18 20

0 5 10 15 20 25 30 35

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-1. Plot of the correlation test data for Facility A. 11 18.20 8.4

12 15.70 4.9

13 13.90 6.3

14 19.70 13.0

15 21.02 9.1

16 22.30 13.9

17 24.10 8.7

18 25.90 0.6

19 24.70 12.6

20 26.40 15.9

21 29.08 16.2

22 29.30 13.5

23 30.50 15.9

24 33.10 18.5

# Y-3 Must I use all of the test data to develop the correlation?

Section 8.6(3) of PS-11 requires a minimum of 15 valid test runs for developing the correlation,

and Section 8.6(3)(iii) allows you to discard up to 5 test runs without explanation. Additional

test runs can be discarded if the basis for rejecting the runs is specified in the manual test

method, in PS-11, or in the quality assurance plan for the source. In this example, an additional

nine test runs were performed, so it is not necessary to use all of the test data. Five of the runs

can be discarded without explanation. As indicat ed in Figure Y-1, the results from test Runs 7,

8, 9, and 18 appear to be outliers. If the four outliers are discarded, there are 20 runs remaining,

which satisfies the minimum requirement of 15 runs for developing the initial correlation. Note

that Section 8.6(3)(ii) of PS-11 still requires you to report all of the data, even for those runs that

are rejected.

How do I develop a linear correlation curve from the data?

To develop a linear correlation model, follow the procedures described in Section 12.3(1) of

PS-11. The form of the linear model is provided in Equation 11-3 of PS-11:

$ y b bx

= + 0 1



# where



# í = the predicted PM concentration

b

0 

# = the intercept for the correlation curve

b

1 

# = the slope of the correlation curve

x = the PM CEMS response value.

To begin, the mean PM CEMS response (x



# G ) and mean PM concentration (y



# G ) are calculated as

follows:



# x

n x ( . . . .... . ) .

i

i n



# = = ∑ + + ++ =

= 

# 1 1

20 4 9 7 6 7 8 331 1853

1



# y

n y ( . . . ....+ . ) .

i

i n



# = = + ++ = ∑

= 

# 1 1

20 21 2 9 51 185 9 47

1



# The variables S

xx 

# and S

xy 

# , which are presented in Equation 11-7 of PS-11, are calculated next as

follows:



# ( ) 

# ( ) 

# ( ) 

# ( ) 

# ( ) S xx

xx i

i n



# = − = − + − ++ − =

= 

# ∑ 2 22 2

1 

# 4 9 1853 7 6 1853 331 1853 1 442 2

. . . . . . ,. K



# ( ) 

# ( ) 

# ( ) 

# ( ) 

# ( ) 

# ( )



# ( ) 

# ( ) S x xy y

xy i i

i n



# = − − = − − + − − ++

− −= = 

# ∑ 4 9 18 53 2 1 9 47 7 6 18 53 2 9 9 47

331 1853 185 9 47 7852 1 

# . . .. . . . .

. . .. . K

# Y-4 The slope of the correlation curve (b

1 

# ) is determined by Equation 11-6 of PS-11:

b S

S xy

xx 1 

# 7852

1 442 2 0544

== = .

, . .



# The intercept of the correlation curve is determined using Equation 11-4 of PS-11:

b ybx

0 1 

# 9 47 0544 1853 0 621

=−⋅ =

− × =

− . (. . ) .



# The resulting equation for the correlation curve is as follows:



# í

=



# ! 0.621 + 0.544 x

How do I determine the linear correlation coefficient?

The linear correlation coefficient (r) is calculated using the parameters S

L 

# and S

y 

# . The parameter

S

L 

# is defined in Equation 11-9 as follows:



# ( ) S

n y y

L ii

i n



# =

− −

= 

# ∑ 1

2 2

1 

# $



# To calculate S

L 

# , the linear correlation equation must be used to determine the predicted PM

concentration ( 

# í ) for each test run. For example, for Run 3, the PM CEMS response is 7.8 and

the predicted PM concentration is



# ( ) $

. . .. y =− + = 0 621 0544 7 8 3626



# Table Y-2 shows the results of the calculations of 

# í . Using these values, the calculation for S

L 

# is

as follows:



# ( ) 

# ( ) 

# ( ) 

# ( ) 

# ( ) S

L 

# =

− − + − + − ++ − = 1

20 2 2 047 21 3517 2 9 3626 51 17 401 185 1910 2 22 2



# .... .. . . . K



# Table Y-2. Predicted PM Concentrations ( 

# í ) for Linear Correlation

Run no. PM CEMS

response, mA PM concentration,

mg/acm Predicted PM

concentration ( 

# í ) a



# 1 4.9 2.1 2.047

2 7.6 2.9 3.517

3 7.8 5.1 3.626

4 9.2 4.8 4.388

5 10.1 2.9 4.878

(continued)

# Table Y-2. (continued)

Run no. PM CEMS

response, mA PM concentration,

mg/acm Predicted PM

concentration ( 

# í ) a



# Y-5 6 11.06 6.3 5.401

10 12 8.4 5.913

11 18.2 8.4 9.289

12 15.7 4.9 7.928

13 13.9 6.3 6.947

14 19.7 13 10.105

15 21.02 9.1 10.824

16 22.3 13.9 11.521

17 24.1 8.7 12.501

19 24.7 12.6 12.828

20 26.4 15.9 13.753

21 29.08 16.2 15.213

22 29.3 13.5 15.332

23 30.5 15.9 15.986

24 33.1 18.5 17.401

a 

# Calculated using the correlation equation: 

# í = -0.621 + 0.544 x

.



# The parameter S

y 

# is calculated using Equation 11-15 of PS-11 as follows:



# ( ) ( )( )( ) ( )

S y y

n y i

i n



# = −

− = − + − + − ++ −

− = = 

# ∑ 2

1 2 22 2



# 1 21 9 47 2 9 9 47 51 9 47 185 9 47

20 1 5095 .. . . .. ..

. K



# With the values for S

L 

# and S

y 

# , the correlation coefficient is calculated using Equation 11-14 as

follows:



# ( )

( ) r S

S L

y 

# =− =− = 1 1 1910

5095 0 927 2

2 2

2 

# .

. .

The correlation coefficient (0.927) is greater than 0.85 and, therefore, satisfies PS-11.

Note: For data sets that show little to no correlation, the value for S

L 

# can be greater than S

y 

# . In

such cases, the correlation coefficient (r) calculated using Equation 11-14 is undefined because it

entails determining the square root of a negative number. Even though the correlation

coefficient cannot be determined, it can be assumed that the value for r does not meet the

criterion of 0.85 specified by PS-11.

# Y-6 How do I calculate the confidence interval half range percentage for the linear correlation?

The half range for the 95 percent confidence inte rval (CI) for the predicted PM concentration at

the mean value of x is determined using Equation 11-8 of PS-11:



# CI t S

n df, a L 

# = ⋅−

− 1 2 

# 1

The variable t df, 1-a/2 

# is obtained from a table of t -statistic values. The degrees of freedom (df)

equals n-2, and the value for the t statistic for this example is 2.101. With this value and the

value for S

L 

# calculated above, CI is calculated as follows:



# ( ) CI

= = 2101 191 1

20 0897

.. .

The emission limit in this example is 20.0 mg/acm. To determine CI as a percentage of the

emission limit, Equation 11-10 is used as follows:



# CI CI

EL % .

. . =× = × = 100% 0897

20 0 100% 4 49%

The resulting value for CI% is 4.49 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I calculate the tolerance interval half range percentage for the linear correlation?

Equation 11-11 of PS-11 is used to calculated the tolerance interval half range ( TI ) at the mean

value of x:

TI k S

T L 

# = ⋅



# The parameter k

T 

# is determined using Equation 11-12 of PS-11:

k uv

T n df



# = ⋅ ′



# The parameters u

n 

# N 

# and v

df 

# are determined from Table 1 of PS-11. In this example, n 

# N is the

number of test runs used for the correlation (20), u

n 

# N 

# = 1.177, and v

df 

# equals 1.384 for n-2 degrees

of freedom. With these values, k

T 

# and TI are calculated as follows:



# ( ) k T u

n v df . . . = ′ ⋅= = 1177 1384 1629



# ( ) TI k S

T L 

# =⋅= = 1629 191 3111

.. .



# With an emission limit of 20.0 mg/acm, the tolerance interval as a percentage of the emission

limit is determined using Equation 11-13 as follows:

# Y-7 

# TI TI

EL % .

. . =× = × = 100% 3111

20 0 100% 156%

The resulting value for TI% is 15.6 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for tolerance interval half range. In summary, the linear

correlation developed from the data satisfies all of the criteria specified in PS-11.

How do I develop a polynomial correlation curve from the data?

A polynomial correlation curve can be expressed by the following:



# $ y b bx bx =+ + 01 2 2



# The procedures for developing a polynomial correlation curve are described in Section 12.3(2) of

PS-11. The first step is to determine the values for the parameters S

1 

# through S

7 

# , as defined in

Equations 11-17 and 11-18 of PS-11. These calculations are summarized as follows:



# ( ) ( ) S x

i

i n

1

1 

# 4 9 7 6 7 8 331 370 7

= = + + ++ =

= 

# ∑

... . . K



# ( ) ( ) S x

i

i n

2 2 222 2

1 

# 4 9 7 6 7 8 331 8 312 = = + + ++ =

= 

# ∑

... . , K



# ( ) ( ) S x

i

i n

3 3

1 333 3 

# 4 9 7 6 7 8 331 208 377 = = + + ++ =

= 

# ∑

... . , K



# ( ) ( ) S x

i

i n

4 4 444 4

1 

# 4 9 7 6 7 8 331 5 574 977 = = + + ++ +

= 

# ∑

. . . . ,, K



# ( ) ( ) S y

i

i n

5

1 

# 21 2 9 51 185 189 4

= = + + ++ =

= 

# ∑

... . . K



# ( ) ( )( ) ( )( ) ( )( ) ( )( ) 

# ( ) S xy

i i

i n

6

1 

# 4 9 21 7 6 2 9 7 8 51 331 185 4 295

= = + + ++ =

= 

# ∑

. . .. .. . . , K



# ( ) ( ) ( ) 

# ( ) ( ) 

# ( ) ( ) 

# ( ) ( ) 

# ( ) S xy

i i

i n

7 2 22 2 2

1 

# 4 9 2 1 7 6 2 0 7 8 51 331 185 108 567 = = + + ++ =

= 

# ∑

. . .. .. . . , K

Next, the parameter det A must be calculated using Equation 11-22 of PS-11. For this example,

the calculation is as follows:

det A nS S S S S S S S S S n S S S S S S

=⋅ ⋅ − ⋅ ⋅ +

⋅ ⋅ − ⋅ ⋅ +

⋅ ⋅ − ⋅⋅ = 2 4 2 2 2 132 33 213 411 

# ( ) 

# ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

20 8 312 5 574 977 8 312 8 312 8 312 370 7 208 377 8 312 208 377 208 377 20

, ,, , , , . , , , , −+ − +



# ( ) ( ) 

# ( )( ) ( )( ) 8 312 370 7 208 377 5 574 977 370 7 370 7 2120 10 9



# , . , ,, . . . − =×



# The values for the correlation curve parameters b

0 

# , b

1 

# , and b

2 

# can now be determined using

Equations 11-19 to 11-21 of PS-11:

# Y-8 

# ( )

b SSS SSS SSS SSS SSS SSS

A 0 524 137 263 722 335 461



# = ⋅⋅ +⋅⋅ + ⋅⋅− ⋅⋅ − ⋅⋅− ⋅⋅

det



# ( ) 

# ( )( ) ( ) 

# ( )( ) ( )( )( )

=

× ++ − 1

212 10 189 8 312 5 574 977 370 7 208 377 108 567 208 377 4 295 208 377

9



# . [ , ,, . , , , , ,



# ( )( )( ) ( ) ( ) ( ) 

# ( ) ( ) ( )

108 567 8 312 8 312 208 377 208 377 189 5 574 977 4 295 370 7 , , , , , , , , .]

− −

= 0 420

.



# ( )

b nS S S S S S S S S S S S S n S S S

A 1 64 532 217 262 73 415



# = ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅− ⋅ ⋅

det



# ( ) 

# ( )( ) ( ) 

# ( )( ) ( ) ( ) 

# ( )

=

× ++ − 1

212 10 20 4 295 5 574 977 189 208 377 8 312 8 312 370 7 108 567

9



# . [ , ,, , , , . ,



# ( )( )( ) ( )( ) ( ) 

# ( ) ( )( ) 8 312 4 295 8 312 108 567 208 377 20 5 574 977 370 7 189

, , , , , ,, . ]

− −



# = 0 406

.



# ( )

b nS S S S S S S S S S S S S n S S S

A 2 2 7 162 513 2 25 36 711



# = ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅− ⋅ ⋅

det



# ( ) 

# ( )( ) ( ) 

# ( )( ) ( )( ) 

# ( )

=

× ++ − 1

212 10 20 8 312 108 567 370 7 4 295 8 312 189 370 7 208 377

9



# . [ , , ., , . ,



# ( )( ) ( ) 

# ( )( ) ( ) 

# ( ) ( )( )

8 312 8 312 189 208 377 4 295 20 108 567 370 7 370 7

, , , , , . .]

− −

= 0 00368

.



# The resulting correlation curve is as follows:



# í

= 0.420 + 0.406 x + 0.00368 x 2



# How do I determine the polynomial correlation coefficient?

The polynomial correlation coefficient (r) is calculated using the parameters S

P 

# and S

y 

# . The

parameter S

P 

# is defined in Equation 11-26 as follows:



# ( ) S

n y y

p ii

i n



# =

− −

= 

# ∑ 1

3 2

1 

# $

To calculate S

P 

# , the predicted PM concentrations must be determined by substituting the PM

CEMS response values into the correlation curve. For example, the predicted PM concentration

for Run 10 is

# Y-9 

# () () $

.. . . y =+ + = 0 420 0 406 12 0 00368 12 5819 2



# Table Y-3 shows the predicted PM concentrations for all of the 20 test runs used to develop the

correlation. Substitute these values into Equation 11-26 to calculate the value for S

p 

# :



# ( )( )( ) ( ) 

# ( ) S

p 

# =

− − + − + − ++ − = 1

20 3 2 497 21 3717 2 9 3809 51 17 884 185 1951 2 22 2



# .... .. . . . K

Table Y-3. Predicted PM Concentrations ( 

# í ) for Polynomial Correlation

Run no. PM CEMS

response, mA PM concentration,

mg/acm Predicted PM

concentration ( 

# í ) a



# 1 4.9 2.1 2.497

2 7.6 2.9 3.717

3 7.8 5.1 3.809

4 9.2 4.8 4.465

5 10.1 2.9 4.894

6 11.06 6.3 5.358

10 12 8.4 5.819

11 18.2 8.4 9.024

12 15.7 4.9 7.698

13 13.9 6.3 6.771

14 19.7 13 9.842

15 21.02 9.1 10.576

16 22.3 13.9 11.299

17 24.1 8.7 12.337

19 24.7 12.6 12.688

20 26.4 15.9 13.698

21 29.08 16.2 15.333

22 29.3 13.5 15.469

23 30.5 15.9 16.221

24 33.1 18.5 17.884

a 

# Calculated using the correlation equation: 

# í = 0.420 + 0.406x + 0.00368 x 2



# .



# The parameter S

y 

# was determined previously to be 5.095. As shown below, substituting S

y 

# and

S

P 

# into Equation 11-33 results in a correlation coefficient of 0.924, which satisfies the minimum

value of 0.85 specified by PS-11.



# ( )

( ) r S

S p

y 

# =− =− = 1 1 1951

5095 0 924 2

2 2

2 

# .

. .

# Y-10 How do I calculate the confidence interval half range percentage for the polynomial correlation?

The half range for the 95 percent CI for the predicted PM concentration must be determined at

the minimum value of the parameter delta ( 

# )

min 

# ). The expression for calculating 

# ) is provided in

Equation 11-25 of PS-11:



# ( ) ∆= + + + + + C Cx C C x Cx Cx

0 1 23 2

4 3

5 4 

# 22 2



# Before calculating 

# ) , it is first necessary to calculate the parameters C

0 

# to C

5 

# , which are defined

in Equation 11-23 of PS-11 as follows:



# ( )

C SS S

D 0 24 3 2



# = ⋅ −



# ( )

C SS SS

D 1 32 14



# = ⋅−⋅

C SS S

D 2 13 2 2



# = ⋅ −



# ( )

C nS S

D 3 4 2 2



# = −



# ( )

C S S nS

D 4 12 3



# = ⋅ −



# ( )

C nS S

D 5 2 1 2



# = −

The variable D in the denominator of these expressions is determined using Equation 11-24. For

this example, this calculation is as follows:



# ( ) ( ) ( ) D nS S S S S S S S S S S S = ⋅− + ⋅−⋅ + ⋅− 24 3 2

13 2 1 4 21 3 2 2



# ( )( ) ( ) 

# ( ) ( )( ) ( ) 

# ( ) 

# ( )

= −+ − + 20 8 312 5 574 977 208 377 370 7 208 377 8 312 370 7 5 574 577 2



# , ,, , . , , .,,



# ( ) 

# ( )( ) 

# ( ) 8 312 370 7 208 377 8 312 2120 10 2 9



# , ., , . − =×



# Substituting the values for D and for the variables S

1 

# to S

4 

# , which were defined previously, you

find the following coefficients for determining 

# ) :



# ( )( ) ( ) 

# ( )

C

0 3

9 

# 8 312 5 574 977 208 377

212 10 1375 , ,, ,

. . −

× =

# Y-11 

# ( ) ( ) ( ) 

# ( ) 

# ( )

C

1 9 

# 208 377 8 312 370 7 5 574 977

212 10 01578

= −

× = − , , .,,

. .



# ( ) 

# ( )( ) 

# ( )

C

2 2

9 

# 370 7 208 377 8 312

212 10 0 003846

= −

× = ., ,

. .



# ( ) 

# ( )( ) 

# ( )

C

3 2

9 

# 20 5 574 977 8 312

212 10 0 02001

= −

× = ,, ,

. .



# ( ) 

# ( ) ( ) 

# ( ) 

# ( )

C

4 9 

# 370 7 8 312 20 208 377

212 10 0 0005126

= −

× = − ., ,

. .



# ( ) 

# ( ) ( ) 

# ( )

C

5 2

9 5 

# 20 8 312 370 7

212 10 1360 10

= −

× = × − 

# , .

. .

When these values are substituted, Equation 11-25 becomes the following:



# ( ) ( )( ) 

# ( ) ( ) ∆= + − + + + − + ×

− 

# 1375 2 01578 2 0 003846 0 02001 2 0 0005126 136 10 2 3 54



# .. . . . . x x xx

= − +

− + × − 

# 1375 0 3155 0 0277 0 001025 136 10 2 3 54



# .. . . . x x x x



# Table Y-4 summarizes the results when the PM CEMS response values for each test run are

substituted into the above equation.

Table Y-4. Values for 

# )

Run

no. PM CEMS

response, mA PM concentration,

mg/acm x 2 

# x 3 

# x 4 

# )

1 4.9 2.1 24.01 118 576 0.3817

2 7.6 2.9 57.76 439 3,336 0.1727

3 7.8 5.1 60.84 475 3,702 0.1634

4 9.2 4.8 84.64 779 7,164 0.1162

5 10.1 2.9 102.01 1,030 10,406 0.0995

6 11.06 6.3 122.32 1,353 14,963 0.0906

10 12.0 8.4 144.00 1,728 20,736 0.0884

11 18.2 8.4 331.24 6,029 109,720 0.1202

12 15.7 4.9 246.49 3,870 60,757 0.1084

13 13.9 6.3 193.21 2,686 37,330 0.0960

14 19.7 13.0 388.09 7,645 150,614 0.1204

15 21.02 9.1 441.84 9,287 195,223 0.1162

16 22.3 13.9 497.29 11,090 247,297 0.1092

(continued)

# Table Y-4. (continued)

Run

no. PM CEMS

response, mA PM concentration,

mg/acm x 2 

# x 3 

# x 4 

# )



# Y-12 17 24.1 8.7 580.81 13,998 337,340 0.0985

19 24.7 12.6 610.09 15,069 372,210 0.0959

20 26.4 15.9 696.96 18,400 485,753 0.0960

21 29.08 16.2 845.65 24,591 715,118 0.1417

22 29.3 13.5 858.49 25,154 737,005 0.1492

23 30.5 15.9 930.25 28,373 865,365 0.2046

24 33.1 18.5 1,095.61 36,265 1,200,361 0.4312



# The minimum value for 

# ) is 0.0884 and corresponds to Run 10. Using this value for 

# )

min 

# , the

value for S

P 

# calculated previously, and the t statistic with n-3 degrees of freedom, you can

calculate the confidence interval half range using Equation 11-27.



# ( )( ) CI t S

df p 

# =⋅ = = ∆

min 

# .. . . 211 1951 0 0884 1223



# The confidence interval half range as a percen tage of the emission limit is calculated using

Equation 11-28 as follows:



# CI CI

EL % .

. =× = × = 100% 1224

20 100% 612%

The resulting value for CI% is 6.12 percent. Because this value is less than10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the polynomial correlation?

To calculate the tolerance interval half range, the parameter n 

# N must be calculated using

Equation 11-31 of PS-11.



# ′

== = n 1 1

0 0884 1131 ∆

min 

# . .

With n 

# N rounded to the nearest integer (11), the values for the variables u

n 

# N 

# and v

df 

# with n-3

degrees of freedom are taken from Table 1 of PS-11 and substituted into Equation 11-30 to

determine k

T 

# :



# ( ) ( ) k uv

T n df



# =⋅= =

′ 

# 1195 1400 1673

.. .



# Using this value for k

T 

# and the value for S

P 

# calculated previously, the tolerance interval half

range is calculated using Equation 11-29 as follows:

# Y-13 

# ( ) ( ) TI k S

T P 

# =⋅= = 1673 1951 3263

.. .



# Finally, to determine the tolerance interval as a percentage of the emission limit (TI%),

Equation 11-32 is used as follows:



# TI TI

EL % .

. . =× = × = 100% 3263

20 0 100% 16 3%

The resulting value for TI% is 16.3 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for tolerance interval half range. In summary, the polynomial

correlation developed from the data satisfies all of the criteria specified in PS-11.

How do I develop a logarithmic correlation curve from the data?

A logarithmic correlation curve is expressed as follows:



# ( ) $ y b b x b bx

=+ =+ ′

01 01 

# ln



# Developing a logarithmic correlation entails following the same procedures used to develop a

linear correlation. However, instead of using the PM CEMS response (x) values, the natural

logarithm of the PM CEMS values (x 

# N values) are used in the calculations. Table Y-5

summarizes the test data, including the corresponding values of x 

# N

.

Table Y-5. Data for Developing Logarithmic Correlation

Run no. PM CEMS

response, mA Logarithm of PM

CEMS response

x' = ln (x) PM concentration,

mg/acm

1 4.9 1.589 2.1

2 7.6 2.028 2.9

3 7.8 2.054 5.1

4 9.2 2.219 4.8

5 10.1 2.313 2.9

6 11.06 2.403 6.3

10 12 2.485 8.4

11 18.2 2.901 8.4

12 15.7 2.754 4.9

13 13.9 2.632 6.3

14 19.7 2.981 13

15 21.02 3.045 9.1

(continued)

# Table Y-5. (continued)

Run no. PM CEMS

response, mA Logarithm of PM

CEMS response

x' = ln (x) PM concentration,

mg/acm



# Y-14 16 22.3 3.105 13.9

17 24.1 3.182 8.7

19 24.7 3.207 12.6

20 26.4 3.273 15.9

21 29.08 3.370 16.2

22 29.3 3.378 13.5

23 30.5 3.418 15.9

24 33.1 3.500 18.5



# Following the same procedures described previously for developing a linear correlation curve,

you can calculate values for the parameters x 

# GN and y



# G , using Equation 11-5; S

x 

# N x 

# N 

# and S

x 

# N y 

# , using

Equation 11-7; b

1 

# , using Equation 11-6; and b

0 

# , using Equation 11-4:



# ( ) ′ = + + ++ = x 1

20 1589 2 028 2 054 3500 2 79

... . . K



# ( ) y = + + ++ = 1

20 21 2 9 51 185 9 47

... . . K



# ( )( )( ) ( ) S

x x ′ ′ 

# = − + − + − ++ − = 1589 2 79 2 028 2 79 2 054 2 79 350 2 79 574 222 2



# . . . . . . .. . K 

# ( )( ) ( ) ( )

( )( ) S

x y ′ ′ 

# = − − + − − ++

− −= 1589 2 79 21 9 47 2 028 2 79 2 9 9 47

350 2 79 185 9 47 47 8 . . .. . . . .

. . .. . K

b

1 

# 47 8

574 8 323

= = .

. .



# ( )( ) b

0 

# 9 47 8 323 2 79 1377

= − =−

... .



# The resulting equation for the logarithmic correlation curve is



# ( ) $

. . y x

=− + ⋅ 138 8 32 ln



# How do I determine the logarithmic correlation coefficient?

Determining the logarithmic correlation coefficient requires calculating the predicted PM

concentration ( 

# í ) for each test run, the parameter S

L 

# , and the parameter S

y 

# following the same

procedure used to calculate the linear correlation coefficient. The only difference is that the

predicted PM concentrations are calculated using the logarithmic correlation equation.

Table Y-6 shows the PM concentrations calculated using the logarithmic correlation coefficient.

# Y-15 Table Y-6. Predicted PM Concentrations for the Logarithmic Correlation

Run no. PM CEMS

response, mA Logarithm of PM

CEMS response

x' = ln (x) PM concentration,

mg/acm Predicted PM

concentration ( 

# í ) a



# 1 4.9 1.589 2.1 -0.539

2 7.6 2.028 2.9 3.114

3 7.8 2.054 5.1 3.330

4 9.2 2.219 4.8 4.704

5 10.1 2.313 2.9 5.481

6 11.06 2.403 6.3 6.237

10 12 2.485 8.4 6.916

11 18.2 2.901 8.4 10.382

12 15.7 2.754 4.9 9.152

13 13.9 2.632 6.3 8.139

14 19.7 2.981 13 11.041

15 21.02 3.045 9.1 11.581

16 22.3 3.105 13.9 12.073

17 24.1 3.182 8.7 12.719

19 24.7 3.207 12.6 12.924

20 26.4 3.273 15.9 13.478

21 29.08 3.370 16.2 14.283

22 29.3 3.378 13.5 14.345

23 30.5 3.418 15.9 14.679

24 33.1 3.500 18.5 15.360

a 

# Calculated using the logarithmic correlation equation: 

# í =



# ! 13.8 + 8.32



# @ ln ( x ).



# With the values for 

# í shown in Table Y-6, the calculation for S

L 

# is as follows:



# ( ) ( )( ) ( ) 

# ( ) S

L 

# =

− − − + − ++ − = 1

20 2 0539 21 3114 2 9 1536 185 2 305 22 2



# .... .. . K

The value for S

y

# is the same as was calculated for the linear correlation (5.095). Substituting the

values for S

L 

# and S

y 

# into Equation 11-14, the logarithmic correlation coefficient is



# ( )

( ) r =− = 1 2 305

5095 0892 2

2 

# .

. .

The logarithmic correlation coefficient (0.892) is greater than the minimum value specified by

PS-11 (0.85) and is acceptable.

# Y-16 How do I determine the confidence interval half range percentage for the logarithmic

correlation?

The confidence interval half range for the logarithmic correlation is determined using the same

procedures used for the linear correlation. Substituting the value for S

L 

# calculated above (2.305)

and the same value for the t -statistic (2.101) used for the linear correlation, Equation 11-8 yields



# ( )( ) CI

= = 2101 2 305 1

20 1083

.. .

The emission limit in this example is 20.0 mg/acm. To determine CI as a percentage of the

emission limit, Equation 11-10 is used as follows:



# 1083

20 100% 541% .

. × =

The resulting value for CI% is 5.41 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the logarithmic correlation?

The tolerance interval half range for the logarithmic correlation is determined using the same

procedures used for the linear correlation. Substituting the value for S

L 

# calculated above (2.305)

and the same value for the k

T 

# used for the linear correlation (1.629), Equation 11-11 yields



# ( ) ( ) TI

= = 1629 2 305 375

.. .



# The emission limit in this example is 20.0 mg/acm. To determine TI as a percentage of the

emission limit, Equation 11-13 is used as follows:



# 375

20 0 100% 188%

.

. . × =

The resulting value for TI% is 18.8 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I develop an exponential correlation curve from the data?

An exponential correlation curve is expressed as follows:

$ y be b x



# = 0 1



# To develop an exponential correlation curve, follow the same procedures used to develop a

linear correlation curve. However, instead of using the PM CEMS concentrations (y) values, use

# Y-17 the natural logarithm of the PM CEMS concentrations (y 

# N values) to perform the calculations.

Table Y-7 summarizes the test data, including the corresponding values of y 

# N

.

Table Y-7. Transformed Data for Developing Exponential Correlation

Run no. PM CEMS

response, mA PM concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y)

1 4.9 2.1 0.742

2 7.6 2.9 1.065

3 7.8 5.1 1.629

4 9.2 4.8 1.569

5 10.1 2.9 1.065

6 11.06 6.3 1.841

10 12 8.4 2.128

11 18.2 8.4 2.128

12 15.7 4.9 1.589

13 13.9 6.3 1.841

14 19.7 13 2.565

15 21.02 9.1 2.208

16 22.3 13.9 2.632

17 24.1 8.7 2.163

19 24.7 12.6 2.534

20 26.4 15.9 2.766

21 29.08 16.2 2.785

22 29.3 13.5 2.603

23 30.5 15.9 2.766

24 33.1 18.5 2.918



# Following the same procedures described previously for developing a linear correlation curve,

calculate values for the parameters x



# G and y 

# GN , using Equation 11-5; S

xx 

# and S

xy 

# N 

# , using

Equation 11-7; b

1 

# N , using Equation 11-6; and b

0 

# N , using Equation 11-4:



# ( )

x = + + ++ = 1

20 4 9 7 6 7 8 331 1853

... . . K



# ( ) ′ = + + ++ = y 1

20 0 742 1065 1629 2 918 2 077

... . . K



# ( )( ) ( ) 

# ( ) S

xx 

# = − + − ++ − = 4 9 1853 7 6 1853 331 1853 1 442 2 22 2



# . . . . . . ,. K



# ( )( ) ( )( )

( )( ) S

xy ′ 

# = − − + − − ++

− −= 

# (

) . .. . . .. .

. .. . . 4 9 1853 0 742 2 077 7 6 1853 1065 2 077

331 1853 2 918 2 077 9658 K

# Y-18 

# b

1 

# 9658

1442 2 0 0670

= = .

. .



# ( )( ) ′

=− = b

0 

# 2 077 0 0670 1853 08357

. . ..



# The resulting equation is expressed in the following form:

$ ′

= ′ + ′ y b bx

0 1



# For this example, the correlation equation becomes

$

. .

′

= + y x 0836 0 0670



# Converting this to the format that yields the predicted PM concentration in the correct units

entails taking the exponential of each side of the equation, as follows:



# ( ) ( ) Exp y Exp x

y e x 

# $

. .

$

. . 

# ′ = +

= 08357 0 0670

2 31 0 0670



# In the resulting correlation equation, the coefficient 2.31 is simply the exponential of b

0 

# N

.

How do I determine the exponential correlation coefficient?

Determining the exponential correlation coefficient requires calculating the predicted PM

concentration ( 

# í N ) for each test run, the parameter S

L 

# , and the parameter S

y 

# N 

# following the same

procedure used to calculate the linear correlation coefficient. Table Y-8 shows the PM

concentrations calculated using the linear form (i.e., 

# í N = b

0 

# N + b

1 

# N x ) of the exponential

correlation equation.

Table Y-8. Predicted PM Concentrations Using Exponential Correlation Equation

Run no. PM CEMS response,

mA PM concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y) Predicted PM

concentration ( 

# í N ) a



# 1 4.9 2.1 0.742 1.164

2 7.6 2.9 1.065 1.345

3 7.8 5.1 1.629 1.358

4 9.2 4.8 1.569 1.452

5 10.1 2.9 1.065 1.512

6 11.06 6.3 1.841 1.576

10 12 8.4 2.128 1.639

11 18.2 8.4 2.128 2.055

12 15.7 4.9 1.589 1.887

(continued)

# Table Y-8. (continued)

Run no. PM CEMS response,

mA PM concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y) Predicted PM

concentration ( 

# í N ) a



# Y-19 13 13.9 6.3 1.841 1.767

14 19.7 13 2.565 2.155

15 21.02 9.1 2.208 2.243

16 22.3 13.9 2.632 2.329

17 24.1 8.7 2.163 2.450

19 24.7 12.6 2.534 2.490

20 26.4 15.9 2.766 2.604

21 29.08 16.2 2.785 2.783

22 29.3 13.5 2.603 2.798

23 30.5 15.9 2.766 2.878

24 33.1 18.5 2.918 3.052

a 

# Calculated using the e xponential correlation equation: 

# í N = 0.836 + 0.067 x

.



# Using the values for 

# í N shown in Table Y-8, calculate S

L 

# as follows:



# ( ) ( )( ) ( ) 

# ( ) S

L 

# −

− − + − ++ − = 1

20 2 1164 0 742 1345 1065 3052 2 918 0 278 22 2



# .. .. . . . K

The value for S

y 

# N 

# is determined using Equation 11-15 with the values for y 

# N substituted for the

values of y, as follows:



# ( )( ) ( ) 

# ( )

( ) S

y 

# = − + − ++ −

− = 0 742 2 077 1065 2 077 2 918 2 077

20 1 0 643 22 2



# .. .. ..

. K

Substitute the values for S

L 

# and S

y 

# N 

# into Equation 11-14 to calculate the exponential correlation

coefficient:



# ( )

( ) r =− = 1 0 278

0 643 0 902 2

2 

# .

. .

The exponential correlation coefficient (0.902) is greater than the minimum value specified by

PS-11 (0.85) and is acceptable.

How do I determine the confidence interval half range percentage for the exponential

correlation?

The confidence interval half range for the exponential correlation is determined using

Equation 11-8, which is the same equation used to calculate the confidence interval half range

# 1 

# The equations presented in this section for expone ntial and power models correspond to the amended

version of PS-11, which was not yet published at the time this was prepared.



# Y-20 for linear correlations. Because the confidence interval half range is calculated on the log scale,

it is designated as CI 

# N for the exponential model. Using Equation 11-8, CI 

# N is calculated as

follows:



# ( )( ) CI t S 1

n df, 1 a 2 L 

# ′

= ⋅= =

− 

# 2101 0 278 1

20 01307

.. .

The lower confidence limit (LCL 

# N ) and upper confidence limit (UCL 

# N ) are calculated using

Equations 11-40 and 11-41, respectively 1



# LCL 

# N = y 

# GN - CI 

# N = 2.077 - 0.1307 = 1.946

UCL 

# N = y 

# GN + CI 

# N = 2.077 + 0.1307 = 2.207

Next, the confidence interval half range (CI) must be calculated on the original PM

concentration scale using Equation 11-42 from PS-11



# CI ee ee UCL LCL



# = −

= −

= ′ ′ 

# ( )( )

. . .



# 2 2 1045 2 207 1 946



# The confidence interval as a percentage of the emission limit is calculated using Equation 11-10

as follows:



# CI % .

. =× = 1045

20 100% 523%

The resulting value for CI% is 5.23 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the exponential correlation?

The tolerance interval half range for the exponential correlation also is calculated using the same

procedure as is used for linear models. Because it is calculated on the log scale, it is denoted

here as TI 

# N . Using Equation 11-11, TI 

# N is calculated as follows:

TI 

# N = k

T 

# @ S

L 

# = (1.629)(0.278) = 0.453

The lower tolerance limit (LTL 

# N ) and upper tolerance limit (UTL 

# N ) are calculated using

Equations11-43 and 11-44, respectively

LTL 

# N = y 

# GN - TI 

# N = 2.077 - 0.453 = 1.624

UTL 

# N = y 

# GN + TI 

# N = 2.077 + 0.453 = 2.530

# Y-21 Next, the tolerance interval half range (TI) must be calculated on the original PM concentration

scale using Equation 11-45 from PS-11



# TI e e ee UTL LTL



# = −

= −

= ′ ′ 

# ( )( )

. . .



# 2 2 374 2 53 1 624



# The tolerance interval as a percentage of the emission limit is calculated using Equation 11-13 as

follows:

TI % .

. =× = 374

20 100% 18 7%



# The resulting value for TI% is 18.7 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range. In summary, the exponential

model satisfies the PS-11 criteria for correlation coefficient, confidence interval half range

percentage, and tolerance interval half range percentage.

How do I develop a power correlation curve from the data?

A power correlation curve has the following form:

$ y bx b



# = 0 1



# Developing a power correlation requires transforming both the PM CEMS response (x) values,

as described previously for a logarithmic correlation, and the PM concentration (y) values, as

was described above for an exponential correlation. The procedures used to develop a linear

correlation are then followed using the transformed values for x and y. Table Y-9 shows the test

data and transformed values that must be used.

Table Y-9. Summary of Test Data and Transformed Values for

Developing Power Correlation

Run no. PM CEMS response,

mA Logarithm of PM

CEMS response

x' = ln (x) PM concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y)

1 4.9 1.589 2.1 0.742

2 7.6 2.028 2.9 1.065

3 7.8 2.054 5.1 1.629

4 9.2 2.219 4.8 1.569

5 10.1 2.313 2.9 1.065

6 11.06 2.403 6.3 1.841

10 12 2.485 8.4 2.128

11 18.2 2.901 8.4 2.128

(continued)

# Table Y-9. (continued)

Run no. PM CEMS response,

mA Logarithm of PM

CEMS response

x' = ln (x) PM concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y)



# Y-22 12 15.7 2.754 4.9 1.589

13 13.9 2.632 6.3 1.841

14 19.7 2.981 13 2.565

15 21.02 3.045 9.1 2.208

16 22.3 3.105 13.9 2.632

17 24.1 3.182 8.7 2.163

19 24.7 3.207 12.6 2.534

20 26.4 3.273 15.9 2.766

21 29.08 3.370 16.2 2.785

22 29.3 3.378 13.5 2.603

23 30.5 3.418 15.9 2.766

24 33.1 3.500 18.5 2.918



# Following the same procedures described previously for developing a linear correlation curve,

calculate values for the parameters x 

# GN and y 

# GN , using Equation 11-5; S

x 

# N x 

# N 

# and S

x 

# N y 

# N 

# , using

Equation 11-7; b

1 

# N , using Equation 11-6; and b

0 

# N , using Equation 11-4:



# ( ) ′ = + + ++ = x 1

20 1589 2 028 2 054 35 2 792

... . K



# ( ) ′ = + + ++ = y 1

20 0 742 1065 1629 2 918 2 077

... . . K



# ( )( ) ( ) 

# ( ) S

x x ′ ′ 

# = − + − ++ − = 1589 2 792 2 028 2 792 350 2 792 574 22 2



# . . . . .. . K 

# ( )( ) ( ) ( )

( )( ) S

x y ′ ′ 

# = − − + − − ++

− −= 1588 2 792 0 742 2 077 2 028 2 792 1065 2 077

350 2 792 2 918 2 077 6 22 . .. . . .. .

.. . . . K

b

1 

# 6 22

574 10839

= = .

. .



# ( )( ) ′

= − =− b

0 

# 2 077 10839 2 792 0 9492

... .



# The resulting equation is expressed in the following form:

$ ′

= ′ + ′ ′ y b bx

0 1



# For this example, the correlation equation becomes

$

. .

′

=

− + ′ y x 0 9492 1084



# Converting this to the format that yields the predicted PM concentration in the correct units

entails taking the exponential of each side of the equation, as follows:

# Y-23 Exp ( 

# í

N ) = Exp (-0.9492 + 1.084 x 

# N )



# í

= 0.387 x 1.084



# In this equation, the coefficient 0.387 is the exponential of -0.9492, the value for b

0 

# N

.

How do I determine the power correlation coefficient?

To determine the power correlation coefficient, the predicted PM concentration ( 

# í N ) must be

calculated for each test run. Table Y-10 shows the predicted PM concentrations calculated using

the linear form (i.e., 

# í N = b

0 

# N + b

1 

# N x 

# N ) of the power correlation equation.

Table Y-10. Predicted PM Concentrations Using the Power Correlation Equation

Run no. PM CEMS

response, mA Logarithm of PM

CEMS response

x' = ln (x) PM

concentration,

mg/acm Logarithm of PM

CEMS response

y' = ln (y) Predicted PM

concentration ( 

# í N ) a



# 1 4.9 1.589 2.1 0.742 0.773

2 7.6 2.028 2.9 1.065 1.249

3 7.8 2.054 5.1 1.629 1.277

4 9.2 2.219 4.8 1.569 1.456

5 10.1 2.313 2.9 1.065 1.557

6 11.06 2.403 6.3 1.841 1.656

10 12 2.485 8.4 2.128 1.744

11 18.2 2.901 8.4 2.128 2.196

12 15.7 2.754 4.9 1.589 2.035

13 13.9 2.632 6.3 1.841 1.903

14 19.7 2.981 13 2.565 2.281

15 21.02 3.045 9.1 2.208 2.352

16 22.3 3.105 13.9 2.632 2.416

17 24.1 3.182 8.7 2.163 2.500

19 24.7 3.207 12.6 2.534 2.527

20 26.4 3.273 15.9 2.766 2.599

21 29.08 3.370 16.2 2.785 2.704

22 29.3 3.378 13.5 2.603 2.712

23 30.5 3.418 15.9 2.766 2.755

24 33.1 3.500 18.5 2.918 2.844

a 

# Calculated using the expone ntial correlation equation: 

# í N =



# ! 0.9492 + 1.084 x 

# N

# Y-24 

# ( )( ) CI ′

= = 2101 0 249 1

20 0117

.. . The parameter S

L 

# is calculated using Equation 11-9 with the predicted values ( 

# í N ) given in the

above table, and the parameter S

y 

# N 

# is calculated following the same procedure used for the

exponential correlation coefficient, as follows:



# ( ) ( )( ) ( ) 

# ( ) S

L 

# =

− − + − ++ − = 1

20 2 0 773 0 742 1249 1065 2 844 2 918 0 249 22 2



# .. .. .. . K



# ( )( ) ( ) 

# ( )

S

y ′ 

# = − + − ++ −

− = 0 742 2 077 1065 2 077 2 918 2 077

20 1 0 643 22 2



# .. .. ..

. K

Substituting the values for S

L 

# and S

y 

# N 

# into Equation 11-14, the power correlation coefficient is



# ( )

( ) r =− = 1 0 249

0 643 0 922 2

2 

# .

. .

The correlation coefficient for the power model (0.922) is greater than the minimum value

specified by PS-11 (0.85) and is acceptable.

How do I determine the confidence interval half range percentage for the power correlation?

The confidence interval half range for the power correlation is determined using the same

procedures used for the exponential correlation. Because the confidence interval half range is

calculated on the log scale, it is designated as CI 

# N for the exponential model. Using

Equation 11-8, CI 

# N is calculated as follows:

The lower confidence limit (LCL 

# N ) and upper confidence limit (UCL 

# N ) are calculated using

Equations 11-40 and 11-41, respectively

LCL 

# N = y 

# GN - CI 

# N = 2.077 - 0.117 = 1.96

UCL 

# N = y 

# GN + CI 

# N = 2.077 + 0.117 = 2.194

Next, the confidence interval half range (CI) must be calculated on the original PM

concentration scale using Equation 11-42 from PS-11

CI ee ee UCL LCL



# = −

= −

= ′ ′ 

# ( )( )

. . .



# 2 2 0 936 2 194 1 96



# The confidence interval as a percentage of the emission limit is calculated using Equation 11-10

as follows:

# Y-25 

# CI % .

. =×= 0 936

20 100% 4 68%

The resulting value for CI% is 4.68 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the power correlation?

The tolerance interval half range for the power correlation also is calculated using the same

procedure used for the exponential model. Because it is calculated on the log scale, it is denoted

here as TI 

# N . Using Equation 11-11, TI 

# N is calculated as follows:

TI 

# N = k

T 

# @ S

L 

# = (1.629)(0.249) = 0.406

The lower tolerance limit (LTL 

# N ) and upper tolerance limit (UTL 

# N ) are calculated using

Equations11-43 and 11-44, respectively

LTL 

# N = y 

# GN - TI 

# N = 2.077 - 0.406 = 1.671

UTL 

# N = y 

# GN + TI 

# N = 2.077 + 0.406 = 2.483

Next, the tolerance interval half range (TI) must be calculated on the original PM concentration

scale using Equation 11-45 from PS-11



# TI ee ee UTL LTL



# = −

= −

= ′ ′ 

# ( )( )

. . .



# 2 2 3327 2 483 1 671



# The tolerance interval as a percentage of the emission limit is calculated using Equation 11-13 as

follows:



# TI % .

. =×= 3327

20 100% 16 6%

The resulting value for TI% is 16.6 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range. In summary, the power

model satisfies the PS-11 criteria for correlation coefficient, confidence interval half range

percentage, and tolerance interval half range percentage.

Which model is the best fit for the data?

Table Y-11 summarizes the results of the analysis for all of the correlation models. As indicated

in the table, all five models met the acceptance criteria specified in Section 13.2 of PS-11. The

linear model had the highest correlation coefficient and would be considered the best fit for the

data. However, any of the models could be used to show compliance. The complete output for

the spreadsheet for this example is provided in Appendix A.1.

# Y-26 Table Y-11. Summary of Results for Facility A Correlation Test

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = -0.621 + 0.544x 0.927 4.49% 15.6%

Polynomial 

# í = 0.420 + 0.406x + 0.00368 x 2 

# 0.924 6.12% 16.3%

Logarithmic 

# í = -13.8 + 8.32Ln(x) 0.892 5.41% 18.8%

Exponential 

# í = 2.31e 0.0670x 

# 0.902 5.23% 18.7%

Power 

# í = 0.387x 1.08 

# 0.922 4.68% 16.6%



# Y.2 EXAMPLE 2: HOW DO I DEVELOP A CORRELATION WITH PAIRED SAMPLING

TRAIN DATA?

Y.2.1 Correlation Test Scenario

A correlation test is performed on an incinerator at Facility B. The incinerator is equipped with

an in situ PM CEMS that responds to PM concentrations at actual conditions. The emission

limit, when converted to units that are consistent with the PM CEMS sampling conditions, is

24.3 mg/acm. Fifteen test runs are performed using Method 5i with paired sampling trains

(Train A and Train B). No problems were reported for the test. Table Y-12 summarizes the

correlation test data. Figure Y-2 shows a plot of the data.

Table Y-12. Summary of Correlation Test Data for Facility B

Run no. PM CEMS

response, mA PM concentration, mg/acm

Sampling Train A Sampling Train B

1 6.4 1.3 1.2

2 7.5 2.3 2.5

3 11.2 2.5 2.8

4 16.4 7.7 7.9

5 29.9 12.9 12.5

6 22.2 7.8 7.3

7 19.5 5.8 5.9

8 27.4 11.7 12.0

9 32.8 20.4 20.9

10 26.5 20.9 21.3

11 33.5 22.3 22.6

12 28.1 18.4 17.58

13 9.6 3.7 3.9

14 5.3 4.2 4.3

15 18.1 6.8 7.1

# Y-27 0 5 10 15 20 25

0 5 10 15 20 25 30 35 40

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-2. Plot of correlation test data for Facility B.

Should I average the two PM concentrations for each test run?

You should not average the two PM concentrations for any of the test runs. Instead, you should

treat each PM concentration as a separate data point, as specified in Section 12.3 of PS-11. This

means that for each run, the data set includes two PM concentrations and one PM CEMS

response value. In other words, the correlation should be developed as if there were 30 test runs

with one PM concentration for each run. Table Y-13 shows the data in the form that should be

used for developing the correlation.

Before proceeding with the correlation, paired sampling train data should be checked for

precision and bias. Guidance on precision and bias is provided separately. For this example, it

is assumed that the data meet the criteria of precision and bias and are acceptable.

# Y-28 Table Y-13. Paired Sampling Train Test Data in Format

for Developing Correlation

Run No. PM CEMS Response,

mA PM Concentration,

mg/acm

1A 6.4 1.3

1B 6.4 1.2

2A 7.5 2.3

2B 7.5 2.5

3A 11.2 2.5

3B 11.2 2.8

4A 16.4 7.7

4B 16.4 7.9

5A 29.9 12.9

5B 29.9 12.5

6A 22.2 7.8

6B 22.2 7.3

7A 19.5 5.8

7B 19.5 5.9

8A 27.4 11.7

8B 27.4 12.0

9A 32.8 20.4

9B 32.8 20.9

10A 26.5 20.9

10B 26.5 21.3

11A 33.5 22.3

11B 33.5 22.6

12A 28.1 18.4

12B 28.1 17.8

13A 9.6 3.7

13B 9.6 3.9

14A 5.3 4.2

14B 5.3 4.3

15A 18.1 6.8

15B 18.1 7.1

# Y-29 What are the results of the correlation analysis?

Table Y-14 summarizes the results of the analysis. The complete output for the spreadsheet for

this example is provided in Appendix A.2.

Table Y-14. Summary of Results for Facility B Correlation Test

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = -3.24 + 0.672 x 0.895 4.98% 20.0%

Polynomial 

# í = 2.85 - 0.173 x + 0.0220 x 2 

# 0.919 5.91% 18.0%

Logarithmic 

# í = -17.7 + 9.81Ln( x ) 0.822 6.37% 25.6%

Exponential 

# í = 1.43e 0.0828 x 

# 0.921 3.80% 15.9%

Power 

# í = 0.201 x 1.27 

# 0.892 4.41% 18.7%



# Which model should I use?

All of the models except the logarithmic model satisfy the acceptance criteria for PS-11. The

logarithmic model fails to meet the minimum allowable correlation coefficient of 0.85. In

addition, the tolerance interval half range percentage for the logarithmic model exceeds the

allowable percentage of 25 percent. Based on the correlation coefficient, the exponential model

is the best fit. However, the linear, polynomial, and power models could also be used.

Y.3 EXAMPLE 3: HOW DO I DEVELOP THE INITIAL CORRELATION?

Y.3.1 Correlation Test Scenario

A correlation test is performed on a hazardous waste incinerator at Facility C. The incinerator is

equipped with an extractive PM CEMS that extracts and heats the samples to 121°C (250°F).

The incinerator is subject to 40 CFR 63, subpart EEE, and must meet an emission limit of

34.0 mg/dscm corrected to 7 percent oxygen. Fifteen test runs are performed using a Method 5

sampling train. The average stack gas pressure, moisture content, and oxygen content during the

test were 760 millimeters of mercury, 12 percent moisture, and 10 percent oxygen, respectively.

No problems were reported for the test, and all data points will be used for the correlation.

Table Y-15 summarizes the correlation test data. Figure Y-3 shows a plot of the data.

# Y-30 0 5 10 15 20 25 30 35

0.00 5.00 10.00 15.00 20.00 25.00 30.00

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-3. Plot of correlation test data for Facility C. Table Y-15. Summary of Correlation Test Data for Facility C

Run no. PM CEMS response, mA PM concentration, mg/acm

1 5.8 3.5

2 7.6 9.0

3 7.8 7.8

4 16.4 25.2

5 10.1 16.6

6 11.1 14.3

7 12.0 19.6

8 13.2 20.1

9 15.7 23.4

10 17.6 26.7

11 21.7 26.8

12 21.0 28.9

13 22.3 25.7

14 24.1 26.2

15 24.7 26.2

# Y-31 Do I need to convert the emission limit to other units for determining compliance with PS-11?

As indicated in Section 13.2 of PS-11, you must convert the emission limit to units that are

consistent with the measurement conditions of your PM CEMS. For this example, you must

convert the emission limit, which is specified in units of mg/dscm at 7 percent oxygen, to match

the conditions under which the PM CEMS is operating. The following calculations show this

conversion:



# ( ) C C T

T P

P SO a a

std std

a 

# mc O @ 

# .

. % 7% 2 

# 1

1

100 13 9

20 9 2 

# =× × × ⎛

⎝ ⎜ ⎞

⎠ ⎟ ×

− −



# where

C

S@7%O 2 

# = PM concentration at standard conditions, corrected to 7 percent oxygen

T

a 

# = stack gas temperature at actual conditions, °K

T

std 

# = stack gas temperature at standard conditions = 293°K

P

std 

# = stack gas pressure at standard conditions = 760 mm of Hg

P

a 

# = stack gas pressure at actual conditions, mm of Hg

mc = stack gas moisture content, %

%O

2 

# = stack gas oxygen content, %.



# ( )

( ) 34 0 121 273

293 760

760 1

1 12

100 139

20 9 10 . .

. = × +

× ⎛

⎝ ⎜ ⎞

⎠ ⎟

×

− ⎛

⎝ ⎜ ⎜ ⎜ ⎞

⎠ ⎟ ⎟ ⎟ ×

− C

a



# 34 0 195

. . = C

a



# C

a 

# = 17 45

.



# The emission limit in units of concentration at the operating conditions of the PM CEMS is

17.45 mg/acm. This is the value that must be used for determining the confidence and tolerance

interval half range percentages.

What are the results of the correlation analysis?

Table Y-16 summarizes the results of the correlation analysis. Only the logarithmic model

satisfies all of the acceptance criteria for PS-11. The correlation coefficient for the exponential

model (0.812) fails to meet the minimum allowable value of 0.85. The confidence and tolerance

interval half range percentages for the linear model (11.3 and 35.6 percent, respectively) exceed

the limits of 10 and 25 percent, respectively. For the power model, the confidence interval

(14.3 percent) and tolerance interval (46.4 percent) half range percentages also exceed the

maximum allowable values.

# Y-32 Plot of Polynomial Regression Curve

y = -0.101x 2 

# + 4.296x - 18.371

0 5 10 15 20 25 30 35

0.00 5.00 10.00 15.00 20.00 25.00 30.00

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-4. Plot of polynomial correlation curve for Facility C. Table Y-16. Summary of Results for Facility C Correlation Test

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = 2.27 + 1.15 x 0.898 11.3% 35.6%

Polynomial 

# í = -18.4 + 4.30 x - 0.101 x 2 

# 0.987 5.56% 13.4%

Logarithmic 

# í = -24.4 + 16.8Ln( x ) 0.959 7.32% 23.1%

Exponential 

# í = 5.26e 0.0785 x 

# 0.812 19.9% 66.5%

Power 

# í = 0.747 x 1.20 

# 0.908 14.3% 46.4%



# The polynomial model has the highest correlation coefficient (0.987) and meets the criteria for

confidence and tolerance interval half range percentages. However, Section 12.4(3) of PS-11

specifies that the maximum value for a polynomial correlation curve must occur at a PM CEMS

response value that is at least 1.25 times the highest PM CEMS response value used to develop

the correlation equation. In this example, the maximum PM CEMS response value is 24.7 mA,

and the maximum value for the correlation curve must occur at a PM CEMS response value of at

least 32.4 mA (24.7 × 1.25 = 30.88). However, the maximum value for the polynomial

correlation curve occurs at 21.2 mA, as is evident from Figure Y-4. Therefore, the polynomial

model cannot be used. The complete output for the spreadsheet for this example is provided in

Appendix A.3.

# Y-33 Y.4 EXAMPLE 4: WHAT CAN I DO IF MY PM CEMS FAILS THE CORRELATION

TEST?

Y.4.1 Correlation Test Scenario

A correlation test is performed on a boiler at Facility D. The boiler is equipped with an in situ

PM CEMS that responds to PM concentrations at actual conditions. The emission limit, when

converted to units that are consistent with the PM CEMS sampling conditions, is 18.5 mg/acm.

The correlation test consists of 15 test runs using a single Method 5 sampling train. Table Y-17

summarizes the correlation test data. Figure Y-5 shows a plot of the data.

Table Y-17. Summary of Test Data for Facility D

Run no. PM CEMS response,

mA PM concentration,

mg/acm

1 6.7 3.1

2 7.6 7.4

3 7.8 2.1

4 9.2 3.7

5 10.1 6.1

6 11.1 2.8

7 12.0 4.9

8 18.2 8.3

9 13.7 4.5

10 14.6 8.2

11 19.7 13.0

12 18.0 11.1

13 20.3 11.9

14 20.2 10.6

15 19.7 9.2

# Y-34 The results of the correlation analysis are summarized in Table Y-18. None of the models satisfy

all of the correlation criteria specified in PS-11. Although all of the models meet the criteria for

confidence and tolerance interval half range percentages, only the polynomial model has an

acceptable correlation coefficient. However, the polynomial correlation curve has a minimum x

value of 7.05 that is greater than the minimum PM CEMS response (6.7 mA for Run 1) value

used to develop the correlation curve. As stated in Section 12.4(3)(iii) of PS-11, a polynomial

correlation curve cannot be used if it has a minimum value that exceeds the minimum PMCEMS response value used to determine the correlation curve. Consequently, the polynomial

correlation also is unacceptable.

Table Y-18. Results of Initial Correlation for Facility D

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = -1.20 + 0.598 x 0.841 5.80% 18.3%

Polynomial 

# í = 6.19 - 0.615 + 0.0436 x 2 

# 0.855 7.68% 18.2%

Logarithmic 

# í = -11.8 + 7.37Ln( x ) 0.801 6.41% 20.2%

Exponential 

# í = 1.69e 0.0934 x 

# 0.806 6.41% 21.3%

Power 

# í = 0.308 x 1.17 

# 0.782 6.76% 22.6% 0 2 4 6 8 10 12 14

0.0 5.0 10.0 15.0 20.0 25.0

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-5. Plot of correlation test data for Facility D.

# Y-35 Several options are available for meeting the requirements of PS-11, including the following:



# # Repeat the correlation test : If you suspect problems with the manual sampling or the

operation of the PM CEMS, you may want to discard all of the test data and conduct

a new correlation test. In such a case, there must be a valid reason for rejecting the

entire data set. For example, you determine that several of the PM samples were

contaminated. You still must report all of the rejected data.



# # Conduct additional test runs . You may want to conduct a limited number of

additional test runs (e.g., five additional runs) if it appears that some of the data are

outliers, problems are suspected for some of the runs, or the range of PM

concentrations measured was not adequate to fully characterize the correlation. In

this case, you can develop the correlation based on all of the data (i.e., the initial 15

runs plus the additional runs), or you can reject some of the test runs, provided you

meet the requirements of Section 8.6(3) of PS-11, which allows you to reject up to

five test runs without explanation and additi onal runs if the basis for rejecting them is

specified in the test method, in PS-11, or in your quality assurance plan.



# # Relocate the PM CEMS and conduct a new correlation test : If the sampling location

is not ideal (e.g., PM concentrations are stratified across the stack at the sampling

point), you may want to consider relocating the PM CEMS to another location that is

more suitable. In this case, you must conduct a full correlation test (i.e., at least 15

valid test runs).



# # Replace the PM CEMS : Certain types of PM CEMS may not be suitable for specific

stack gas conditions (e.g., an in situ light scattering instrument in a wet stack). If that

is the case, you may decide to install a PM CEMS that uses a different measurement

principle. If so, you must conduct a full correlation test with the new instrument.



# # Consider alternative solutions : Section 12.4(4) allows you to petition the regulatory

authority for approval of alternatives to the correlation requirements of PS-11.

For this example, we will assume that you decide to conduct five additional test runs. These

runs are identified as Runs 16 to 20. Table Y-19 shows the results of the additional runs. Figure

Y-6 shows the results of all 20 test runs.

Table Y-19. Results of Additional Test Runs

Run no. PM CEMS response,

mA PM concentration,

mg/acm

16 5.2 2.4

17 8.6 4.8

18 14.1 6.2

19 14.5 7.0

20 17.3 8.6

# Y-36 Table Y-20 shows the results of the correlation analysis on all 20 test runs. The analysis shows

that the linear and polynomial models meet the correlation criteria of PS-11, and no further

analyses or testing is required. With the additional data included, the minimum value for the

polynomial correlation curve now occurs at 4.2 mA, which is less than the minimum PM CEMS

value used to develop the curve (5.2 mA). However, a review of Figure Y-6 indicates that Run 2

(7.4 mA, 7.4 mg/acm) may be an outlier. You may decide to repeat the correlation analysis with

Run 2 excluded. Table Y-21 shows the results of the analysis without the data from Run 2. In

the revised analysis, all five models meet the criteria of PS-11. The correlation coefficients for

the linear and polynomial models also are considerably greater for the revised analysis. You

have the option of using the results of either analyses. In either case, the polynomial model

provides the best fit; however, any of the models that satisfy all of the correlation criteria in PS11 can be used.

Table Y-20. Results of Revised Correlation Analysis with All 20 Runs

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = -0.905 + 0.573 x 0.861 4.25% 14.7%

Polynomial 

# í = 3.94 - 0.263 x + 0.0314 x 2 

# 0.875 5.47% 14.4%

Logarithmic 

# í = -9.79 + 6.57Ln( x ) 0.816 4.82% 16.7%

Exponential 

# í = 1.70e 0.0935 x 

# 0.840 4.52% 16.3%

Power 

# í = 0.367 x 1.11 

# 0.824 4.73% 17.0% 

# 0 2 4 6 8 10 12 14

0 5 10 15 20 25

PM CEMS Response, mAPM Concentration, mg/acm



# Figure Y-6. Plot of original 15 runs plus 5 additional runs for Facility D. Run 2

# Y-37 Table Y-21. Results of Revised Correl ation Analysis with Run 2 Excluded

Model type Correlation equation Correlation

coefficient (r) Confidence

interval half

range Tolerance

interval half

range

Acceptance Criteria for PS-11 

# $ 0.85 

# # 10% 

# # 25%

Linear 

# í = -1.88 + 0.629 x 0.912 3.63% 12.4%

Polynomial 

# í = 2.19 - 0.067 x + 0.026 x 2 

# 0.921 4.54% 12.1%

Logarithmic 

# í = -11.7 + 7.24Ln( x ) 0.867 4.40% 15.0%

Exponential 

# í = 1.41e 0.104 x 

# 0.909 3.63% 12.6%

Power 

# í = 0.251 x 1.24 

# 0.894 3.90% 13.6%



# The complete output of the spreadsheet for the initial correlation, the correlation with all five

additional test runs, and the correlation with Run 2 excluded are provided in Appendices A.4

through A.6.

# Appendix A.1

Complete Results of Example No. 1 –

Initial Correlation TestCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: FacilityAresponse mg/acmLocation:Anytown USARun x yEmission Unit: Boiler No. 31 4.9 2.1Test Dates: March 6 to 15, 20042 7.6 2.93 7.8 5.1ValueUnits4 9.2 4.8Emission limit:20 mg/acm5 10.1 2.96 11.1 6.310 12.0 8.411 18.2 8.412 15.7 4.913 13.9 6.314 19.7 13.015 21.0 9.116 22.3 13.917 24.1 8.719 24.7 12.620 26.4 15.921 29.1 16.222 29.3 13.523 30.5 15.924 33.1 18.5024681012141618200.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21 4.9 2.1 185.86 54.32 100.48 2.047 0.0032 7.6 2.9 119.53 43.16 71.83 3.517 0.3813 7.8 5.1 115.20 19.10 46.90 3.626 2.1724 9.2 4.8 87.10 21.81 43.59 4.388 0.1695 10.1 2.9 71.12 43.16 55.40 4.878 3.9146 11.06 6.3 55.85 10.05 23.69 5.401 0.80810 12 8.4 42.68 1.14 6.99 5.913 6.18511 18.2 8.4 0.11 1.14 0.36 9.289 0.79012 15.7 4.9 8.03 20.88 12.95 7.928 9.16613 13.9 6.3 21.46 10.05 14.69 6.947 0.41914 19.7 13 1.36 12.46 4.12 10.105 8.37915 21.02 9.1 6.19 0.14 -0.92 10.824 2.97316 22.3 13.9 14.19 19.62 16.69 11.521 5.66017 24.1 8.7 30.99 0.59 -4.29 12.501 14.44819 24.7 12.6 38.03 9.80 19.30 12.828 0.05220 26.4 15.9 61.89 41.34 50.58 13.753 4.60821 29.08 16.2 111.24 45.29 70.98 15.213 0.97522 29.3 13.5 115.93 16.24 43.39 15.332 3.35723 30.5 15.9 143.21 41.34 76.95 15.986 0.00724 33.1 18.5 212.20 81.54 131.54 17.401 1.207PS-11 Example 1-R 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004Correlation equation: y = -0.621 + 0.544 xVariable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =18.53 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =1,442 Correlation coefficient 0.927>0.85yesy~y~ = 1/n*(Sum of (yi)) =9.47 Confidence interval 4.49%<10%yesSyySyy= Sum((yi-y~)^2) =493 Tolerance interval 15.6%<25%yesSxySxy= Sum((xi-x~)(yi-y~)) =785.2 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =-0.621b1b1= Sxy/Sxx=0.544SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.910y^meany^ at mean x value = 9.470tftffrom table =2.101CICI = tf*SL*SQRT(1/n) =0.897EL Emission Limit = 20.0CI% CI% =CI/EL*100% = 4.49%n' n' = n = 20vfvffrom table =1.384un'un'from table =1.177kTkt= un'*vf=1.629TITI = kt*SL=3.111TI% TI% = TI/EL*100% = 15.6%SySy= SQRT(Syy/(n-1)) =5.095PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.859Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.927Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 0.5445x - 0.6207051015200.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility A Emission Unit: Boiler No. 3Location: Anytown USA Test Dates: March 6 to 15, 2004PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1 4.9 2.1 2411857610502.4970.1570.381754.32 2.542 7.6 2.9 584393,336221683.7170.6670.172743.16 1.713 7.8 5.1 614753,702403103.8091.6670.163419.10 1.664 9.2 4.8 857797,164444064.4650.1120.116221.81 1.405 10.1 2.9 1021,03010,406292964.8943.9750.099543.16 1.306 11.06 6.3 1221,35314,963707715.3580.8870.090610.05 1.2410 12 8.4 1441,72820,7361011,2105.8196.6600.08841.14 1.2211 18.2 8.4 3316,029109,7201532,7829.0240.3900.12021.14 1.4312 15.7 4.9 2463,87060,757771,2087.6987.8280.108420.88 1.3613 13.9 6.3 1932,68637,330881,2176.7710.2220.096010.05 1.2714 19.7 13 3887,645150,6142565,0459.8429.9720.120412.46 1.4315 21.02 9.1 4429,287195,2231914,02110.5762.1780.11620.14 1.4016 22.3 13.9 49711,090247,2973106,91211.2996.7640.109219.62 1.3617 24.1 8.7 58113,998337,3402105,05312.33713.2290.09850.59 1.2919 24.7 12.6 61015,069372,2103117,68712.6880.0080.09599.80 1.2720 26.4 15.9 69718,400485,75342011,08213.6984.8490.096041.34 1.2721 29.08 16.2 84624,591715,11847113,69915.3330.7520.141745.29 1.5522 29.3 13.5 85825,154737,00539611,59015.4693.8790.149216.24 1.5923 30.5 15.9 93028,373865,36548514,79116.2210.1030.204641.34 1.8624 33.1 18.5 1,09636,2651,200,36161220,26917.8840.3790.431281.54 2.70PS-11 Example 1-R 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004Variable Equation ValueCorrelation equation: y = 0.420 + 0.406 x + 0.00368 x2n Number of data points = 20S1S1 = Sum (xi) = 371Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 8,312 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 208,377 Correlation coefficient 0.924>0.85yesS4S4 = Sum (xi4) = 5,574,977 Confidence interval 6.12%<10%yesS5S5 = Sum (yi) = 189 Tolerance interval 16.3%<25%yesS6S6 = Sum (xiyi) = 4,295 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 108,567det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 2.120E+09Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 0.420 Correlation curve minimum point -55.08b1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = 0.406 Minimum allowable x value 4.9b2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.00368 Is correlation curve minimum < minimum x value? yesSpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =1.951 Correlation curve maximum point NADD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =2.120E+09 Extrapolation limit for x (125% of maximum x value) NAC0C0 = (S2S4-S32)/D =1.375 Is correlation curve maximum > extrapolation limit? NAC1C1 = (S3S2-S1S4)/D = -0.1578C2C2 = (S1S3-S22)/D =3.846E-03C3C3 = (nS4-S22)/D =2.001E-02C4C4 = (S1S2-nS3)/D = -5.126E-04C5C5 = (nS2-S12)/D =1.360E-05tftffrom table =2.110EL Emission Limit = 20.0CICI = tf*Sp*sqrt(delta-min) =1.223CI% CI% = CI/EL*100= 6.12%vfvffrom table =1.400un'un'from table =1.195n' n' = 1/delta = 11.31kTkt = un'*vf = 1.673TI TI = kt*sp = 3.263TI% TI% = TI/EL *100% = 16.3%y~ y~ = 1/n*(Sum of (yi)) = 9.47SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 5.09r2r2= 1-(Sp^2/Sy^2) =0.853rr = sqrt((1-Sp^2/Sy^2)) =0.924Max-min?b2> 0?minimumPS-11 Correlation Testxmax-miny = -b1/2b2=-55.08Calculation Spreadsheet1.25xmax41.38Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = 0.0037x2+ 0.4057x + 0.4204024681012141618200.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21 4.9 1.589 2.1 1.446 54.317 8.863 -0.539 6.9642 7.6 2.028 2.9 0.583 43.165 5.017 3.114 0.0463 7.8 2.054 5.1 0.544 19.097 3.224 3.330 3.1324 9.2 2.219 4.8 0.328 21.809 2.674 4.704 0.0095 10.1 2.313 2.9 0.230 43.165 3.149 5.481 6.6616 11.06 2.403 6.3 0.151 10.049 1.231 6.237 0.00410 12 2.485 8.4 0.094 1.145 0.328 6.916 2.20311 18.2 2.901 8.4 0.012 1.145 -0.117 10.382 3.92912 15.7 2.754 4.9 0.001 20.885 0.174 9.152 18.08313 13.9 2.632 6.3 0.026 10.049 0.507 8.139 3.38214 19.7 2.981 13 0.036 12.461 0.666 11.041 3.83615 21.02 3.045 9.1 0.064 0.137 -0.094 11.581 6.15616 22.3 3.105 13.9 0.098 19.625 1.386 12.073 3.33817 24.1 3.182 8.7 0.152 0.593 -0.301 12.719 16.15419 24.7 3.207 12.6 0.172 9.797 1.299 12.924 0.10520 26.4 3.273 15.9 0.232 41.345 3.096 13.478 5.86721 29.08 3.370 16.2 0.334 45.293 3.891 14.283 3.67722 29.3 3.378 13.5 0.343 16.241 2.361 14.345 0.71423 30.5 3.418 15.9 0.392 41.345 4.025 14.679 1.49024 33.1 3.500 18.5 0.501 81.541 6.391 15.360 9.858PS-11 Example 1-R 10-27-04.xls Logarithmic CalcsPage 1 of 110/29/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004Correlation equation: y = -13.766 + 8.323 Ln(x)Variable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.79 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =5.74 Correlation coefficient 0.892>0.85yesy~y~ = 1/n*(Sum of (yi)) =9.5 Confidence interval 5.41%<10%yesSyySyy= Sum((yi-y~)^2) =493 Tolerance interval 18.8%<25%yesSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =47.8 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-13.766b1b1= Sx'y/Sx'x'=8.323SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =2.305y^meany^ at mean x value = 9.470tftffrom table =2.101CICI = tf*SL*SQRT(1/n) =1.083EL Emission Limit = 20.0CI% CI% =CI/EL*100% = 5.41%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkt= un'*vf=1.629TITI = kt*SL =3.75TI% TI% = TI/EL*100% = 18.8%SySy= SQRT(Syy/(n-1)) =5.095PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.795Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.892Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 8.3229Ln(x) - 13.766-5051015200.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21 4.9 2.1 0.742 185.86 1.782 18.198 1.164 0.1782 7.6 2.9 1.065 119.53 1.024 11.065 1.345 0.0783 7.8 5.1 1.629 115.20 0.200 4.804 1.358 0.0744 9.2 4.8 1.569 87.10 0.258 4.743 1.452 0.0145 10.1 2.9 1.065 71.12 1.024 8.535 1.512 0.2006 11.06 6.3 1.841 55.85 0.056 1.766 1.576 0.07010 12 8.4 2.128 42.68 0.003 -0.336 1.639 0.23911 18.2 8.4 2.128 0.11 0.003 -0.017 2.055 0.00512 15.7 4.9 1.589 8.03 0.238 1.381 1.887 0.08913 13.9 6.3 1.841 21.46 0.056 1.095 1.767 0.00514 19.7 13 2.565 1.36 0.238 0.570 2.155 0.16815 21.02 9.1 2.208 6.19 0.017 0.327 2.243 0.00116 22.3 13.9 2.632 14.19 0.308 2.091 2.329 0.09217 24.1 8.7 2.163 30.99 0.007 0.482 2.450 0.08219 24.7 12.6 2.534 38.03 0.209 2.818 2.490 0.00220 26.4 15.9 2.766 61.89 0.475 5.424 2.604 0.02621 29.08 16.2 2.785 111.24 0.502 7.469 2.783 0.00022 29.3 13.5 2.603 115.93 0.277 5.662 2.798 0.03823 30.5 15.9 2.766 143.21 0.475 8.251 2.878 0.01324 33.1 18.5 2.918 212.20 0.707 12.250 3.052 0.018PS-11 Example 1-R 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004Correlation equation: y = 2.306 e0.0670 xVariable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =18.53 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =1,442.2 Correlation coefficient 0.902>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =2.077 Confidence interval 5.23%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =7.86 Tolerance interval 18.7%<25%yesSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =96.58 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =0.836b0b0= e^b0' =2.306b1b1= Sxy'/Sxx=0.067SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.278y'^meany'^ at mean x value = 2.077tftffrom table =2.101CI'CI' = tf*SL*SQRT(1/n) =0.1307LCL' LCL' = y~' - CI' 1.946UCL' UCL' = y~' + CI' 2.207CI CI = (e^UCL' - e^LCL')/2 1.045EL Emission Limit = 20.0CI% CI% =CI/EL*100% = 5.23%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkT= un'*vf=1.629TI'TI' = kt*SL =0.453LTL' LTL' = y~' - TI' 1.624UTL' UTL' = y~' + TI' 2.530TI TI = (e^UTL' - e^LTL')/2 3.740TI% TI% = TI/EL*100% = 18.7%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.643PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.813Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.902Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 2.3064e0.067x05101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xlsExponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21 4.9 1.589 2.1 0.742 1.446 1.782 1.605 0.773 0.00102 7.6 2.028 2.9 1.065 0.583 1.024 0.773 1.249 0.03403 7.8 2.054 5.1 1.629 0.544 0.200 0.330 1.277 0.12394 9.2 2.219 4.8 1.569 0.328 0.258 0.291 1.456 0.01265 10.1 2.313 2.9 1.065 0.230 1.024 0.485 1.557 0.24276 11.06 2.403 6.3 1.841 0.151 0.056 0.092 1.656 0.034210 12 2.485 8.4 2.128 0.094 0.003 -0.016 1.744 0.147511 18.2 2.901 8.4 2.128 0.012 0.003 0.006 2.196 0.004512 15.7 2.754 4.9 1.589 0.001 0.238 0.019 2.035 0.199113 13.9 2.632 6.3 1.841 0.026 0.056 0.038 1.903 0.004014 19.7 2.981 13 2.565 0.036 0.238 0.092 2.281 0.080415 21.02 3.045 9.1 2.208 0.064 0.017 0.033 2.352 0.020616 22.3 3.105 13.9 2.632 0.098 0.308 0.174 2.416 0.046717 24.1 3.182 8.7 2.163 0.152 0.007 0.034 2.500 0.113319 24.7 3.207 12.6 2.534 0.172 0.209 0.190 2.527 0.000120 26.4 3.273 15.9 2.766 0.232 0.475 0.332 2.599 0.028121 29.08 3.370 16.2 2.785 0.334 0.502 0.410 2.704 0.006622 29.3 3.378 13.5 2.603 0.343 0.277 0.308 2.712 0.011923 30.5 3.418 15.9 2.766 0.392 0.475 0.432 2.755 0.000124 33.1 3.500 18.5 2.918 0.501 0.707 0.595 2.844 0.0055PS-11 Example 1-R 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004Correlation equation: y = 0.3870 x1.0839Variable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.792 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =5.74 Correlation coefficient 0.922>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =2.077 Confidence interval 4.68%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =7.86 Tolerance interval 16.6%<25%yesSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =6.22 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-0.949b0b0= e^b0' =0.387b1b1= Sx'y'/Sx'x'=1.084SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.249y'^meany'^ at mean x value = 2.077tftffrom table =2.101CI'CI = tf*SL*SQRT(1/n) =0.117LCL' LCL' = y~' - CI' 1.960UCL' UCL' = y~' + CI' 2.194CI CI = (e^UCL' - e^LCL')/2 0.936EL Emission Limit = 20.0CI% CI% =CI/EL*100% = 4.68%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkT= un'*vf=1.629TI'TI' = kTSL=0.406LTL' LTL' = y~' - TI' 1.671UTL' UTL' = y~' + TI' 2.483TI TI = (e^UTL' - e^LTL')/2 3.327TI% TI% = TI/EL*100% = 16.6%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.643PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.850Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.922Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.387x1.0839024681012141618200.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0PM CEMS Response

PM ConcentrationPS-11 Example 1-R 10-27-04.xlsPower Summary10/29/2004SELECTION OF BEST MODELFacility: FacilityAEmission Unit: Boiler No. 3Location: Anytown USATest Dates: March 6 to 15, 2004ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.927 Yes 4.49% Yes 15.6% Yes (b) YesPolynomial 0.924 Yes 6.12% Yes 16.3% Yes Yes YesLogarithmic 0.892 Yes 5.41% Yes 18.8% Yes (b) YesExponential 0.902 Yes 5.23% Yes 18.7% Yes (b) YesPower 0.922 Yes 4.68% Yes 16.6% Yes (b) Yes(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: LinearPS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 1-R 10-27-04.xls Best Model 10/29/2004

# Appendix A.2

Complete Results of Example No. 2 –

Initial Correlation TestCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: Facility Bresponse mg/acmLocation:Anytown USARun x yEmission Unit: Incinerator1A 6.4 1.3Test Dates:April 3 to 6, 20031B 6.4 1.22A 7.5 2.3ValueUnits2B 7.5 2.5Emission limit:24.3 mg/acm3A 11.2 2.53B 11.2 2.84A 16.4 7.74B 16.4 7.95A 29.9 12.95B 29.9 12.56A 22.2 7.86B 22.2 7.37A 19.5 5.87B 19.5 5.98A 27.4 11.78B 27.4 129A 32.8 20.49B 32.8 20.910A 26.5 20.910B 26.5 21.311A 33.5 22.311B 33.5 22.612A 28.1 18.412B 28.1 17.813A 9.6 3.713B 9.6 3.914A5.34.214B 5.3 4.315A 18.1 6.815B 18.1 7.105101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21A 6.4 1.3 174.94 74.94 114.50 1.065 0.0551B 6.4 1.2 174.94 76.68 115.82 1.065 0.0182A 7.5 2.3 147.06 58.62 92.85 1.805 0.2452B 7.5 2.5 147.06 55.60 90.42 1.805 0.4833A 11.2 2.5 71.01 55.60 62.83 4.292 3.2113B 11.2 2.8 71.01 51.22 60.31 4.292 2.2264A 16.4 7.7 10.41 5.09 7.28 7.788 0.0084B 16.4 7.9 10.41 4.23 6.64 7.788 0.0135A 29.9 12.9 105.54 8.66 30.24 16.863 15.7035B 29.9 12.5 105.54 6.47 26.13 16.863 19.0346A 22.2 7.8 6.62 4.65 -5.55 11.687 15.1056B 22.2 7.3 6.62 7.06 -6.84 11.687 19.2427A 19.5 5.8 0.02 17.28 0.53 9.872 16.5777B 19.5 5.9 0.02 16.46 0.51 9.872 15.7738A 27.4 11.7 60.42 3.04 13.55 15.182 12.1268B 27.4 12 60.42 4.18 15.88 15.182 10.1269A 32.8 20.4 173.54 109.06 137.57 18.812 2.5219B 32.8 20.9 173.54 119.76 144.16 18.812 4.35910A 26.5 20.9 47.24 119.76 75.22 14.577 39.97810B 26.5 21.3 47.24 128.67 77.97 14.577 45.19711A 33.5 22.3 192.47 152.36 171.24 19.283 9.10311B 33.5 22.6 192.47 159.85 175.41 19.283 11.00412A 28.1 18.4 71.80 71.29 71.54 15.653 7.54712B 28.1 17.8 71.80 61.52 66.46 15.653 4.61113A 9.6 3.7 100.53 39.15 62.73 3.216 0.23413B 9.6 3.9 100.53 36.68 60.73 3.216 0.46714A 5.3 4.2 205.25 33.14 82.47 0.326 15.01014B 5.3 4.3 205.25 32.00 81.04 0.326 15.79415A 18.1 6.8 2.33 9.96 4.82 8.930 4.53915B 18.1 7.1 2.33 8.16 4.36 8.930 3.350PS-11 Example 2-R 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003Correlation equation: y = -3.237 + 0.672 xVariable Equation Valuen Number of data points = 30Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =19.63 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =2,738 Correlation coefficient 0.895>0.85yesy~y~ = 1/n*(Sum of (yi)) =9.96 Confidence interval 4.98%<10%yesSyySyy= Sum((yi-y~)^2) =1,531 Tolerance interval 20.0%<25%yesSxySxy= Sum((xi-x~)(yi-y~)) =1,840.8 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =-3.237b1b1= Sxy/Sxx=0.672SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =3.238y^meany^ at mean x value = 9.957tftffrom table =2.048CICI = tf*SL*SQRT(1/n) =1.211EL Emission Limit = 24.3CI% CI% =CI/EL*100% = 4.98%n' n' = n = 30vfvffrom table =1.286un'un'from table =1.168kTkt= un'*vf=1.503TITI = kt*SL=4.866TI% TI% = TI/EL*100% = 20.0%SySy= SQRT(Syy/(n-1)) =7.266PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.801Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.895Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 0.6722x - 3.237105101520250.0 10.0 20.0 30.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1A 6.4 1.3 412621,6788532.6491.8190.132674.94 2.151B 6.4 1.2 412621,6788492.6492.0990.132676.68 2.152A 7.5 2.3 564223,164171292.7950.2450.100958.62 1.872B 7.5 2.5 564223,164191412.7950.0870.100955.60 1.873A 11.2 2.5 1251,40515,735283143.6801.3920.064655.60 1.503B 11.2 2.8 1251,40515,735313513.6800.7740.064651.22 1.504A 16.4 7.7 2694,41172,3391262,0715.9423.0900.08525.09 1.724B 16.4 7.9 2694,41172,3391302,1255.9423.8330.08524.23 1.725A 29.9 12.9 89426,731799,25438611,53317.37920.0610.07568.66 1.625B 29.9 12.5 89426,731799,25437411,17517.37923.8040.07566.47 1.626A 22.2 7.8 49310,941242,8911733,8449.8724.2920.08234.65 1.696B 22.2 7.3 49310,941242,8911623,5989.8726.6140.08237.06 1.697A 19.5 5.8 3807,415144,5901132,2057.8584.2360.090617.28 1.777B 19.5 5.9 3807,415144,5901152,2437.8583.8340.090616.46 1.778A 27.4 11.7 75120,571563,6413218,78414.6558.7320.05933.04 1.448B 27.4 12 75120,571563,6413299,00914.6557.0490.05934.18 1.449A 32.8 20.4 1,07635,2881,157,43266921,94720.8840.2340.1573109.06 2.349B 32.8 20.9 1,07635,2881,157,43268622,48520.8840.0000.1573119.76 2.3410A 26.5 20.9 70218,610493,15555414,67713.74251.2390.0604119.76 1.4510B 26.5 21.3 70218,610493,15556414,95813.74257.1250.0604128.67 1.4511A 33.5 22.3 1,12237,5951,259,44574725,02621.7850.2650.1920152.36 2.5811B 33.5 22.6 1,12237,5951,259,44575725,36321.7850.6640.1920159.85 2.5812A 28.1 18.4 79022,188623,48451714,52915.3909.0600.060571.29 1.4512B 28.1 17.8 79022,188623,48450014,05515.3905.8080.060561.52 1.4513A 9.6 3.7 928858,493363413.2230.2270.070039.15 1.5613B 9.6 3.9 928858,493373593.2230.4580.070036.68 1.5614A 5.3 4.2 28149789221182.5562.7040.178533.14 2.4914B 5.3 4.3 28149789231212.5563.0430.178532.00 2.4915A 18.1 6.8 3285,930107,3281232,2286.9400.0200.09019.96 1.7715B 18.1 7.1 3285,930107,3281292,3266.9400.0250.09018.16 1.77PS-11 Example 2-R 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: Facility BEmission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003Variable Equation ValueCorrelation equation: y = 2.854 + -0.173 x + 0.02203 x2n Number of data points = 30S1S1 = Sum (xi) = 589Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 14,295 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 385,603 Correlation coefficient 0.919>0.85yesS4S4 = Sum (xi4) = 10,986,838 Confidence interval 5.91%<10%yesS5S5 = Sum (yi) = 299 Tolerance interval 18.0%<25%yesS6S6 = Sum (xiyi) = 7,703 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 216,158det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 1.198E+10Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 2.854 Correlation curve minimum point 3.93b1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.173 Minimum allowable x value 5.3b2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.02203 Is correlation curve minimum < minimum x value? yesSpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =2.873 Correlation curve maximum point NADD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =1.198E+10 Extrapolation limit for x (125% of maximum x value) NAC0C0 = (S2S4-S32)/D =0.698 Is correlation curve maximum > extrapolation limit? NAC1C1 = (S3S2-S1S4)/D = -0.0799C2C2 = (S1S3-S22)/D =1.895E-03C3C3 = (nS4-S22)/D =1.045E-02C4C4 = (S1S2-nS3)/D = -2.630E-04C5C5 = (nS2-S12)/D =6.855E-06tftffrom table =2.052EL Emission Limit = 24.3CICI = tf*Sp*sqrt(delta-min) =1.436CI% CI% = CI/EL*100= 5.91%vfvffrom table =1.293un'un'from table =1.181n' n' = 1/delta = 16.86kTkt = un'*vf = 1.527TI TI = kt*sp = 4.385TI% TI% = TI/EL *100% = 18.0%y~ y~ = 1/n*(Sum of (yi)) = 9.96SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 7.27r2r2= 1-(Sp^2/Sy^2) =0.844rr = sqrt((1-Sp^2/Sy^2)) =0.919Max-min?b2> 0?minimumPS-11 Correlation Testxmax-miny = -b1/2b2=3.93Calculation Spreadsheet1.25xmax41.88Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = 0.022x2- 0.1731x + 2.853905101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21A 6.4 1.856 1.3 0.929 74.938 8.345 0.504 0.6341B 6.4 1.856 1.2 0.929 76.679 8.441 0.504 0.4852A 7.5 2.015 2.3 0.649 58.625 6.166 2.059 0.0582B 7.5 2.015 2.5 0.649 55.602 6.005 2.059 0.1943A 11.2 2.416 2.5 0.163 55.602 3.015 5.992 12.1913B 11.2 2.416 2.8 0.163 51.218 2.894 5.992 10.1864A 16.4 2.797 7.7 0.001 5.093 0.052 9.732 4.1274B 16.4 2.797 7.9 0.001 4.230 0.047 9.732 3.3555A 29.9 3.398 12.9 0.334 8.663 1.700 15.621 7.4055B 29.9 3.398 12.5 0.334 6.469 1.469 15.621 9.7426A 22.2 3.100 7.8 0.078 4.651 -0.604 12.701 24.0216B 22.2 3.100 7.3 0.078 7.058 -0.743 12.701 29.1727A 19.5 2.970 5.8 0.023 17.278 -0.624 11.429 31.6907B 19.5 2.970 5.9 0.023 16.457 -0.609 11.429 30.5748A 27.4 3.311 11.7 0.240 3.039 0.855 14.765 9.3948B 27.4 3.311 12 0.240 4.175 1.002 14.765 7.6459A 32.8 3.490 20.4 0.449 109.063 6.999 16.529 14.9859B 32.8 3.490 20.9 0.449 119.757 7.334 16.529 19.10610A 26.5 3.277 20.9 0.209 119.757 5.000 14.437 41.76510B 26.5 3.277 21.3 0.209 128.671 5.183 14.437 47.09511A 33.5 3.512 22.3 0.478 152.358 8.533 16.736 30.95711B 33.5 3.512 22.6 0.478 159.854 8.740 16.736 34.38612A 28.1 3.336 18.4 0.266 71.290 4.353 15.012 11.47612B 28.1 3.336 17.8 0.266 61.518 4.043 15.012 7.77113A 9.6 2.262 3.7 0.312 39.146 3.494 4.480 0.60813B 9.6 2.262 3.9 0.312 36.683 3.382 4.480 0.33614A 5.3 1.668 4.2 1.328 33.139 6.635 -1.346 30.75614B 5.3 1.668 4.3 1.328 31.998 6.519 -1.346 31.87515A 18.1 2.896 6.8 0.006 9.965 -0.239 10.699 15.20015B 18.1 2.896 7.1 0.006 8.161 -0.216 10.699 12.951PS-11 Example 2-R 10-27-04.xls Logarithmic CalcsPage 1 of 111/1/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003Correlation equation: y = -17.700 + 9.807 Ln(x)Variable Equation Valuen Number of data points = 30Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.82 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =10.93 Correlation coefficient 0.822>0.85noy~y~ = 1/n*(Sum of (yi)) =10.0 Confidence interval 6.37%<10%yesSyySyy= Sum((yi-y~)^2) =1,531 Tolerance interval 25.6%<25%noSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =107.2 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-17.700b1b1= Sx'y/Sx'x'=9.807SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =4.141y^meany^ at mean x value = 9.957tftffrom table =2.048CICI = tf*SL*SQRT(1/n) =1.549EL Emission Limit = 24.3CI% CI% =CI/EL*100% = 6.37%n'n' = n =30vfvffrom table =1.286un'un'from table =1.168kTkt= un'*vf=1.503TITI = kt*SL =6.22TI% TI% = TI/EL*100% = 25.6%SySy= SQRT(Syy/(n-1)) =7.266PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.675Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.822Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 9.8067Ln(x) - 17.7-505101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21A 6.4 1.3 0.262 174.94 2.957 22.744 0.886 0.3891B 6.4 1.2 0.182 174.94 3.239 23.803 0.886 0.4962A 7.5 2.3 0.833 147.06 1.320 13.934 0.978 0.0212B 7.5 2.5 0.916 147.06 1.136 12.923 0.978 0.0043A 11.2 2.5 0.916 71.01 1.136 8.980 1.284 0.1353B 11.2 2.8 1.030 71.01 0.907 8.025 1.284 0.0654A 16.4 7.7 2.041 10.41 0.004 -0.191 1.715 0.1074B 16.4 7.9 2.067 10.41 0.007 -0.274 1.715 0.1245A 29.9 12.9 2.557 105.54 0.331 5.910 2.833 0.0765B 29.9 12.5 2.526 105.54 0.296 5.587 2.833 0.0946A 22.2 7.8 2.054 6.62 0.005 0.186 2.195 0.0206B 22.2 7.3 1.988 6.62 0.000 0.015 2.195 0.0437A 19.5 5.8 1.758 0.02 0.050 0.028 1.971 0.0467B 19.5 5.9 1.775 0.02 0.043 0.026 1.971 0.0398A 27.4 11.7 2.460 60.42 0.228 3.713 2.626 0.0288B 27.4 12 2.485 60.42 0.253 3.910 2.626 0.0209A 32.8 20.4 3.016 173.54 1.068 13.616 3.073 0.0039B 32.8 20.9 3.040 173.54 1.119 13.935 3.073 0.00110A 26.5 20.9 3.040 47.24 1.119 7.271 2.551 0.23910B 26.5 21.3 3.059 47.24 1.159 7.401 2.551 0.25811A 33.5 22.3 3.105 192.47 1.260 15.575 3.131 0.00111B 33.5 22.6 3.118 192.47 1.291 15.760 3.131 0.00012A 28.1 18.4 2.912 71.80 0.866 7.884 2.684 0.05212B 28.1 17.8 2.879 71.80 0.805 7.603 2.684 0.03813A 9.6 3.7 1.308 100.53 0.454 6.754 1.151 0.02513B 9.6 3.9 1.361 100.53 0.386 6.226 1.151 0.04414A 5.3 4.2 1.435 205.25 0.299 7.835 0.795 0.40914B 5.3 4.3 1.459 205.25 0.274 7.497 0.795 0.44015A 18.1 6.8 1.917 2.33 0.004 0.099 1.855 0.00415B 18.1 7.1 1.960 2.33 0.000 0.033 1.855 0.011PS-11 Example 2-R 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003Correlation equation: y = 1.428 e0.0828 xVariable Equation Valuen Number of data points = 30Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =19.63 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =2,738.4 Correlation coefficient 0.921>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =1.982 Confidence interval 3.80%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =22.02 Tolerance interval 15.9%<25%yesSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =226.81 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =0.356b0b0= e^b0' =1.428b1b1= Sxy'/Sxx=0.083SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.340y'^meany'^ at mean x value = 1.982tftffrom table =2.048CI'CI' = tf*SL*SQRT(1/n) =0.1270LCL' LCL' = y~' - CI' 1.855UCL' UCL' = y~' + CI' 2.109CI CI = (e^UCL' - e^LCL')/2 0.924EL Emission Limit = 24.3CI% CI% =CI/EL*100% = 3.80%n'n' = n =30vfvffrom table =1.286un'un'from table =1.168kTkT= un'*vf=1.503TI'TI' = kt*SL =0.510LTL' LTL' = y~' - TI' 1.472UTL' UTL' = y~' + TI' 2.492TI TI = (e^UTL' - e^LTL')/2 3.866TI% TI% = TI/EL*100% = 15.9%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.871PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.848Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.921Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 1.4281e0.0828x05101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xlsExponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21A 6.4 1.856 1.3 0.262 0.929 2.957 1.658 0.756 0.24331B 6.4 1.856 1.2 0.182 0.929 3.239 1.735 0.756 0.32872A 7.5 2.015 2.3 0.833 0.649 1.320 0.925 0.957 0.01552B 7.5 2.015 2.5 0.916 0.649 1.136 0.858 0.957 0.00173A 11.2 2.416 2.5 0.916 0.163 1.136 0.431 1.468 0.30393B 11.2 2.416 2.8 1.030 0.163 0.907 0.385 1.468 0.19184A 16.4 2.797 7.7 2.041 0.001 0.004 -0.001 1.953 0.00784B 16.4 2.797 7.9 2.067 0.001 0.007 -0.002 1.953 0.01305A 29.9 3.398 12.9 2.557 0.334 0.331 0.332 2.717 0.02555B 29.9 3.398 12.5 2.526 0.334 0.296 0.314 2.717 0.03656A 22.2 3.100 7.8 2.054 0.078 0.005 0.020 2.338 0.08066B 22.2 3.100 7.3 1.988 0.078 0.000 0.002 2.338 0.12267A 19.5 2.970 5.8 1.758 0.023 0.050 -0.034 2.173 0.17237B 19.5 2.970 5.9 1.775 0.023 0.043 -0.031 2.173 0.15848A 27.4 3.311 11.7 2.460 0.240 0.228 0.234 2.606 0.02138B 27.4 3.311 12 2.485 0.240 0.253 0.247 2.606 0.01469A 32.8 3.490 20.4 3.016 0.449 1.068 0.693 2.835 0.03289B 32.8 3.490 20.9 3.040 0.449 1.119 0.709 2.835 0.042110A 26.5 3.277 20.9 3.040 0.209 1.119 0.483 2.563 0.227110B 26.5 3.277 21.3 3.059 0.209 1.159 0.492 2.563 0.245511A 33.5 3.512 22.3 3.105 0.478 1.260 0.776 2.861 0.059111B 33.5 3.512 22.6 3.118 0.478 1.291 0.785 2.861 0.065812A 28.1 3.336 18.4 2.912 0.266 0.866 0.480 2.638 0.075412B 28.1 3.336 17.8 2.879 0.266 0.805 0.463 2.638 0.058313A 9.6 2.262 3.7 1.308 0.312 0.454 0.376 1.271 0.001413B 9.6 2.262 3.9 1.361 0.312 0.386 0.347 1.271 0.008014A 5.3 1.668 4.2 1.435 1.328 0.299 0.630 0.516 0.845314B 5.3 1.668 4.3 1.459 1.328 0.274 0.603 0.516 0.889115A 18.1 2.896 6.8 1.917 0.006 0.004 -0.005 2.078 0.026015B 18.1 2.896 7.1 1.960 0.006 0.000 -0.002 2.078 0.0140PS-11 Example 2-R 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003Correlation equation: y = 0.2007 x1.2722Variable Equation Valuen Number of data points = 30Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.820 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =10.93 Correlation coefficient 0.892>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =1.982 Confidence interval 4.41%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =22.02 Tolerance interval 18.7%<25%yesSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =13.90 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-1.606b0b0= e^b0' =0.201b1b1= Sx'y'/Sx'x'=1.272SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.393y'^meany'^ at mean x value = 1.982tftffrom table =2.048CI'CI = tf*SL*SQRT(1/n) =0.147LCL' LCL' = y~' - CI' 1.835UCL' UCL' = y~' + CI' 2.129CI CI = (e^UCL' - e^LCL')/2 1.071EL Emission Limit = 24.3CI% CI% =CI/EL*100% = 4.41%n'n' = n =30vfvffrom table =1.286un'un'from table =1.168kTkT= un'*vf=1.503TI'TI' = kTSL=0.591LTL' LTL' = y~' - TI' 1.391UTL' UTL' = y~' + TI' 2.573TI TI = (e^UTL' - e^LTL')/2 4.540TI% TI% = TI/EL*100% = 18.7%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.871PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.796Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.892Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.2007x1.272205101520250.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0PM CEMS Response

PM ConcentrationPS-11 Example 2-R 10-27-04.xlsPower Summary10/29/2004SELECTION OF BEST MODELFacility: Facility B Emission Unit: IncineratorLocation: Anytown USATest Dates:April 3 to 6, 2003ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.895 Yes 4.98% Yes 20.0% Yes (b) YesPolynomial 0.919 Yes 5.91% Yes 18.0% Yes Yes YesLogarithmic 0.822 No 6.37% Yes 25.6% No (b) NoExponential 0.921 Yes 3.80% Yes 15.9% Yes (b) YesPower 0.892 Yes 4.41% Yes 18.7% Yes (b) Yes(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: ExponentialPS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 2-R 10-27-04.xls Best Model 10/29/2004

# Appendix A.3

Complete Results of Example No. 3 –

Initial Correlation TestCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: Facility Cresponse mg/acmLocation:Anytown USARun x yEmission Unit: HWI1 5.8 3.5Test Dates: November 1 to 4, 20022 7.6 93 7.8 7.8ValueUnits4 16.4 25.2Emission limit:17.45 mg/acm5 10.1 16.66 11.1 14.37 12.0 19.68 13.2 20.19 15.7 23.410 17.6 26.711 21.7 26.812 21.0 28.913 22.3 25.714 24.1 26.215 24.7 26.2051015202530350.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21 5.8 3.5 92.29 272.25 158.51 8.942 29.6212 7.6 9 60.94 121.00 85.87 11.014 4.0583 7.8 7.8 57.86 148.84 92.80 11.245 11.8654 16.4 25.2 0.99 27.04 5.17 21.143 16.4565 10.1 16.6 28.16 11.56 18.04 13.892 7.3346 11.1 14.3 18.55 32.49 24.55 15.043 0.5527 12 19.6 11.61 0.16 1.36 16.079 12.3998 13.2 20.1 4.87 0.01 -0.22 17.460 6.9699 15.7 23.4 0.09 11.56 1.00 20.338 9.37810 17.6 26.7 4.81 44.89 14.70 22.525 17.43411 21.7 26.8 39.61 46.24 42.79 27.244 0.19712 21 28.9 31.29 79.21 49.78 26.438 6.06113 22.3 25.7 47.52 32.49 39.29 27.934 4.99314 24.1 26.2 75.57 38.44 53.90 30.006 14.48715 24.7 26.2 86.37 38.44 57.62 30.697 20.222PS-11 Example 3-R 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = 2.267 + 1.151 xVariable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =15.41 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =561 Correlation coefficient 0.898>0.85yesy~y~ = 1/n*(Sum of (yi)) =20.00 Confidence interval 11.29%<10%noSyySyy= Sum((yi-y~)^2) =905 Tolerance interval 35.6%<25%noSxySxy= Sum((xi-x~)(yi-y~)) =645.2 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =2.267b1b1= Sxy/Sxx=1.151SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =3.530y^meany^ at mean x value = 20.000tftffrom table =2.160CICI = tf*SL*SQRT(1/n) =1.969EL Emission Limit = 17.5CI% CI% =CI/EL*100% = 11.29%n' n' = n = 15vfvffrom table =1.485un'un'from table =1.184kTkt= un'*vf=1.759TITI = kt*SL=6.210TI% TI% = TI/EL*100% = 35.6%SySy= SQRT(Syy/(n-1)) =8.038PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.807Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.898Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 1.151x + 2.2665051015202530350.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1 5.8 3.5 341951,132201183.1380.1310.4679272.25 1.912 7.6 9 584393,336685208.4290.3260.2223121.00 1.313 7.8 7.8 614753,702614758.9761.3840.2060148.84 1.274 16.4 25.2 2694,41172,3394136,77824.8510.1210.165027.04 1.135 10.1 16.6 1021,03010,4061681,69314.6893.6530.121411.56 0.976 11.1 14.3 1231,36815,1811591,76216.8386.4440.122432.49 0.987 12 19.6 1441,72820,7362352,82218.6001.0000.13200.16 1.018 13.2 20.1 1742,30030,3602653,50220.6940.3520.14880.01 1.089 15.7 23.4 2463,87060,7573675,76824.1190.5170.167211.56 1.1410 17.6 26.7 3105,45295,9514708,27125.8760.6790.154144.89 1.0911 21.7 26.8 47110,218221,73758212,62027.1770.1420.137446.24 1.0312 21 28.9 4419,261194,48160712,74527.1962.9030.126579.21 0.9913 22.3 25.7 49711,090247,29757312,78027.0821.9110.156532.49 1.1014 24.1 26.2 58113,998337,34063115,21726.3600.0260.294938.44 1.5115 24.7 26.2 61015,069372,21064715,98425.9730.0510.377538.44 1.71PS-11 Example 3-R 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: Facility CEmission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002Variable Equation ValueCorrelation equation: y = -18.371 + 4.296 x + -0.10123 x2n Number of data points = 15S1S1 = Sum (xi) = 231Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 4,121 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 80,903 Correlation coefficient 0.987>0.85yesS4S4 = Sum (xi4) = 1,686,966 Confidence interval 5.56%<10%yesS5S5 = Sum (yi) = 300 Tolerance interval 13.4%<25%yesS6S6 = Sum (xiyi) = 5,267 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 101,055det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 1.168E+08Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = -18.371 Correlation curve minimum point NAb1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = 4.296 Minimum allowable x value NAb2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = -0.10123 Is correlation curve minimum < minimum x value? NASpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =1.279 Correlation curve maximum point 21.2DD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =1.168E+08 Extrapolation limit for x (125% of maximum x value) 30.9C0C0 = (S2S4-S32)/D =3.481 Is correlation curve maximum > extrapolation limit? noC1C1 = (S3S2-S1S4)/D = -0.4833C2C2 = (S1S3-S22)/D =1.467E-02C3C3 = (nS4-S22)/D =7.124E-02C4C4 = (S1S2-nS3)/D = -2.236E-03C5C5 = (nS2-S12)/D =7.197E-05tftffrom table =2.179EL Emission Limit = 17.5CICI = tf*Sp*sqrt(delta-min) =0.971CI% CI% = CI/EL*100= 5.56%vfvffrom table =1.515un'un'from table =1.208n' n' = 1/delta = 8.24kTkt = un'*vf = 1.831TI TI = kt*sp = 2.342TI% TI% = TI/EL *100% = 13.4%y~ y~ = 1/n*(Sum of (yi)) = 20.00SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 8.04r2r2= 1-(Sp^2/Sy^2) =0.975rr = sqrt((1-Sp^2/Sy^2)) =0.987Max-min?b2> 0?maximumPS-11 Correlation Testxmax-miny = -b1/2b2=21.22Calculation Spreadsheet1.25xmax30.88Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = -0.1012x2+ 4.2957x - 18.371051015202530350.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21 5.8 1.758 3.5 0.785 272.250 14.620 5.106 2.5782 7.6 2.028 9 0.379 121.000 6.774 9.649 0.4213 7.8 2.054 7.8 0.348 148.840 7.196 10.086 5.2254 16.4 2.797 25.2 0.024 27.040 0.797 22.578 6.8775 10.1 2.313 16.6 0.110 11.560 1.127 14.429 4.7116 11.1 2.407 14.3 0.056 32.490 1.351 16.016 2.9467 12 2.485 19.6 0.025 0.160 0.064 17.327 5.1678 13.2 2.580 20.1 0.004 0.010 -0.006 18.929 1.3719 15.7 2.754 23.4 0.012 11.560 0.373 21.844 2.42010 17.6 2.868 26.7 0.050 44.890 1.501 23.765 8.61611 21.7 3.077 26.8 0.188 46.240 2.947 27.285 0.23512 21 3.045 28.9 0.160 79.210 3.565 26.734 4.69313 22.3 3.105 25.7 0.212 32.490 2.626 27.743 4.17514 24.1 3.182 26.2 0.290 38.440 3.337 29.048 8.11115 24.7 3.207 26.2 0.317 38.440 3.490 29.461 10.637PS-11 Example 3-R 10-27-04.xls Logarithmic CalcsPage 1 of 110/29/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = -24.443 + 16.809 Ln(x)Variable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.64 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.96 Correlation coefficient 0.959>0.85yesy~y~ = 1/n*(Sum of (yi)) =20.0 Confidence interval 7.32%<10%yesSyySyy= Sum((yi-y~)^2) =905 Tolerance interval 23.1%<25%yesSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =49.8 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-24.443b1b1= Sx'y/Sx'x'=16.809SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =2.290y^meany^ at mean x value = 20.000tftffrom table =2.160CICI = tf*SL*SQRT(1/n) =1.277EL Emission Limit = 17.5CI% CI% =CI/EL*100% = 7.32%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkt= un'*vf=1.759TITI = kt*SL =4.03TI% TI% = TI/EL*100% = 23.1%SySy= SQRT(Syy/(n-1)) =8.038PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.919Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.959Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 16.809Ln(x) - 24.443051015202530350.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21 5.8 3.5 1.253 92.29 2.614 15.533 2.115 0.7442 7.6 9 2.197 60.94 0.452 5.249 2.256 0.0043 7.8 7.8 2.054 57.86 0.665 6.203 2.272 0.0484 16.4 25.2 3.227 0.99 0.128 0.355 2.948 0.0785 10.1 16.6 2.809 28.16 0.004 0.320 2.453 0.1276 11.1 14.3 2.660 18.55 0.044 0.902 2.531 0.0177 12 19.6 2.976 11.61 0.011 -0.361 2.602 0.1398 13.2 20.1 3.001 4.87 0.017 -0.289 2.696 0.0939 15.7 23.4 3.153 0.09 0.080 0.083 2.893 0.06810 17.6 26.7 3.285 4.81 0.172 0.910 3.042 0.05911 21.7 26.8 3.288 39.61 0.175 2.635 3.364 0.00612 21 28.9 3.364 31.29 0.244 2.764 3.309 0.00313 22.3 25.7 3.246 47.52 0.142 2.598 3.411 0.02714 24.1 26.2 3.266 75.57 0.157 3.444 3.552 0.08215 24.7 26.2 3.266 86.37 0.157 3.681 3.600 0.111PS-11 Example 3-R 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = 5.256 e0.0785 xVariable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =15.41 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =560.5 Correlation coefficient 0.812>0.85noy'~y'~ = 1/n*(Sum of (y'i)) =2.870 Confidence interval 19.93%<10%noSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.06 Tolerance interval 66.5%<25%noSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =44.03 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =1.659b0b0= e^b0' =5.256b1b1= Sxy'/Sxx=0.079SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.351y'^meany'^ at mean x value = 2.870tftffrom table =2.160CI'CI' = tf*SL*SQRT(1/n) =0.1960LCL' LCL' = y~' - CI' 2.674UCL' UCL' = y~' + CI' 3.066CI CI = (e^UCL' - e^LCL')/2 3.477EL Emission Limit = 17.5CI% CI% =CI/EL*100% = 19.93%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkT= un'*vf=1.759TI'TI' = kt*SL =0.618LTL' LTL' = y~' - TI' 2.252UTL' UTL' = y~' + TI' 3.488TI TI = (e^UTL' - e^LTL')/2 11.602TI% TI% = TI/EL*100% = 66.5%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.601PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.659Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.812Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 5.2565e0.0785x05101520253035400.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xlsExponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21 5.8 1.758 3.5 1.253 0.785 2.614 1.433 1.810 0.31052 7.6 2.028 9 2.197 0.379 0.452 0.414 2.133 0.00413 7.8 2.054 7.8 2.054 0.348 0.665 0.481 2.164 0.01214 16.4 2.797 25.2 3.227 0.024 0.128 0.055 3.053 0.03025 10.1 2.313 16.6 2.809 0.110 0.004 0.020 2.473 0.11296 11.1 2.407 14.3 2.660 0.056 0.044 0.050 2.586 0.00557 12 2.485 19.6 2.976 0.025 0.011 -0.017 2.679 0.08778 13.2 2.580 20.1 3.001 0.004 0.017 -0.008 2.793 0.04309 15.7 2.754 23.4 3.153 0.012 0.080 0.031 3.001 0.023110 17.6 2.868 26.7 3.285 0.050 0.172 0.093 3.137 0.021711 21.7 3.077 26.8 3.288 0.188 0.175 0.181 3.388 0.009912 21 3.045 28.9 3.364 0.160 0.244 0.198 3.349 0.000213 22.3 3.105 25.7 3.246 0.212 0.142 0.174 3.421 0.030314 24.1 3.182 26.2 3.266 0.290 0.157 0.213 3.513 0.061315 24.7 3.207 26.2 3.266 0.317 0.157 0.223 3.543 0.0767PS-11 Example 3-R 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = 0.7466 x1.1959Variable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.644 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.96 Correlation coefficient 0.908>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =2.870 Confidence interval 14.28%<10%noSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.06 Tolerance interval 46.4%<25%noSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =3.54 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-0.292b0b0= e^b0' =0.747b1b1= Sx'y'/Sx'x'=1.196SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.253y'^meany'^ at mean x value = 2.870tftffrom table =2.160CI'CI = tf*SL*SQRT(1/n) =0.141LCL' LCL' = y~' - CI' 2.729UCL' UCL' = y~' + CI' 3.011CI CI = (e^UCL' - e^LCL')/2 2.492EL Emission Limit = 17.5CI% CI% =CI/EL*100% = 14.28%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkT= un'*vf=1.759TI'TI' = kTSL=0.444LTL' LTL' = y~' - TI' 2.425UTL' UTL' = y~' + TI' 3.314TI TI = (e^UTL' - e^LTL')/2 8.093TI% TI% = TI/EL*100% = 46.4%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.601PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.824Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.908Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.7466x1.195905101520253035400.0 5.0 10.0 15.0 20.0 25.0 30.0PM CEMS Response

PM ConcentrationPS-11 Example 3-R 10-27-04.xlsPower Summary10/29/2004SELECTION OF BEST MODELFacility: Facility C Emission Unit: HWILocation: Anytown USATest Dates: November 1 to 4, 2002ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.898 Yes 11.29% No 35.6% No (b) NoPolynomial 0.987 Yes 5.56% Yes 13.4% Yes No NoLogarithmic 0.959 Yes 7.32% Yes 23.1% Yes (b) YesExponential 0.812 No 19.93% No 66.5% No (b) NoPower 0.908 Yes 14.28% No 46.4% No (b) No(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: LogarithmicPS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 3-R 10-27-04.xls Best Model 10/29/2004

# Appendix A.4

Complete Results of Example No. 4

Original Test DataCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: Facility Dresponse mg/acmLocation:Anytown USARun x yEmission Unit: Boiler1 6.7 3.1Test Dates: November 1 to 4, 20022 7.6 7.43 7.8 2.1ValueUnits4 9.2 3.7Emission limit:18.5 mg/acm5 10.1 6.16 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.510 14.6 8.211 19.7 13.012 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.2024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21 6.7 3.1 52.22 16.21 29.10 2.808 0.0852 7.6 7.4 40.03 0.07 -1.73 3.346 16.4373 7.8 2.1 37.54 25.27 30.80 3.465 1.8644 9.2 3.7 22.34 11.74 16.20 4.302 0.3625 10.1 6.1 14.64 1.05 3.93 4.840 1.5886 11.1 2.8 7.99 18.72 12.23 5.437 6.9567 12 4.9 3.71 4.96 4.29 5.975 1.1568 18.2 8.3 18.26 1.38 5.01 9.680 1.9069 13.7 4.5 0.05 6.90 0.60 6.991 6.20610 14.6 8.2 0.45 1.15 0.72 7.529 0.45011 19.7 13 33.33 34.50 33.91 10.577 5.87212 18 11.1 16.59 15.79 16.18 9.561 2.36913 20.3 11.9 40.62 22.78 30.42 10.935 0.93014 20.2 10.6 39.35 12.06 21.79 10.876 0.07615 19.7 9.2 33.33 4.30 11.97 10.577 1.896PS-11 Example 4A-R 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = -1.196 + 0.598 xVariable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.93 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =360 Correlation coefficient 0.841>0.85noy~y~ = 1/n*(Sum of (yi)) =7.13 Confidence interval 5.80%<10%yesSyySyy= Sum((yi-y~)^2) =177 Tolerance interval 18.3%<25%yesSxySxy= Sum((xi-x~)(yi-y~)) =215.4 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =-1.196b1b1= Sxy/Sxx=0.598SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.925y^meany^ at mean x value = 7.127tftffrom table =2.160CICI = tf*SL*SQRT(1/n) =1.074EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 5.80%n' n' = n = 15vfvffrom table =1.485un'un'from table =1.184kTkt= un'*vf=1.759TITI = kt*SL=3.386TI% TI% = TI/EL*100% = 18.3%SySy= SQRT(Syy/(n-1)) =3.555PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.707Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.841Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 0.5976x - 1.196024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1 6.7 3.1 453012,015211394.0240.8540.411116.21 2.572 7.6 7.4 584393,336564274.03211.3460.24120.07 1.973 7.8 2.1 614753,702161284.0433.7750.215825.27 1.874 9.2 3.7 857797,164343134.2190.2700.129611.74 1.455 10.1 6.1 1021,03010,406626224.4232.8120.13071.05 1.456 11.1 2.8 1231,36815,181313454.7333.7350.155918.72 1.597 12 4.9 1441,72820,736597065.0850.0340.18384.96 1.728 18.2 8.3 3316,029109,7201512,7499.4371.2920.12531.38 1.429 13.7 4.5 1882,57135,228628455.9452.0880.21456.90 1.8610 14.6 8.2 2133,11245,4371201,7486.5022.8830.21031.15 1.8411 19.7 13 3887,645150,6142565,04510.9934.0270.182534.50 1.7212 18 11.1 3245,832104,9762003,5969.2443.4440.126215.79 1.4313 20.3 11.9 4128,365169,8182424,90411.6710.0530.252322.78 2.0214 20.2 10.6 4088,242166,4972144,32511.5560.9130.238212.06 1.9615 19.7 9.2 3887,645150,6141813,57010.9933.2160.18254.30 1.72PS-11 Example 4A-R 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: Facility DEmission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002Variable Equation ValueCorrelation equation: y = 6.189 + -0.615 x + 0.04361 x2n Number of data points = 15S1S1 = Sum (xi) = 209Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 3,270 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 55,562 Correlation coefficient 0.855>0.85yesS4S4 = Sum (xi4) = 995,443 Confidence interval 7.68%<10%yesS5S5 = Sum (yi) = 107 Tolerance interval 18.2%<25%yesS6S6 = Sum (xiyi) = 1,704 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 29,463det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 2.107E+07Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 6.189 Correlation curve minimum point 7.05b1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.615 Minimum allowable x value 6.7b2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.04361 Is correlation curve minimum < minimum x value? noSpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =1.843 Correlation curve maximum point NADD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =2.107E+07 Extrapolation limit for x (125% of maximum x value) NAC0C0 = (S2S4-S32)/D =7.963 Is correlation curve maximum > extrapolation limit? NAC1C1 = (S3S2-S1S4)/D = -1.2473C2C2 = (S1S3-S22)/D =4.346E-02C3C3 = (nS4-S22)/D =2.013E-01C4C4 = (S1S2-nS3)/D = -7.138E-03C5C5 = (nS2-S12)/D =2.567E-04tftffrom table =2.179EL Emission Limit = 18.5CICI = tf*Sp*sqrt(delta-min) =1.421CI% CI% = CI/EL*100= 7.68%vfvffrom table =1.515un'un'from table =1.208n' n' = 1/delta = 7.98kTkt = un'*vf = 1.831TI TI = kt*sp = 3.373TI% TI% = TI/EL *100% = 18.2%y~ y~ = 1/n*(Sum of (yi)) = 7.13SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.55r2r2= 1-(Sp^2/Sy^2) =0.731rr = sqrt((1-Sp^2/Sy^2)) =0.855Max-min?b2> 0?minimumPS-11 Correlation Testxmax-miny = -b1/2b2=7.05Calculation Spreadsheet1.25xmax25.38Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = 0.0436x2- 0.6153x + 6.189024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21 6.7 1.902 3.1 0.439 16.214 2.669 2.244 0.7322 7.6 2.028 7.4 0.288 0.075 -0.147 3.173 17.8713 7.8 2.054 2.1 0.261 25.267 2.568 3.364 1.5974 9.2 2.219 3.7 0.120 11.742 1.185 4.580 0.7745 10.1 2.313 6.1 0.064 1.054 0.259 5.267 0.6946 11.1 2.407 2.8 0.025 18.720 0.684 5.962 10.0007 12 2.485 4.9 0.006 4.958 0.178 6.536 2.6788 18.2 2.901 8.3 0.113 1.377 0.395 9.604 1.7009 13.7 2.617 4.5 0.003 6.899 -0.138 7.512 9.07310 14.6 2.681 8.2 0.013 1.152 0.124 7.981 0.04811 19.7 2.981 13 0.173 34.496 2.441 10.187 7.91212 18 2.890 11.1 0.106 15.787 1.293 9.523 2.48813 20.3 3.011 11.9 0.199 22.785 2.127 10.408 2.22614 20.2 3.006 10.6 0.194 12.064 1.530 10.372 0.05215 19.7 2.981 9.2 0.173 4.299 0.862 10.187 0.975PS-11 Example 4A-R 10-27-04.xls Logarithmic CalcsPage 1 of 110/29/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = -11.764 + 7.365 Ln(x)Variable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.57 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.18 Correlation coefficient 0.801>0.85noy~y~ = 1/n*(Sum of (yi)) =7.1 Confidence interval 6.41%<10%yesSyySyy= Sum((yi-y~)^2) =177 Tolerance interval 20.2%<25%yesSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =16.0 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-11.764b1b1= Sx'y/Sx'x'=7.365SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =2.127y^meany^ at mean x value = 7.127tftffrom table =2.160CICI = tf*SL*SQRT(1/n) =1.187EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 6.41%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkt= un'*vf=1.759TITI = kt*SL =3.74TI% TI% = TI/EL*100% = 20.2%SySy= SQRT(Syy/(n-1)) =3.555PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.642Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.801Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 7.3647Ln(x) - 11.764024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21 6.7 3.1 1.131 52.22 0.480 5.007 1.149 0.0002 7.6 7.4 2.001 40.03 0.031 -1.121 1.233 0.5903 7.8 2.1 0.742 37.54 1.171 6.631 1.252 0.2604 9.2 3.7 1.308 22.34 0.266 2.438 1.383 0.0065 10.1 6.1 1.808 14.64 0.000 0.061 1.467 0.1176 11.1 2.8 1.030 7.99 0.631 2.246 1.560 0.2817 12 4.9 1.589 3.71 0.055 0.453 1.644 0.0038 18.2 8.3 2.116 18.26 0.085 1.248 2.223 0.0119 13.7 4.5 1.504 0.05 0.102 0.073 1.803 0.08910 14.6 8.2 2.104 0.45 0.078 0.188 1.887 0.04711 19.7 13 2.565 33.33 0.549 4.276 2.364 0.04112 18 11.1 2.407 16.59 0.340 2.374 2.205 0.04113 20.3 11.9 2.477 40.62 0.426 4.157 2.420 0.00314 20.2 10.6 2.361 39.35 0.288 3.367 2.410 0.00215 19.7 9.2 2.219 33.33 0.156 2.280 2.364 0.021PS-11 Example 4A-R 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = 1.687 e0.0934 xVariable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.93 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =360.5 Correlation coefficient 0.806>0.85noy'~y'~ = 1/n*(Sum of (y'i)) =1.824 Confidence interval 6.41%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =4.66 Tolerance interval 21.3%<25%yesSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =33.68 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =0.523b0b0= e^b0' =1.687b1b1= Sxy'/Sxx=0.093SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.341y'^meany'^ at mean x value = 1.824tftffrom table =2.160CI'CI' = tf*SL*SQRT(1/n) =0.1903LCL' LCL' = y~' - CI' 1.634UCL' UCL' = y~' + CI' 2.015CI CI = (e^UCL' - e^LCL')/2 1.187EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 6.41%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkT= un'*vf=1.759TI'TI' = kt*SL =0.600LTL' LTL' = y~' - TI' 1.224UTL' UTL' = y~' + TI' 2.424TI TI = (e^UTL' - e^LTL')/2 3.947TI% TI% = TI/EL*100% = 21.3%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.577PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.650Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.806Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 1.6872e0.0934x024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xls Exponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21 6.7 1.902 3.1 1.131 0.439 0.480 0.459 1.049 0.00682 7.6 2.028 7.4 2.001 0.288 0.031 -0.095 1.196 0.64863 7.8 2.054 2.1 0.742 0.261 1.171 0.553 1.227 0.23484 9.2 2.219 3.7 1.308 0.120 0.266 0.178 1.420 0.01245 10.1 2.313 6.1 1.808 0.064 0.000 0.004 1.529 0.07816 11.1 2.407 2.8 1.030 0.025 0.631 0.126 1.639 0.37177 12 2.485 4.9 1.589 0.006 0.055 0.019 1.730 0.01998 18.2 2.901 8.3 2.116 0.113 0.085 0.098 2.218 0.01039 13.7 2.617 4.5 1.504 0.003 0.102 -0.017 1.885 0.145410 14.6 2.681 8.2 2.104 0.013 0.078 0.032 1.960 0.020811 19.7 2.981 13 2.565 0.173 0.549 0.308 2.310 0.064812 18 2.890 11.1 2.407 0.106 0.340 0.190 2.205 0.040913 20.3 3.011 11.9 2.477 0.199 0.426 0.291 2.345 0.017214 20.2 3.006 10.6 2.361 0.194 0.288 0.236 2.340 0.000415 19.7 2.981 9.2 2.219 0.173 0.156 0.164 2.310 0.0083PS-11 Example 4A-R 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002Correlation equation: y = 0.3084 x1.1698Variable Equation Valuen Number of data points = 15Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.565 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.18 Correlation coefficient 0.782>0.85noy'~y'~ = 1/n*(Sum of (y'i)) =1.824 Confidence interval 6.76%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =4.66 Tolerance interval 22.6%<25%yesSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =2.55 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-1.177b0b0= e^b0' =0.308b1b1= Sx'y'/Sx'x'=1.170SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.360y'^meany'^ at mean x value = 1.824tftffrom table =2.160CI'CI = tf*SL*SQRT(1/n) =0.201LCL' LCL' = y~' - CI' 1.624UCL' UCL' = y~' + CI' 2.025CI CI = (e^UCL' - e^LCL')/2 1.251EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 6.76%n'n' = n =15vfvffrom table =1.485un'un'from table =1.184kTkT= un'*vf=1.759TI'TI' = kTSL=0.632LTL' LTL' = y~' - TI' 1.192UTL' UTL' = y~' + TI' 2.457TI TI = (e^UTL' - e^LTL')/2 4.187TI% TI% = TI/EL*100% = 22.6%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.577PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.612Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.782Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.3084x1.1698024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4A-R 10-27-04.xlsPower Summary10/29/2004SELECTION OF BEST MODELFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: November 1 to 4, 2002ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.841 No 5.80% Yes 18.3% Yes (b) NoPolynomial 0.855 Yes 7.68% Yes 18.2% Yes No NoLogarithmic 0.801 No 6.41% Yes 20.2% Yes (b) NoExponential 0.806 No 6.41% Yes 21.3% Yes (b) NoPower 0.782 No 6.76% Yes 22.6% Yes (b) No(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: None acceptablePS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 4A-R 10-27-04.xls Best Model 10/29/2004

# Appendix A.5

Complete Results of Example No. 4

With Five Additional Test RunsCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: Facility Dresponse mg/acmLocation:Anytown USARun x yEmission Unit: Boiler1 6.7 3.1Test Dates: 11/1-11/4/02, 1/7-1/8/032 7.6 7.43 7.8 2.1ValueUnits4 9.2 3.7Emission limit:18.5 mg/acm5 10.1 6.16 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.510 14.6 8.211 19.7 13.012 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.216 5.2 2.417 8.6 4.818 14.1 6.219 14.5 7.020 17.3 8.6024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21 6.7 3.1 45.29 13.65 24.87 2.936 0.0272 7.6 7.4 33.99 0.37 -3.53 3.452 15.5833 7.8 2.1 31.70 22.04 26.43 3.567 2.1524 9.2 3.7 17.89 9.58 13.09 4.370 0.4495 10.1 6.1 11.09 0.48 2.31 4.886 1.4746 11.1 2.8 5.43 15.96 9.31 5.459 7.0717 12 4.9 2.04 3.59 2.71 5.975 1.1568 18.2 8.3 22.75 2.27 7.18 9.530 1.5129 13.7 4.5 0.07 5.27 -0.62 6.950 6.00210 14.6 8.2 1.37 1.97 1.64 7.466 0.53911 19.7 13 39.31 38.50 38.91 10.390 6.81312 18 11.1 20.88 18.53 19.67 9.415 2.83913 20.3 11.9 47.20 26.06 35.07 10.734 1.36014 20.2 10.6 45.83 14.48 25.76 10.677 0.00615 19.7 9.2 39.31 5.78 15.08 10.390 1.41616 5.2 2.4 67.73 19.32 36.17 2.076 0.10517 8.6 4.8 23.33 3.98 9.64 4.026 0.59918 14.1 6.2 0.45 0.35 -0.40 7.179 0.95919 14.5 7 1.14 0.04 0.22 7.408 0.16720 17.3 8.6 14.98 3.26 6.99 9.014 0.171PS-11 Example 4B-R 20 Runs 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = -0.905 + 0.573 xVariable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.43 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =472 Correlation coefficient 0.861>0.85yesy~y~ = 1/n*(Sum of (yi)) =6.80 Confidence interval 4.25%<10%yesSyySyy= Sum((yi-y~)^2) =205 Tolerance interval 14.7%<25%yesSxySxy= Sum((xi-x~)(yi-y~)) =270.5 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =-0.905b1b1= Sxy/Sxx=0.573SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.673y^meany^ at mean x value = 6.795tftffrom table =2.101CICI = tf*SL*SQRT(1/n) =0.786EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 4.25%n' n' = n = 20vfvffrom table =1.384un'un'from table =1.177kTkt= un'*vf=1.629TITI = kt*SL=2.726TI% TI% = TI/EL*100% = 14.7%SySy= SQRT(Syy/(n-1)) =3.289PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.741Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.861Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 0.5733x - 0.905024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1 6.7 3.1 453012,015211393.5780.2290.203513.65 1.522 7.6 7.4 584393,336564273.74513.3600.13400.37 1.233 7.8 2.1 614753,702161283.7892.8520.124022.04 1.194 9.2 3.7 857797,164343134.1670.2180.09379.58 1.035 10.1 6.1 1021,03010,406626224.4742.6430.09720.48 1.056 11.1 2.8 1231,36815,181313454.8764.3100.109015.96 1.117 12 4.9 1441,72820,736597065.2910.1530.11973.59 1.168 18.2 8.3 3316,029109,7201512,7499.5311.5160.09822.27 1.059 13.7 4.5 1882,57135,228628456.2142.9370.12595.27 1.1910 14.6 8.2 2133,11245,4371201,7486.7762.0290.11951.97 1.1611 19.7 13 3887,645150,6142565,04510.9194.3290.172538.50 1.4012 18 11.1 3245,832104,9762003,5969.3573.0390.094718.53 1.0413 20.3 11.9 4128,365169,8182424,90411.5140.1490.235126.06 1.6314 20.2 10.6 4088,242166,4972144,32511.4130.6620.223014.48 1.5915 19.7 9.2 3887,645150,6141813,57010.9192.9560.17255.78 1.4016 5.2 2.4 2714173112653.4131.0270.443319.32 2.2417 8.6 4.8 746365,470413553.9900.6570.09963.98 1.0618 14.1 6.2 1992,80339,525871,2336.4570.0660.12380.35 1.1819 14.5 7 2103,04944,2051021,4726.7110.0840.12050.04 1.1720 17.3 8.6 2995,17889,5751492,5748.7660.0280.09033.26 1.01PS-11 Example 4B-R 20 Runs 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: Facility DEmission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Variable Equation ValueCorrelation equation: y = 3.935 + -0.263 x + 0.03137 x2n Number of data points = 20S1S1 = Sum (xi) = 269Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 4,079 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 67,368 Correlation coefficient 0.875>0.85yesS4S4 = Sum (xi4) = 1,174,949 Confidence interval 5.47%<10%yesS5S5 = Sum (yi) = 136 Tolerance interval 14.4%<25%yesS6S6 = Sum (xiyi) = 2,096 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 35,161det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 6.865E+07Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 3.935 Correlation curve minimum point 4.20b1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.263 Minimum allowable x value 5.2b2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.03137 Is correlation curve minimum < minimum x value? yesSpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =1.595 Correlation curve maximum point NADD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =6.865E+07 Extrapolation limit for x (125% of maximum x value) NAC0C0 = (S2S4-S32)/D =3.705 Is correlation curve maximum > extrapolation limit? NAC1C1 = (S3S2-S1S4)/D = -0.5942C2C2 = (S1S3-S22)/D =2.121E-02C3C3 = (nS4-S22)/D =9.993E-02C4C4 = (S1S2-nS3)/D = -3.667E-03C5C5 = (nS2-S12)/D =1.375E-04tftffrom table =2.110EL Emission Limit = 18.5CICI = tf*Sp*sqrt(delta-min) =1.011CI% CI% = CI/EL*100= 5.47%vfvffrom table =1.400un'un'from table =1.195n' n' = 1/delta = 11.07kTkt = un'*vf = 1.673TI TI = kt*sp = 2.668TI% TI% = TI/EL *100% = 14.4%y~ y~ = 1/n*(Sum of (yi)) = 6.80SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.29r2r2= 1-(Sp^2/Sy^2) =0.765rr = sqrt((1-Sp^2/Sy^2)) =0.875Max-min?b2> 0?minimumPS-11 Correlation Testxmax-miny = -b1/2b2=4.20Calculation Spreadsheet1.25xmax25.38Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = 0.0314x2- 0.2634x + 3.9349024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21 6.7 1.902 3.1 0.385 13.653 2.292 2.717 0.1462 7.6 2.028 7.4 0.244 0.366 -0.299 3.546 14.8543 7.8 2.054 2.1 0.219 22.043 2.198 3.717 2.6144 9.2 2.219 3.7 0.092 9.579 0.938 4.802 1.2145 10.1 2.313 6.1 0.044 0.483 0.146 5.416 0.4686 11.1 2.407 2.8 0.013 15.960 0.461 6.036 10.4737 12 2.485 4.9 0.001 3.591 0.071 6.549 2.7188 18.2 2.901 8.3 0.144 2.265 0.570 9.287 0.9749 13.7 2.617 4.5 0.009 5.267 -0.218 7.420 8.52510 14.6 2.681 8.2 0.025 1.974 0.223 7.838 0.13111 19.7 2.981 13 0.210 38.502 2.844 9.808 10.19012 18 2.890 11.1 0.135 18.533 1.584 9.214 3.55513 20.3 3.011 11.9 0.238 26.061 2.493 10.005 3.59114 20.2 3.006 10.6 0.234 14.478 1.839 9.973 0.39415 19.7 2.981 9.2 0.210 5.784 1.102 9.808 0.36916 5.2 1.649 2.4 0.763 19.316 3.840 1.051 1.82017 8.6 2.152 4.8 0.137 3.980 0.739 4.359 0.19518 14.1 2.646 6.2 0.015 0.354 -0.074 7.609 1.98519 14.5 2.674 7 0.023 0.042 0.031 7.793 0.62920 17.3 2.851 8.6 0.108 3.258 0.593 8.954 0.125PS-11 Example 4B-R 20 Runs 10-27-04.xls Logarithmic CalcsPage 1 of 110/29/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = -9.788 + 6.574 Ln(x)Variable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.52 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =3.25 Correlation coefficient 0.816>0.85noy~y~ = 1/n*(Sum of (yi)) =6.8 Confidence interval 4.82%<10%yesSyySyy= Sum((yi-y~)^2) =205 Tolerance interval 16.7%<25%yesSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =21.4 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-9.788b1b1= Sx'y/Sx'x'=6.574SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.900y^meany^ at mean x value = 6.795tftffrom table =2.101CICI = tf*SL*SQRT(1/n) =0.893EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 4.82%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkt= un'*vf=1.629TITI = kt*SL =3.09TI% TI% = TI/EL*100% = 16.7%SySy= SQRT(Syy/(n-1)) =3.289PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.666Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.816Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 6.5743Ln(x) - 9.7878024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21 6.7 3.1 1.131 45.29 0.429 4.409 1.157 0.0012 7.6 7.4 2.001 33.99 0.046 -1.253 1.241 0.5783 7.8 2.1 0.742 31.70 1.091 5.881 1.260 0.2684 9.2 3.7 1.308 17.89 0.229 2.023 1.391 0.0075 10.1 6.1 1.808 11.09 0.000 -0.073 1.475 0.1116 11.1 2.8 1.030 5.43 0.573 1.763 1.569 0.2907 12 4.9 1.589 2.04 0.039 0.282 1.653 0.0048 18.2 8.3 2.116 22.75 0.109 1.573 2.233 0.0149 13.7 4.5 1.504 0.07 0.080 -0.076 1.812 0.09510 14.6 8.2 2.104 1.37 0.101 0.372 1.896 0.04311 19.7 13 2.565 39.31 0.606 4.881 2.373 0.03712 18 11.1 2.407 20.88 0.385 2.836 2.214 0.03713 20.3 11.9 2.477 47.20 0.476 4.741 2.429 0.00214 20.2 10.6 2.361 45.83 0.330 3.889 2.420 0.00315 19.7 9.2 2.219 39.31 0.187 2.713 2.373 0.02416 5.2 2.4 0.875 67.73 0.830 7.498 1.017 0.02017 8.6 4.8 1.569 23.33 0.047 1.052 1.335 0.05518 14.1 6.2 1.825 0.45 0.001 0.026 1.849 0.00119 14.5 7 1.946 1.14 0.025 0.171 1.887 0.00420 17.3 8.6 2.152 14.98 0.133 1.414 2.148 0.000PS-11 Example 4B-R 20 Runs 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = 1.700 e0.0935 xVariable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.43 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =471.8 Correlation coefficient 0.840>0.85noy'~y'~ = 1/n*(Sum of (y'i)) =1.786 Confidence interval 4.52%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.72 Tolerance interval 16.3%<25%yesSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =44.12 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =0.531b0b0= e^b0' =1.700b1b1= Sxy'/Sxx=0.094SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.298y'^meany'^ at mean x value = 1.786tftffrom table =2.101CI'CI' = tf*SL*SQRT(1/n) =0.1398LCL' LCL' = y~' - CI' 1.647UCL' UCL' = y~' + CI' 1.926CI CI = (e^UCL' - e^LCL')/2 0.837EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 4.52%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkT= un'*vf=1.629TI'TI' = kt*SL =0.485LTL' LTL' = y~' - TI' 1.302UTL' UTL' = y~' + TI' 2.271TI TI = (e^UTL' - e^LTL')/2 3.007TI% TI% = TI/EL*100% = 16.3%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.549PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.706Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.840Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 1.7e0.0935x024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Exponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21 6.7 1.902 3.1 1.131 0.385 0.429 0.406 1.100 0.00102 7.6 2.028 7.4 2.001 0.244 0.046 -0.106 1.240 0.58023 7.8 2.054 2.1 0.742 0.219 1.091 0.489 1.269 0.27734 9.2 2.219 3.7 1.308 0.092 0.229 0.145 1.451 0.02045 10.1 2.313 6.1 1.808 0.044 0.000 -0.005 1.554 0.06456 11.1 2.407 2.8 1.030 0.013 0.573 0.087 1.659 0.39597 12 2.485 4.9 1.589 0.001 0.039 0.007 1.745 0.02438 18.2 2.901 8.3 2.116 0.144 0.109 0.125 2.206 0.00809 13.7 2.617 4.5 1.504 0.009 0.080 -0.027 1.892 0.150210 14.6 2.681 8.2 2.104 0.025 0.101 0.050 1.962 0.020211 19.7 2.981 13 2.565 0.210 0.606 0.357 2.293 0.073712 18 2.890 11.1 2.407 0.135 0.385 0.228 2.194 0.045513 20.3 3.011 11.9 2.477 0.238 0.476 0.337 2.327 0.022514 20.2 3.006 10.6 2.361 0.234 0.330 0.278 2.321 0.001615 19.7 2.981 9.2 2.219 0.210 0.187 0.198 2.293 0.005516 5.2 1.649 2.4 0.875 0.763 0.830 0.796 0.820 0.003117 8.6 2.152 4.8 1.569 0.137 0.047 0.081 1.377 0.036918 14.1 2.646 6.2 1.825 0.015 0.001 0.005 1.923 0.009819 14.5 2.674 7 1.946 0.023 0.025 0.024 1.954 0.000120 17.3 2.851 8.6 2.152 0.108 0.133 0.120 2.150 0.0000PS-11 Example 4B-R 20 Runs 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = 0.3665 x1.1062Variable Equation Valuen Number of data points = 20Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.522 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =3.25 Correlation coefficient 0.824>0.85noy'~y'~ = 1/n*(Sum of (y'i)) =1.786 Confidence interval 4.73%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.72 Tolerance interval 17.0%<25%yesSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =3.60 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-1.004b0b0= e^b0' =0.367b1b1= Sx'y'/Sx'x'=1.106SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.311y'^meany'^ at mean x value = 1.786tftffrom table =2.101CI'CI = tf*SL*SQRT(1/n) =0.146LCL' LCL' = y~' - CI' 1.640UCL' UCL' = y~' + CI' 1.933CI CI = (e^UCL' - e^LCL')/2 0.875EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 4.73%n'n' = n =20vfvffrom table =1.384un'un'from table =1.177kTkT= un'*vf=1.629TI'TI' = kTSL=0.507LTL' LTL' = y~' - TI' 1.280UTL' UTL' = y~' + TI' 2.293TI TI = (e^UTL' - e^LTL')/2 3.154TI% TI% = TI/EL*100% = 17.0%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.549PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.679Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.824Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.3665x1.1062024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4B-R 20 Runs 10-27-04.xls Power Summary10/29/2004SELECTION OF BEST MODELFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.861 Yes 4.25% Yes 14.7% Yes (b) YesPolynomial 0.875 Yes 5.47% Yes 14.4% Yes Yes YesLogarithmic 0.816 No 4.82% Yes 16.7% Yes (b) NoExponential 0.840 No 4.52% Yes 16.3% Yes (b) NoPower 0.824 No 4.73% Yes 17.0% Yes (b) No(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: PolynomialPS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 4B-R 20 Runs 10-27-04.xls Best Model 10/29/2004

# Appendix A.6

Complete Results of Example No. 4 –

With Run 2 ExcludedCORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATAPM CEMS PM concentrationFacility: Facility Dresponse mg/acmLocation:Anytown USARun x yEmission Unit: Boiler1 6.7 3.1Test Dates: 11/1-11/4/02, 1/7-1/8/033 7.8 2.14 9.2 3.7ValueUnits5 10.1 6.1Emission limit:18.5 mg/acm6 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.510 14.6 8.211 19.7 1312 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.216 5.2 2.417 8.6 4.818 14.1 6.219 14.5 720 17.3 8.6024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Test DataPage 1 of 110/29/2004CALCULATIONS FOR LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y(xi-x~)2(yi-y~)2(xi-x~)(yi-y~)y^(y^i-yi)21 6.7 3.1 49.52 13.42 25.78 2.338 0.5813 7.8 2.1 35.25 21.75 27.68 3.029 0.8644 9.2 3.7 20.58 9.38 13.90 3.910 0.0445 10.1 6.1 13.23 0.44 2.41 4.476 2.6386 11.1 2.8 6.95 15.71 10.45 5.105 5.3127 12 4.9 3.02 3.47 3.24 5.671 0.5948 18.2 8.3 19.92 2.36 6.86 9.570 1.6139 13.7 4.5 0.00 5.12 0.08 6.740 5.01810 14.6 8.2 0.75 2.06 1.24 7.306 0.79911 19.7 13 35.56 38.90 37.19 10.513 6.18312 18 11.1 18.17 18.81 18.49 9.444 2.74113 20.3 11.9 43.08 26.39 33.71 10.891 1.01914 20.2 10.6 41.77 14.72 24.80 10.828 0.05215 19.7 9.2 35.56 5.94 14.53 10.513 1.72516 5.2 2.4 72.88 19.04 37.25 1.394 1.01117 8.6 4.8 26.39 3.85 10.08 3.533 1.60618 14.1 6.2 0.13 0.32 -0.20 6.992 0.62719 14.5 7 0.58 0.06 0.18 7.243 0.05920 17.3 8.6 12.70 3.37 6.54 9.004 0.163PS-11 Example 4C-R 19 Runs 10-27-04.xls Linear CalcsPage 1 of 110/29/2004RESULTS OF LINEAR CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = -1.876 + 0.629 xVariable Equation Valuen Number of data points = 19Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.74 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =436 Correlation coefficient 0.912>0.85yesy~y~ = 1/n*(Sum of (yi)) =6.76 Confidence interval 3.63%<10%yesSyySyy= Sum((yi-y~)^2) =205 Tolerance interval 12.4%<25%yesSxySxy= Sum((xi-x~)(yi-y~)) =274.2 * Indicates correlation coefficient is undefined.b0b0= y~-b1x~ =-1.876b1b1= Sxy/Sxx=0.629SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.386y^meany^ at mean x value = 6.763tftffrom table =2.110CICI = tf*SL*SQRT(1/n) =0.671EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 3.63%n' n' = n = 19vfvffrom table =1.400un'un'from table =1.178kTkt= un'*vf=1.649TITI = kt*SL=2.285TI% TI% = TI/EL*100% = 12.4%SySy= SQRT(Syy/(n-1)) =3.376PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.831Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.912Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Linear Regression Liney = 0.6289x - 1.8759024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Linear Summary Linear SummaryCALCULATIONS FOR POLYNOMIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS PM concentrationresponse mg/acm CALCULATED STATISTICAL PARAMETERSRun x y x2x3x4xy x2y y^ (y^-y)2delta (yi-y~)2CI1 6.7 3.1 453012,015211392.9060.0380.232913.42 1.353 7.8 2.1 614753,702161283.2461.3130.143221.75 1.064 9.2 3.7 857797,164343133.7700.0050.10399.38 0.905 10.1 6.1 1021,03010,406626224.1613.7610.10350.44 0.906 11.1 2.8 1231,36815,181313454.6443.4000.112515.71 0.947 12 4.9 1441,72820,736597065.1230.0500.12153.47 0.978 18.2 8.3 3316,029109,7201512,7499.5691.6100.09832.36 0.889 13.7 4.5 1882,57135,228628456.1442.7020.12625.12 0.9910 14.6 8.2 2133,11245,4371201,7486.7452.1180.11952.06 0.9711 19.7 13 3887,645150,6142565,04510.9444.2270.172538.90 1.1612 18 11.1 3245,832104,9762003,5969.3942.9100.094818.81 0.8613 20.3 11.9 4128,365169,8182424,90411.5270.1390.235126.39 1.3614 20.2 10.6 4088,242166,4972144,32511.4280.6860.223014.72 1.3215 19.7 9.2 3887,645150,6141813,57010.9443.0420.17255.94 1.1616 5.2 2.4 2714173112652.5430.0200.492419.04 1.9617 8.6 4.8 746365,470413553.5331.6060.11313.85 0.9418 14.1 6.2 1992,80339,525871,2336.4060.0420.12390.32 0.9819 14.5 7 2103,04944,2051021,4726.6760.1050.12050.06 0.9720 17.3 8.6 2995,17889,5751492,5748.8000.0400.09043.37 0.84PS-11 Example 4C-R 19 Runs 10-27-04.xls Polynomial CalcsPage 1 of 110/29/2004RESULTS OF POLYNOMIAL CORRELATIONFacility: Facility DEmission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Variable Equation ValueCorrelation equation: y = 2.189 + -0.067 x + 0.02596 x2n Number of data points = 19S1S1 = Sum (xi) = 261Summary of Acceptance Criteria for PS-11S2S2 = Sum (xi2) = 4,021 Criterion Actual Allowable Acceptable?S3S3 = Sum (xi3) = 66,929 Correlation coefficient 0.921>0.85yesS4S4 = Sum (xi4) = 1,171,613 Confidence interval 4.54%<10%yesS5S5 = Sum (yi) = 129 Tolerance interval 12.1%<25%yesS6S6 = Sum (xiyi) = 2,039 * Indicates correlation coefficient is undefined.S7S7 = Sum (xi2yi) = 34,734det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 5.945E+07Check for Correlation Curve Minimum/Maximumb0b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 2.189 Correlation curve minimum point 1.29b1b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.067 Minimum allowable x value 5.2b2b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.02596 Is correlation curve minimum < minimum x value? yesSpSp= Sqrt((1/(n-3)Sum of (y^-Y)2) =1.318 Correlation curve maximum point NADD = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) =5.945E+07 Extrapolation limit for x (125% of maximum x value) NAC0C0 = (S2S4-S32)/D =3.902 Is correlation curve maximum > extrapolation limit? NAC1C1 = (S3S2-S1S4)/D = -0.6164C2C2 = (S1S3-S22)/D =2.182E-02C3C3 = (nS4-S22)/D =1.024E-01C4C4 = (S1S2-nS3)/D = -3.736E-03C5C5 = (nS2-S12)/D =1.394E-04tftffrom table =2.120EL Emission Limit = 18.5CICI = tf*Sp*sqrt(delta-min) =0.840CI% CI% = CI/EL*100= 4.54%vfvffrom table =1.418un'un'from table =1.195n' n' = 1/delta = 11.06kTkt = un'*vf = 1.694TI TI = kt*sp = 2.233TI% TI% = TI/EL *100% = 12.1%y~ y~ = 1/n*(Sum of (yi)) = 6.76SySy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.38r2r2= 1-(Sp^2/Sy^2) =0.847rr = sqrt((1-Sp^2/Sy^2)) =0.921Max-min?b2> 0?minimumPS-11 Correlation Testxmax-miny = -b1/2b2=1.29Calculation Spreadsheet1.25xmax25.38Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Polynomial Regression Curvey = 0.026x2- 0.067x + 2.189024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Polynomial Summary11/1/2004CALCULATIONS FOR LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM CEMS response PM concentrationMeasured Transformed mg/acm CALCULATED STATISTICAL PARAMETERSRun x x' = ln (x) y(xi'-x'~)2(yi-y~)2(x'i-x'~)(yi-y~)y^(y^i-yi)21 6.7 1.902 3.1 0.418 13.419 2.367 2.082 1.0373 7.8 2.054 2.1 0.244 21.745 2.305 3.183 1.1734 9.2 2.219 3.7 0.108 9.383 1.008 4.379 0.4615 10.1 2.313 6.1 0.056 0.440 0.156 5.055 1.0926 11.1 2.407 2.8 0.020 15.707 0.560 5.739 8.6367 12 2.485 4.9 0.004 3.471 0.118 6.303 1.9708 18.2 2.901 8.3 0.125 2.362 0.543 9.321 1.0429 13.7 2.617 4.5 0.005 5.122 -0.156 7.263 7.63510 14.6 2.681 8.2 0.018 2.065 0.191 7.724 0.22611 19.7 2.981 13 0.187 38.898 2.696 9.894 9.64512 18 2.890 11.1 0.117 18.808 1.483 9.241 3.45713 20.3 3.011 11.9 0.214 26.387 2.375 10.112 3.19814 20.2 3.006 10.6 0.209 14.721 1.755 10.076 0.27515 19.7 2.981 9.2 0.187 5.938 1.053 9.894 0.48216 5.2 1.649 2.4 0.809 19.037 3.926 0.246 4.64117 8.6 2.152 4.8 0.157 3.854 0.779 3.890 0.82818 14.1 2.646 6.2 0.010 0.317 -0.055 7.472 1.61719 14.5 2.674 7 0.016 0.056 0.030 7.674 0.45520 17.3 2.851 8.6 0.091 3.374 0.555 8.953 0.125PS-11 Example 4C-R 19 Runs 10-27-04.xls Logarithmic CalcsPage 1 of 110/29/2004RESULTS OF LOGARITHMIC CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = -11.697 + 7.244 Ln(x)Variable Equation Valuen Number of data points = 19Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.55 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.99 Correlation coefficient 0.867>0.85yesy~y~ = 1/n*(Sum of (yi)) =6.8 Confidence interval 4.40%<10%yesSyySyy= Sum((yi-y~)^2) =205 Tolerance interval 15.0%<25%yesSx'ySx'y= Sum((x'i-x'~)(yi-y~)) =21.7 * Indicates correlation coefficient is undefined.b0b0= y~-b1x'~ =-11.697b1b1= Sx'y/Sx'x'=7.244SLSL= SQRT(1/(n-2)(Sum(yi^-yi)^2)) =1.680y^meany^ at mean x value = 6.763tftffrom table =2.110CICI = tf*SL*SQRT(1/n) =0.813EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 4.40%n'n' = n =19vfvffrom table =1.400un'un'from table =1.178kTkt= un'*vf=1.649TITI = kt*SL =2.77TI% TI% = TI/EL*100% = 15.0%SySy= SQRT(Syy/(n-1)) =3.376PS-11 Correlation Testr2r2= 1-(SL^2/Sy^2) =0.752Calculation Spreadsheetrr = SQRT((1-SL^2/Sy^2)) =0.867Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Logarithmic Regression Curvey = 7.244Ln(x) - 11.697024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Logarithmic Summary10/29/2004CALCULATIONS FOR EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM concentrationPM CEMS mg/acmresponse Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x y y' = Ln(y)(xi-x~)2(y'i-y'~)2(xi-x~)(y'i-y'~)y'^(y'^i-y'i)21 6.7 3.1 1.131 49.52 0.414 4.530 1.042 0.0083 7.8 2.1 0.742 35.25 1.068 6.134 1.156 0.1724 9.2 3.7 1.308 20.58 0.218 2.118 1.302 0.0005 10.1 6.1 1.808 13.23 0.001 -0.120 1.396 0.1706 11.1 2.8 1.030 6.95 0.556 1.966 1.500 0.2227 12 4.9 1.589 3.02 0.035 0.323 1.594 0.0008 18.2 8.3 2.116 19.92 0.116 1.522 2.240 0.0159 13.7 4.5 1.504 0.00 0.073 0.010 1.771 0.07110 14.6 8.2 2.104 0.75 0.108 0.284 1.865 0.05711 19.7 13 2.565 35.56 0.624 4.710 2.397 0.02812 18 11.1 2.407 18.17 0.399 2.693 2.219 0.03513 20.3 11.9 2.477 43.08 0.492 4.603 2.459 0.00014 20.2 10.6 2.361 41.77 0.343 3.785 2.449 0.00815 19.7 9.2 2.219 35.56 0.197 2.648 2.397 0.03116 5.2 2.4 0.875 72.88 0.809 7.681 0.886 0.00017 8.6 4.8 1.569 26.39 0.043 1.061 1.240 0.10818 14.1 6.2 1.825 0.13 0.002 0.018 1.813 0.00019 14.5 7 1.946 0.58 0.029 0.130 1.855 0.00820 17.3 8.6 2.152 12.70 0.142 1.342 2.146 0.000PS-11 Example 4C-R 19 Runs 10-27-04.xls Exponential CalcsPage 1 of 110/29/2004RESULTS OF EXPONENTIAL CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = 1.410 e0.1042 xVariable Equation Valuen Number of data points = 19Summary of Acceptance Criteria for PS-11x~x~ = 1/n*(Sum of (xi)) =13.74 Criterion Actual Allowable Acceptable?SxxSxx= Sum((xi-x~)^2) =436.0 Correlation coefficient 0.909>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =1.775 Confidence interval 3.63%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.67 Tolerance interval 12.6%<25%yesSxy'Sxy'= Sum((xi-x~)(y'i-y'~)) =45.44 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =0.344b0b0= e^b0' =1.410b1b1= Sxy'/Sxx=0.104SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.235y'^meany'^ at mean x value = 1.775tftffrom table =2.110CI'CI' = tf*SL*SQRT(1/n) =0.1135LCL' LCL' = y~' - CI' 1.662UCL' UCL' = y~' + CI' 1.889CI CI = (e^UCL' - e^LCL')/2 0.671EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 3.63%n'n' = n =19vfvffrom table =1.400un'un'from table =1.178kTkT= un'*vf=1.649TI'TI' = kt*SL =0.387LTL' LTL' = y~' - TI' 1.388UTL' UTL' = y~' + TI' 2.162TI TI = (e^UTL' - e^LTL')/2 2.340TI% TI% = TI/EL*100% = 12.6%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.561PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.825Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.909Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Exponential Regression Curvey = 1.4101e0.1042x024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Exponential Summary10/29/2004CALCULATIONS FOR POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03PM concentrationPM CEMS response mg/acmMeasured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERSRun x x' = Ln(x) y y' = Ln(y)(x'i-x'~)2(y'i-y'~)2(x'i-x'~)(y'i-y'~)y'^(y'I^-y'i)21 6.7 1.902 3.1 1.131 0.418 0.414 0.416 0.975 0.02453 7.8 2.054 2.1 0.742 0.244 1.068 0.511 1.163 0.17734 9.2 2.219 3.7 1.308 0.108 0.218 0.154 1.367 0.00355 10.1 2.313 6.1 1.808 0.056 0.001 -0.008 1.483 0.10586 11.1 2.407 2.8 1.030 0.020 0.556 0.105 1.600 0.32537 12 2.485 4.9 1.589 0.004 0.035 0.012 1.697 0.01158 18.2 2.901 8.3 2.116 0.125 0.116 0.120 2.212 0.00929 13.7 2.617 4.5 1.504 0.005 0.073 -0.019 1.861 0.127110 14.6 2.681 8.2 2.104 0.018 0.108 0.044 1.939 0.027111 19.7 2.981 13 2.565 0.187 0.624 0.341 2.311 0.064712 18 2.890 11.1 2.407 0.117 0.399 0.216 2.199 0.043313 20.3 3.011 11.9 2.477 0.214 0.492 0.324 2.348 0.016614 20.2 3.006 10.6 2.361 0.209 0.343 0.268 2.342 0.000415 19.7 2.981 9.2 2.219 0.187 0.197 0.192 2.311 0.008316 5.2 1.649 2.4 0.875 0.809 0.809 0.809 0.661 0.046117 8.6 2.152 4.8 1.569 0.157 0.043 0.082 1.284 0.081018 14.1 2.646 6.2 1.825 0.010 0.002 0.005 1.896 0.005119 14.5 2.674 7 1.946 0.016 0.029 0.021 1.931 0.000220 17.3 2.851 8.6 2.152 0.091 0.142 0.114 2.150 0.0000PS-11 Example 4C-R 19 Runs 10-27-04.xls Power CalcsPage 1 of 110/29/2004RESULTS OF POWER CORRELATIONFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03Correlation equation: y = 0.2513 x1.2385Variable Equation Valuen Number of data points = 19Summary of Acceptance Criteria for PS-11x'~x'~ = 1/n*(Sum of (x'i)) =2.548 Criterion Actual Allowable Acceptable?Sx'x'Sx'x'= Sum((x'i-x'~)^2) =2.99 Correlation coefficient 0.894>0.85yesy'~y'~ = 1/n*(Sum of (y'i)) =1.775 Confidence interval 3.90%<10%yesSy'y'Sy'y'= Sum((y'i-y'~)^2) =5.67 Tolerance interval 13.6%<25%yesSx'y'Sx'y'= Sum((x'i-x'~)(y'i-y'~)) =3.71 * Indicates correlation coefficient is undefined.b0' b0' = y'~-b1x~ =-1.381b0b0= e^b0' =0.251b1b1= Sx'y'/Sx'x'=1.239SLSL= SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) =0.252y'^meany'^ at mean x value = 1.775tftffrom table =2.110CI'CI = tf*SL*SQRT(1/n) =0.122LCL' LCL' = y~' - CI' 1.653UCL' UCL' = y~' + CI' 1.897CI CI = (e^UCL' - e^LCL')/2 0.721EL Emission Limit = 18.5CI% CI% =CI/EL*100% = 3.90%n'n' = n =19vfvffrom table =1.400un'un'from table =1.178kTkT= un'*vf=1.649TI'TI' = kTSL=0.415LTL' LTL' = y~' - TI' 1.360UTL' UTL' = y~' + TI' 2.190TI TI = (e^UTL' - e^LTL')/2 2.521TI% TI% = TI/EL*100% = 13.6%Sy'Sy'= SQRT(Sy'y'/(n-1)) =0.561PS-11 Correlation Testr2r2= 1-(SL^2/Sy'^2) =0.799Calculation Spreadsheetrr = SQRT((1-SL^2/Sy'^2)) =0.894Version 2-6 10/25/04* Indicates correlation coefficient is undefined.Plot of Power Function Regression Curvey = 0.2513x1.2385024681012140.0 5.0 10.0 15.0 20.0 25.0PM CEMS Response

PM ConcentrationPS-11 Example 4C-R 19 Runs 10-27-04.xls Power Summary10/29/2004SELECTION OF BEST MODELFacility: Facility D Emission Unit: BoilerLocation: Anytown USATest Dates: 11/1-11/4/02, 1/7-1/8/03ModelCorrelationcoefficient (a)>0.85 ?Confidence intervalhalf range percentage<10% ?Tolerance interval halfrange percentage<25% ?Min/max withinallowable range?Does model meet allcriteria?Linear 0.912 Yes 3.63% Yes 12.4% Yes (b) YesPolynomial 0.921 Yes 4.54% Yes 12.1% Yes Yes YesLogarithmic 0.867 Yes 4.40% Yes 15.0% Yes (b) YesExponential 0.909 Yes 3.63% Yes 12.6% Yes (b) YesPower 0.894 Yes 3.90% Yes 13.6% Yes (b) Yes(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.Best model: PolynomialPS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04PS-11 Example 4C-R 19 Runs 10-27-04.xls Best Model 10/29/2004