United States
Environmental Protection
Agency
Office of Prevention. Pesticides
and Toxic Substances
Washington, DC 20460
EPA747-R-93-010
August 1993
vvEPA
Office of Prevention, Pesticides, and Toxic Substances
SAMPLING GUIDANCE FOR SCRAP
METAL SHREDDERS
Technical Background
Grid superimposed over
material to be sampled
1
Take samples from approximate
centers of squares in the grid.
XXX
XXX
XXX
All samples
combined
in one
bucket
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SAMPLING GUIDANCE FOR SCRAP
METAL SHREDDERS
Technical Background
August 1993
United States Environmental Protection Agency
Office of Prevention, Pesticides
and Toxic Substances
Washington, DC 20460
-------
DISCLAIMER
This document has been reviewed and approved for
publication by the Office of Pollution Prevention and Toxics, U.S.
Environmental Protection Agency. The use of trade names or
commercial products does not constitute Agency endorsement or
recommendation for use.
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Authors and Contributors
Westat Project Staff:
James Bethel
Ralph DiGaetano
Mary Peppier
EPA Project Staff:
Edith Sterrett, Project Officer
Susan Dillman, Task Manager
-in-
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Acknowledgments
The Office of Pollution Prevention and Toxics wishes to thank everyone involved
with this project at Westat and at the Evironmental Protection Agency, as well as others that have
contributed to the document
In particular, we acknowledge the technical help of John Rogers and William
Devlin of Westat in preparing this document. Mary Lou Pieranunzi, Angelia Murphy, Maida
Montes and Anna Page also helped to prepare and proof-read the manuscript
We are grateful to the many reviewers in the Office of Pollution Prevention and
Toxics and in other branches of the Environmental Protection Agency for reading the document.
In particular, we thank Brad Schultz and Dan Reinhart of the Exposure Evaluation Division.
We owe special thanks to Mitchell D. Erickson of Argonne National Laboratory,
David N. Speis of Environmental Testing and Certification Corporation, and Herschel Cutler of the
Institute of Scrap Recycling Industries for reviewing this document and contributing many useful
comments about it Their review does not constitute approval.
-IV-
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TABLE OF CONTENTS
TECHNICAL BACKGROUND
1. Overview 1
2. Modeling Sampling and Analytical Errors 1
3. Standard Error Calculations 13
3.1 Data Sources 13
3.2 Numerical Methods 17
4. Confidence Intervals and Relative Error 20
4.1 Numerical Methods 20
4.2 Alternative Confidence Levels 25
5. Hypothesis Testing 25
5.1 Monitoring 25
5.2 Clean-up Verification 28
6. Power Calculations 30
6.1 Monitoring 30
6.2 Clean-up Verification 36
LIST OF TABLES
Table Page
1 Summary of preliminary data 15
2 Relative errors for estimating PCB levels with sample sizes of 2 to 25 21
3 t-values for confidence interval and hypothesis tests 23
4 Relative errors for estimating PCB levels with sample sizes of 2 to 25
with 90% confidence 26
5 Relative errors for estimating PCB levels with sample sizes of 2 to 25
with 99% confidence 27
6 Cut-off values for monitoring 29
7 Cut-off values for clean-up verification 31
8 Chance of finding violations in monitoring with a 25 ppm standard 32
-v-
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LIST OF TABLES (Continued)
Table Page
9 Chance of finding violations in monitoring with a 50 ppm standard 33
10 Chance of finding violations in monitoring with a 100 ppm standard 34
11 Chance of finding violations in monitoring with a 50 ppm standard
with 90% confidence 37
12 Chance of finding violations in monitoring with a 50 ppm standard
with 99% confidence 38
13 Chance of requiring additional clean-up with a 25 ppm standard 39
14 Chance of requiring additional clean-up with a 50 ppm standard 40
15 Chance of requiring additional clean-up with a 100 ppm standard 41
16 Chance of requiring additional clean-up with a 50 ppm standard
with 90% confidence 43
17 Chance of requiring additional clean-up with a 50 ppm standard
with 99% confidence 44
-VI-
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SAMPLING GUIDANCE FOR SCRAP METAL SHREDDERS
Technical Background
1. Overview
The purpose of this volume is to provide additional detail on the
assumptions, background data, and numerical calculations that support PCB Sampling
Guidance for Scrap Metal Shredders. The discussion is somewhat technical at times, but in
general can be read by anyone with a modest background in mathematical statistics. The
primary emphases are on the following areas:
Assumptions regarding sampling and analytical errors;
Background data for estimating standard errors;
The effects of compositing on standard errors;
Calculations used to estimate relative error in confidence intervals;
Calculations for finding cut-off values and power in hypothesis
testing; and
Effects of alternative confidence and significance levels on power
and relative error.
For the convenience of the reader, tables and definitions are reproduced here, wherever
possible, so that it should not be necessary to refer back to the Field Manual while reading
this volume.
2. Modeling Sampling and Analytical Errors
Variance Components. The sampling model involves three
components:
Sampling over time, which will generally mean the relatively short
period of time in which the samples are being collected, such as
several hours or, at most, several days. In some monitoring
programs, the reference time period might be longer, if certification
is determined based on samples collected over several weeks or
months.
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Sampling over space, such as taking samples from a fluff pile or
from volume of fresh fluff that has been spread out for sampling.
Potential errors in analysis of PCS content.
The first stage may not be applicable in all cases, but the second and third elements will
always be present.
Sampling over time can actually have two different meanings. If a site is
visited for 8 hours and samples of fresh output are taken for 10-minute intervals once an
hour, then there is some sampling variability due to the fact that some 10-minute intervals
were selected and not others. On the other hand, if a site is visited over a longer period of
time, say several months, then there is sampling variability because of the days that are
selected to visit the site.
Sampling in space requires grid sampling or some other method that insures
that samples are distributed over the volume of material that is to be assessed. Whenever
grid sampling is involved, the steps suggested in the manual require that all cells have
samples taken from them. Since samples are taken from all cells, there is no "between-cell"
variance component, but there is still some variability due to the fact that we are selecting
one of the many different samples that could be selected from each cell.
Finally, besides this uncertainty about exactly which samples of material
might be selected, there is uncertainty about the accuracy of laboratory measurements.
(Note that because of this uncertainty, it is recommended in the Field Manual that at least
10% of the samples be analyzed in duplicate. See M.D. Erickson, Analytical Chemistry of
PCBs, 1986, Butterworth, Stoneham, Massachusetts, p. 308.)
All these components are reflected, directly or indirectly, in making
estimates of variance calculations for the sampling guidance document Each component of
variance corresponds to one of the possible sources of error discussed in Section 4.1. In
the following sections we will describe the sampling and variance models in detail.
Sampling Model. The sampling model discussed in this section is
intended to give a formal theoretical model for the grid sampling and other sampling
procedures discussed in Chapter 2. It should be noted, however, that the procedures
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recommended in Chapter 2 are not based on theoretical considerations but rather their
apparently successful use in a study conducted for EPA.*
In its most general form, the sampling model consists of samples collected
at m points in time, with n samples being selected at each time period, for a total of mn
samples. Usually, when sampling over time, we would have n = 1, so that m samples
would be collected. On the other hand, when sampling stored fluff, m = 1, and n samples
would be collected. Furthermore, each of the collected samples would be divided into c
subsamples for compositing. Finally, each composite sample would be divided into r
replicates for laboratory analysis.
First, let us assume that output is sampled over some period of time, which
may consist of several hours or several days. This will be achieved by stopping the
shredder and taking samples at, say, m points over this period of time. For example,
shredder output may be sampled over an 8-hour period, with samples being taken each half
hour. In this case, there would be 16 samples over time, one for each half hour. If grid
samples are being taken, then at time i, 1< i < 16, output would be collected and spread
into a 3x3 grid. Within the /-th cell of the grid, 1 < / < 9, a sample of material would be
selected, after which all 9 samples would be combined and mixed. If grid sampling is not
used (e.g., if no front loader is available), then grab samples would be taken from the pile
of fresh fluff as described in Section 2 of the Field Manual.
When stored fluff is being sampled, there is only one time period (i.e., m =
1). If the steps described in Section 2 of the Field Manual are followed, then 20 samples
would be taken, so that we would have n = 20.
If desired, the m samples over time, or n samples taken at a single time,
may be randomly sorted into groups of c samples each and then composited into q (where
q = mlc or q = n/c) composite samples. Finally, each of the q composite samples is
separated into r replicates for laboratory analysis.
* USEPA Fluff Pilot Program, conducted by Westat, Inc., Rockville, MD, and Midwest Research Institute,
Kansas City, MO. for the Office of Pollution Prevention and Toxics (see PCB. Lead, and Cadmium
Levels in Shredder Waste Materials: A Pilot Study. USEPA, Office of Toxic Substances. EPA 560/5-
90-008B. 1991).
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Thus the sampling model involves four steps: sampling over time, selecting
a grid sample (or several grab samples*), compositing, and the formation and sampling of
replicates. Three of the steps in this sampling model involve randomization: sampling over
time, grid sampling, and sampling of replicates. In sampling over time, it is theoretically
possible to eliminate the between-time sampling component by taking samples from every
time period. For example, this would be the case if the shredder operation were shut
down each half-hour and all output were arranged in a grid. In most applications,
however, the output from even as brief a period as a half-hour would be too large in
volume for such a procedure, so that for each half-hour only a fraction of the material
produced will actually be selected and arranged in a grid. In the discussion below, it is
assumed that this fraction is small, so that sampling over time is considered to be like
sampling from an infinite population of possible times. Similarly, in selecting samples of
material from grid cells, it is assumed that there are many possible samples of material, and
this stage of sampling is also assumed to be like sampling from an infinite population of
such samples. In sampling replicates, one or two of the eight replicates will be selected at
random for laboratory analysis.
To facilitate the discussion of expectations and variances, it will be
convenient to indicate the first stage of sampling - the selection of times - by S p the
second stage - the selection of grid samples - by S2, and the third stage - the selection of
replicates - by S3. That is, Sj will denote the actual sample of times selected, S2 will
denote the actual grid samples selected, and S3 will denote the actual replicates selected.
Furthermore, the following notations will be useful:
9 = actual average concentration among all material generated over the
target time period,
6- = actual average concentration among all material at all time i, and
6U = actual average concentration at time i in cell /.
Other notations will be introduced as necessary in the discussion which follows.
* As we noted earlier, grab samples may be taken instead of grid samples. However, the grab sampling
procedures described in Section 2 are relatively similar to grid sampling, in that several samples are taken
from a variety of positions in the body of material and then combined into a single sample for analysis.
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Variance Model. Let
X-, = actual concentration in they-th sample taken from the /-th grid cell at
y*
time/.
Since X-^ is the concentration of a sample of material that is selected at random, it is
assumed that
E(xijl\sl) = eu.
That is, it is assumed that conditional on the time periods selected, the expected value of the
concentration of a randomly selected sample of material is the average concentration of all
the material. Unfortunately, this assumption requires something of a leap of faith. For
example, if the sampling process is not done carefully, then "fines" - small particles of
fluff - may be less likely to be selected. If, as is suspected, fines are more likely to contain
PCBs, or other toxic materials, then the concentration of toxic materials in a sample might
tend to be smaller than the overall average concentration.
The variance of Xij{ is defined as
This term represents the variance of picking a single sample at random from cell /, at time i.
As noted above, the population of possible samples is finite, but since the size of this
population will usually be quite large, the finite population correction will be neglected in
our calculations.
When samples from the nine grid cells have been selected, they are
combined and mixed. The resulting concentration is denoted as:
1 9
Xi, = g2 KM = average concentration among the mixed grid samples.
-------
Notice that
9
X
actual average concentration throughout all grid cells at
time i.
(This calculation assumes implicitly that the grids are constructed so that each cell has an
equal volume of material.) Furthermore, because the sampling is done independently in
each cell,
1 9 1 9 2 3.
|o i J XT 2^ * \r* ii/^1 / = Q i ^ ^i/ = i*
1 si/=1 y ei/=1
This term represents the variance created by testing only selected samples within each cell,
rather that testing all material in each cell.
As discussed above, compositing may also be used. We will index the
composite samples as i/", denoting the mean concentration in the y"-th composite a&X^-.
The mean concentration in the /y"-th composite sample can be written as follows:
For example, if c = 2 and n = 8, then./ = 1,2 ..... 8, and;' = 1, 2, ..., 4. In particular, for
4 _
= *
;=3
Notice that this notation includes the case where c - 1, i.e., where there is no compositing.
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This notation implies that the compositing is done only within time periods,
since i indexes time periods and; indexes samples within time periods. However, samples
over time can also be composited. The notation above could be adapted by allowing i" to
index composite samples over time. However, we recommend that samples over time be
composited only when n = 1. In this case, it is mathematically equivalent to set i = 1 and to
let) index the samples over time. Thus we will treat only the case shown above where j'
indexes composite samples.
The final stage of sampling involves forming and selecting replicates.
Denote the concentration in the fc-th replicate as X^. Assuming that the selection of
replicates is random,
(Notice that this expectation is conditional on both the selection of time periods and the
samples within grid cells.) The notation
will be convenient for indicating the variance between replicates.
Measurement Error. When samples of material are submitted to
laboratory analysis, some measurement error is inevitable. Thus it is reasonable to let
Y( .k = measured concentration in the fc-th replicate of the
i/"-th composite sample
= (Actual Concentration) + (Measurement Error)
£ij'k
where
= 0
-------
and where Xirlf and £-,. are assumed to be independent.
V * V *
Total Variance. As discussed earlier, there are three stages of sampling,
consisting of m time periods, n/c composite samples, and r replicates. To estimate the
overall concentration, 6, the average measured concentration among all samples is
calculated. The overall average measured concentration can be written as
m/i7?£ 2 V*'
Notice that
(1) £(F) = EEF SlfS2.S3
Conditional on Sp 52, and S3 - that is, conditional on the time periods, grid samples and
replicates that were selected -X-tyk is constant and £.i}-k has an expected value of zero,
independent of Slt S2, and 53. Thus
irk\ Slt S2,
It follows that
-------
elvvV
n n r% ^ I
_ J/l^v y r-fiv Y I C C
~mii ' r /r* * 2
Now, conditional on S1 and 52> as discussed earlier,
so that
Notice however, that
C]' ~
so that
- I
;
*s^ I
= ft
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(The validity of the last step depends on the assumption that the time periods are randomly
selected.) This shows that F is an unbiased estimator of 6, and thus the notation
will be used in the sections which follow.
Using the conditional expectation formula for variances,
Conditional on 5lf 52, and 53, X^ is constant, while e^ has mean 0 and variance
neither of which depends on Sv S2, or S3. Thus
and
Combining these expressions, we have
(2)
10
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l el 2 Jfl c\z _ nii-l . !/l c v
= m«7^ + ^m nj ? ^ ~f) +V\m n ?
n r
y y
, m « - r
where
Similarly,
is V
Since the mean concentration of the y"-th composite sample is
.., = I X-Jc.
J '- '
we have
Thus
v - £ I vi I,
11
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Setting
(where 0 = j-rlfl,- ) and
yields
The first term represents the variability due to sampling over space, while the second
represents the variability due to sampling over time. Combining this expression with (2)
yields
Assuming that the variance between replicates, 7-, is relatively constant over time and
between composite samples, i.e., fly = fi, leads to the simplification
m mn m n r m n r
nmr
where
This expression for the variance of the overall average sample concentration
includes terms for the variability over time tf/m), the variability between samples within
12
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grid cells (erV/n/i), and the variability due to measurement errors tfc/nmr). In most cases,
the between-time component (T2) is probably larger than variability of samples within grid
cells (a2). One exception would be in sampling stored fluff, where in most cases there
would be no between-time component (i.e., m = 1) and formula (3) would reduce to
On the other hand, when sampling from fresh output streams, it will generally be the case
that n = 1, in which case formula (3) reduces to
m mr
In this case, however, there may be compositing over time, where the m samples are
grouped at random into q (= m/c) samples for analysis. Finally, it would generally be
expected that r=l, except when unusual concentrations are found. These considerations
would lead to the variance formula
A T2 + a2 T^
(4) V(8) = jjp- + -£-.
In fact, as we will see in Section B.3.1, our preliminary estimates of
sampling variability for stored fluff are somewhat lower than those for fresh fluff, so that
assuming the model in (4) is probably conservative. Moreover, based on the available
data, we are not able to obtain distinct estimates for T2 and o2 which are sufficiently
accurate. For these two reasons, we will assume the model in (4) for standard error
calculations.
3. Standard Error Calculations
3.1 Data Sources
Data for estimating the components of the variance model in (4) were
available from two sources. First, a pilot study for a proposed national survey of shredder
sites gathered samples of fluff and other materials at seven shredder sites distributed across
13
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the United States.* These samples were collected under controlled, monitored
circumstances, with adequate quality control and documentation on the methods used in
collecting the data. In addition, other data on fluff were available from several states where
samples of fluff had been collected and analyzed. Unfortunately, for the latter data, no
information is available on the sampling methods or on the types of materials sampled.
Both sources of data are summarized in Table 1, which gives the averages, standard
deviations, and coefficients of variation (CV's) found in various types of material.
The EPA data show exploratory research on PCB levels in fluff from shredded automobiles
and white goods, as well as stored fluff, soil, and other materials. The data presented
include estimated standard deviations for analytical error and sampling error, and the
estimated overall average PCB concentrations. For example, for automobiles, the
estimated standard deviation for analytical error is 9.28, indicating that
y = 9.28
might be used as an estimate of the standard deviation for measurement error in analyzing
fluff from automobiles. In the EPA data, the standard deviations for analytical error range
are 9.28 to 19.27, while the standard deviations for sampling error range from 23.90 to
155.37. In the State data, no standard deviations are available for analytical error, and the
standard deviations for sampling error, which range from 22.32 to 345.55, are not
identified by the type of material.
In the EPA study, procedures similar to those described in the Sampling
Guidance were used to collect multiple samples of stored fluff, soil, and fresh fluff. In
sampling fresh fluff, however, only one sample was taken for each time period, so that the
sampling error term includes both T2 and o2, as indicated in formula (4) shown in
Section 2. Thus it is not possible to make separate estimates of the parameters T2 and o2.
Another source of confounding is that the estimates of CV's in the "Sampling Error"
column also contain measurement error. Thus the estimates for standard errors and CV's
have a slight upward bias, although this bias amounts to only a few percentage points at
most.
* USEPA Fluff Pilot Program, conducted by Westat, Inc., RockviUe, MD, and Midwest Research Institute,
Kansas City, MO, for the Office of Pollution Prevention and Toxics (see PCB, Lead, and Cadmium
Levels in Shredder Waste Materials: A Pilot Study. USEPA, Office of Toxic Substances. EPA 560/5-
90-008B. 1991).
14
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Table 1: Summary of preliminary data
Data from EPA Pilot Study
Analytical Error* Sampling Error
Overall Standard Standard Number
Type of Material PCS Level Deviation CV Deviation CV of Samples
Automobile Fluff 24.64 9.28
White Goods Fluff 57.16 19.27
Other Fresh Fluff 176.56
Stored Fluff 77.13
Soil 44.14
0.30
0.34
23.90
71.45
155.37
47.82
41.05
0.97
1.25
0.88
0.62
0.93
37
21
9
8
8
Overall
55.69
17.82
0.32 57.36
1.03
83
Data from State Samples
Analytical Error* Sampling Error
State
Massachusetts
Rhode Island
Maryland
California
Indiana
Arizona
Overall Standard
PCB Level Deviation
33.85
21.52
228.84
58.73
87.13
190.50
Standard
CV Deviation
29.45
25.18
345.55
22.32
29.62
72.39
Number
CV of Samples
0.87
1.17
1.51
0.38
0.34
0.38
80
80
12
13
8
12
Estimates for analytical error were based on repeated analysis of eight samples of automobile fluff and
four samples of white goods fluff.
15
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Notice that the larger the estimated overall PCB level, the higher the
standard deviation. This phenomenon is frequently observed in sampling and it is often
taken into account by using CV rather than the standard deviation as a measure of
variability. For a random variable x, the coefficient of variation is generally defined as
E(x) '
(See, for example, Snedecor and Cochran, Statistical Methods, Iowa State University
Press, Sixth Edition, 1967, p. 62.) In this case, we will be interested in the coefficient of
variation for analytical error, which will be defined as
CV =
CV -
The data in Table 1 show estimated CV's for measurement error for automobile and white
goods as .30 and .34, respectively. As an overall figure, we might estimate the CV for
measurement error as
- 17-82 _
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The estimates of variances that were used for calculating sample sizes and
power were based on the numbers in Table 1. In all cases, estimates were rounded off.
Because of the tendency toward over-estimation discussed above, estimates for CV's were
rounded down somewhat. In all cases, estimates are based on the numbers shown in the
first and second rows in Table 1. The estimate for CVy was obtained by averaging the
CV's for automobile and white goods; the estimates for overall PCB levels and for CV's
for sampling error were based directly on the respective entries in Table 1. With that in
mind, the following estimates were assumed for use in standard error calculations:
CV (CV for analytical error): 32%
y
A
CVOTl (CV for sampling error) for automobiles: 100%
Overall PCB level for automobiles: 25 ppm
cY^ (CV for sampling error) for white goods: 100%
Overall PCB level for white goods: 57 ppm.
In the next section we will discuss in more detail how these numerical values were used,
3.2 Numerical Methods
Mixture Distributions. A mixture distribution occurs when two
different quantities are combined at random. For example, in the output at shredder sites,
fluff from shredding automobiles and white goods is combined. One model for this
process can be formulated as follows. Let U, Z,, and Z2 be independent random variables,
where U takes the values 1 and 0, with mean and variance
E(U) = n V(E7) = it(\ - it),
and where
E(Zf.) = m V(z,0 = f??-
If we define Was
W = UZ1 + (1 - U)Z2
17
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then the distribution of W is a mixture of the distributions of Zl and Z2. The mean and
variance of W are given by
(5)
and
(6) V(W) =
= E (v(uzl + (i - t/)z2 It/)) + v (E(UZ{ + (i - toz2
= £
This model can be applied to samples of shredder output by letting
1 if the sample was generated by shredding
white goods
U
0 if the sample was generated by shredding
automobiles
and letting Zj and Z2 represent the distributions of PCB levels in samples of white goods
and automobile fluff, respectively. Thus, in this model, n is the percent of white goods
that is mixed with automobiles, while ^ and ^ are the average PCB concentrations in
white goods and automobiles, respectively. Similarly, fy and rj2 represent the variability
of PCB levels in white goods and automobiles.
Combining the notation in Section 2 with formula (5), the overall
concentration of PCBs would be denoted as
(7) 6 = JT/J, + (1 -
18
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To obtain a formula for V(0), remember that, from (4),
m q
As discussed earlier, the "i2 + o2" term represents sampling variability, while the / term
represents measurement error. Since, in formula (6), V(W) represents sampling error
only, we have
iW-
which yields
Recall from our discussion of analytical error that ? appears to be larger when the overall
PCB level is larger. On the other hand, the coefficient of variation, CVy appears to be
relatively constant Notice that
CVV =*
7 6
implies
(9)
This gives us an alternative formula for variability due to measurement error, one which
reflects the change in ^ relative to the change in the overall concentration of PCBs.
Combining formulas (4), (8) and (9), we have the following expression:
m
19
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Formula (10) was used for estimating variances for calculating relative errors, while a
simpler variation of it was used for power calculations. Since the actual parameter values
are not known, estimates were derived based on the data in Table 1. For example, the CV
for analytical error and the overall PCB concentrations were estimated to be
CVy = 32%
£j = 57ppm
A
1*2 = 25ppm
although /Zj and /^ were sometimes allowed to vary (in power calculations, for example).
The values for 7]^ and rfc were derived from the CV's shown on page 15:
= d-00)(57) = 57
= (1.00)(25) = 25.
The value for n was either taken to be 10% or allowed to vary between 0 to 100%.
4. Confidence Intervals and Relative Error
4.1 Numerical Methods
In the Field Manual, Section 4.2 discusses confidence intervals and relative
error, Table 2 in the Field Manual shows the relative errors for various sample sizes and
compositing strategies. Here, Table 2 shows the relative errors for various mixtures of
white goods and automobiles. Note that Table 2 in the Field Manual corresponds to the
"25% White Goods" column. In this section, we will discuss how the relative errors were
calculated.
20
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Table 2: Relative errors for estimating PCB levels with sample sizes of 2 to 25
Total
samples
collected
m
Number of
composites
analyzed
q = mJc
Subsamples
in each
composite
c
Relative error*
Percent white goods
0%
10%
25% 1 50%
85%
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
944%
167%
1 81%
56%
43%
698%
123%
2 60%
41%
32%
534%
94%
4 46%
32%
24%
429%
76%
8 37%
25%
20%
1038%
184%
89%
62%
48%
762%
135%
65%
45%
35%
576%
102%
49%
34%
26%
455%
81%
39%
27%
21%
1084%
192%
93%
64%
50%
793%
140%
68%
47%
36%
597%
106%
51%
35%
27%
468%
83%
40%
28%
21%
1066%
189%
91%
63%
49%
781%
138%
67%
46%
36%
588%
104%
50%
35%
27%
463%
82%
40%
27%
21%
984%
174%
84%
58%
45%
725%
128%
62%
43%
33%
551%
98%
47%
33%
25%
440%
78%
38%
26%
20%
*A relative error of 50% means that with 95% certainty, the estimated average concentration will be
within 50% of the actual average concentration.
-------
The formula for a confidence interval for 0 is given by
(11)
where tgft is the percentile corresponding to l-a/2 of a Student's r-distribution with (q -
1) degrees of freedom. Since sample sizes will be small in most cases, the r-distribution is
more appropriate than the standard normal. Unfortunately, the distribution of PCB levels
in samples of fluff is highly skewed and, as a result, the normal distribution gives only a
rough approximation to the actual coverage probabilities of confidence intervals. Based on
simulations done with data from the EPA Pilot Study, 95% confidence intervals may be
accurate as little as 75% of the time with sample sizes of 15 to 25. In nearly all cases,
when confidence intervals are incorrect they fall below the actual PCB level.
In any case, the relative error is given by
(12) Relative Error =
&
Combining formulas (7), (10), and (12), we have
(13) Relative Error =
As an example calculation, suppose that K = .10, c = 2, m = 8, and a = .05 (i.e., the
mixture of white goods is 10%, there are 8 total samples consisting of 4 composites of 2
subsamples each, and the confidence level is 95%). Notice from Table 3, t^ = 3.18. For
the other parameters, we will use the estimates discussed in Section 3.2:
CVr = 32%
= 57 = 25
22
-------
Table 3: t-values for confidence intervals and hypothesis tests
Number of
composite
samples
q = mlc
t-values
Confidence intervals
90% 95% 99%
Hypothesis tests
90% 95%
99%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
50
75
100
>100
6.31
2.92
2.35
2.13
2.02
1.94
1.89
1.86
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.75
1.74
1.73
1.73
1.73
1.72
1.72
1.71
1.71
1.70
1.68
1.67
1.66
1.65
12.71
4.30
3.18
2.78
2.57
2.45
2.36
2.31
2.26
2.23
2.20
2.18
2.16
2.15
2.13
2.12
2.11
2.10
2.09
2.09
2.08
2.07
2.07
2.06
2.05
2.01
1.99
1.98
1.96
63.66
9.93
5.84
4.60
4.03
3.71
3.50
3.35
3.25
3.17
3.11
3.05
3.01
2.98
2.95
2.92
2.90
2.88
2.86
2.85
2.83
2.82
2.81
2.80
2.76
2.68
2.64
2.63
2.58
3.08
1.89
1.64
1.53
1.48
1.44
1.42
1.40
1.38
1.37
1.36
1.36
1.35
1.35
1.34
1.34
1.33
1.33
1.33
1.33
1.32
1.32
1.32
1.32
1.31
1.30
1.29
1.29
1.28
6.31
2.92
2.35
2.13
2.02
1.94
1.89
1.86
1.83
1.81
1.80
1.78
1.77
1.76
1.75
1.75
1.74
1.73
1.73
1.73
1.72
1.72
1.71
1.71
1.70
1.68
1.67
1.66
1.65
31.82
6.97
4.54
3.75
3.37
3.14
3.00
2.90
2.82
2.76
2.72
2.68
2.65
2.63
2.60
2.58
2.57
2.55
2.54
2.53
2.52
2.51
2.50
2.49
2.46
2.41
2.38
2.37
2.33
23
-------
A A
7j, =57 r/2 = 25.
Substituting these estimates for their respective parameter values in formula (13) gives a
relative error of
_ . . _ 3.18 /324.9 + 562.5 + 92.16 A 81.43
RelativeError = ^ A/ - g - + ~T~
= 135%
as shown in Table 2.
Notice that the Relative Error depends on the parameters a, m, and on c as follows:
As a increases, i^ decreases, making the Relative Error smaller,
As m increases, V(Q) decreases, making the Relative Error smaller,
As c increases, V(&) increases, making the Relative Error larger.
In the latter cases, the sample size is altered and the sampling and/or measurement errors
actually change. When a changes, however, the sampling and measurement errors remain
the same, but the possibility of error in our conclusions changes. For example, suppose
that the level of confidence is changed from 95% to 90% - i.e., a changes from .05 to . 10.
This means that we are willing to accept a 10% chance of being wrong rather than a 5%
chance. In exchange for this sacrifice, both r^ and the resulting confidence interval will
be smaller, so that we can make a closer estimate the actual level of PCBs. Thus the
Relative Error, as we have defined it in formula (12), is smaller, although the chance that
this error occurs has increased.
24
-------
4.2 Alternative Confidence Levels
Tables 4 and 5 show what the relative errors would be under the alternative confidence
levels of 90% and 99%. When the confidence level is decreased from 95% to 90%, the
decrease in relative error is quite striking for very small sample sizes (e.g., 2 < q < 4), but
relatively modest for larger sample sizes (e.g., 16 < q < 25). Similarly, increasing the level
of confidence from 95% to 99% sharply increases the relative errors when q is small, but
the increase is much smaller when the sample sizes are larger.
5. Hypothesis Testing
5.1 Monitoring
To find cut-off values in monitoring, we employ the standard statistical
procedures for testing the hypothesis
HQ: n<[iQ
against the alternative
where /-!<, is taken to be 25, 50 or 100. That is, we compare the standardized statistic
with r05, the 95-th percentile of the r-distribution with m-\ degrees of freedom, rejecting
HQ if r* > r05. For simplicity, it is suggested that the user implement this procedure by
comparing x with a cut-off value given by
(14) Cut-off Value = /^ + fosTF-
25
-------
Table 4: Relative errors for estimating PCS levels with sample sizes of 2 to 25 and 90% confidence
Total
samples
collected
m
Number of
composites
analyzed
mtc
Subsamples
in each
composite
c
Relative error*
Percent white goods
0%
10%
25% 1 50%
85%
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
469%
124%
1 65%
46%
36%
347%
91%
2 48%
34%
27%
265%
70%
4 37%
26%
20%
213%
56%
8 30%
21%
16%
516%
136%
72%
51%
40%
378%
100%
53%
37%
29%
286%
75%
40%
28%
22%
226%
60%
31%
22%
17%
539%
142%
75%
53%
41%
394%
104%
55%
39%
30%
296%
78%
41%
29%
23%
233%
61%
32%
23%
18%
530%
140%
74%
52%
41%
388%
102%
54%
38%
30%
292%
77%
41%
29%
22%
230%
61%
32%
23%
18%
489%
129%
68%
48%
37%
360%
95%
50%
35%
28%
274%
72%
38%
27%
21%
218%
58%
30%
21%
17%
*A relative error of 50% means that with 95% certainty, the estimated average concentration will be
within 50% of the actual average concentration.
-------
Table 5: Relative errors for estimating PCB levels with sample sizes of 2 to 25 and 99% confidence
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Relative error*
Percent white goods
0% 10%
25% 1 50%
85%
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
4726%
307%
1 117%
77%
59%
3494%
227%
2 87%
57%
43%
2672%
173%
4 66%
44%
33%
2146%
139%
8 53%
35%
27%
5199%
337%
129%
85%
65%
3815%
248%
95%
62%
47%
2883%
187%
72%
47%
36%
2279%
148%
57%
37%
28%
5430%
352%
135%
89%
67%
3972%
258%
99%
65%
49%
2988%
194%
74%
49%
37%
2345%
152%
58%
38%
29%
5339%
346%
133%
87%
66%
3910%
254%
97%
64%
49%
2947%
191%
73%
48%
37%
2319%
150%
58%
38%
29%
4927%
320%
122%
81%
61%
3630%
236%
90%
59%
45%
2761%
179%
69%
45%
34%
2202%
143%
55%
36%
27%
*A relative error of 50% means that with 95% certainty, the estimated average concentration will be
within 50% of the actual average concentration.
-------
Alternatively, the cut-off values can be approximated by using the values in Table A-l
(reproduced here as Table 6). The values in this table were calculated using formula (14).
For example, for q = 9, s = 150 and /XQ = SO, we have
Cut-off Value = 50 + 1.86 - = 143.0
V9
as shown in Table 6.
5.2 Clean-Up Verification
To find cut-off values for clean-up verification, we reverse the direction of
the hypothesis test for monitoring. Here we want to test
against the alternative
We reverse the direction in clean-up verification because it is known that the site, or output
at the site, has been found to be contaminated. Thus, we assume that the site is
contaminated until the data demonstrate otherwise.
In this case, the cut-off value is calculated as
'.05!
(15) Cut-off Value = LL. - tn*-j=
and HQ is rejected if x is smaller than the cut-off value. Notice that when s is large, it may
occur that
in which case it will be impossible to reject the null hypothesis and further clean-up will be
required.
28
-------
Table 6: Cut-off values for monitoring*
Safety
Standard
Standard
Deviation
Number of Composite Samples Analyzed (q=m/c)
2
4
9
« 1
25
25
50
100
20
35
50
75
100
150
250
20
35
50
75
100
150
250
20
35
50
75
100
150
250
114.2
181.2
248.1
359.6
471.2
694.3
1,140.5
139.2
206.2
273.1
384.6
496.2
719.3
1,165.5
189.2
256.2
323.1
434.6
546.2
769.3
1,215.5
48.5
66.1
83.8
113.1
142.5
201.3
318.8
73.5
91.1
108.8
138.1
167.5
226.3
343.8
123.5
141.1
158.8
188.1
217.5
276.3
393.8
37.4
46.7
56.0
71.5
87.0
118.0
180.0
62.4
71.7
81.0
96.5
112.0
143.0
205.0
112.4
121.7
131.0
146.5
162.0
193.0
255.0
33.8
40.3
46.9
57.8
68.8
90.6
134.4
58.8
65.3
71.9
82.8
93.8
115.6
159.4
108.8
115.3
121.9
132.8
143.8
165.6
209.4
31.8
37.0
42.1
50.7
59.2
76.3
110.5
56.8
62.0
67.1
75.7
84.2
101.3
135.5
106.8
112.0
117.1
125.7
134.2
151.3
185.5
*If the average of the analyzed samples is larger than the cut-off value in the table, then conclude
that the shredder output violates the given standard. Otherwise, assume that the output meets the
standard. The chance of incorrectly finding a violation is 5%.
-------
in which case it will be impossible to reject the null hypothesis and further clean-up will be
required.
The cut-off values can be approximated by using the values in Table A-5
(reproduced here as Table 7). The values in this table were calculated using formula (15).
For example, for q = 4, s = 10, and ^ = 50, we have
Cut-off Value = 50 - 2.35 ^ = 38.3
V4
as shown in Table 7.
6. Power Calculations
6.1 Monitoring
Power Calculations. In hypothesis tests for monitoring, the power of
the test against a specific alternative, say \i = //a, is given by
(16) Power = P(X > Cut-off Value /z = ^a
'.os
To calculate estimates for power shown in Tables A-2 through A-4 (reproduced here as
Tables 8 - 10), the probability in formula (16) was approximated by using the standard
normal distribution. That is, we used the approximation
Power* ' -"' ""/*a
- -
where 0is the cumulative distribution function of the standard normal distribution.
30
-------
Table 7: Cut-off values for clean-up verification
Safety
Standard
Standard
deviation
Number of composite samples analyzed (q=m/c)
2
4
9
16
25
25
50
100
10
15
20
25
35
50
65
10
20
30
50
60
75
125
15
25
50
75
100
150
250
13.3
7.4
1.5
_
- -
5.4 38.3
26.5
14.8
_
_
- -
33.1 82.4
70.6
41.3
11.9
18.8
15.7
12.6
9.5
3.3
-
43.8
37.6
31.4
19.0
12.8
3.5
-
90.7
84.5
69.0
53.5
38.0
7.0
20.6
18.4
16.3
14.1
9.7
3.1
45.6
41.3
36.9
28.1
23.8
17.2
93.4
89.1
78.1
67.2
56.3
34.4
21.6
19.9
18.2
16.5
13.0
7.9
2.8
46.6
43.2
39.7
32.9
29.5
24.4
7.3
94.9
91.5
82.9
74.4
65.8
48.7
14.5
*A dash (-) indicates that the standard deviation is too large to establish that the site is clean.
-------
Table 8: Chance of finding violations in monitoring with a 25 ppm standard
to
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Chance of detecting violation*
Actual PCB concentration
30
35
40
50
60
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.00
0.02
1 0.08
0.13
0.18
0.00
0.03
2 0.11
0.19
0.26
0.00
0.04
4 0.15
0.26
0.38
0.00
0.05
8 0.21
0.36
0.51
0.00
0.04
0.15
0.25
0.36
0.00
0.05
0.22
0.39
0.55
0.00
0.08
0.34
0.57
0.76
0.00
0.12
0.48
0.74
0.90
0.00
0.05
0.22
0.37
0.53
0.00
0.08
0.34
0.57
0.76
0.00
0.14
0.51
0.78
0.93
0.00
0.22
0.69
0.92
0.99
0.00
0.08
0.33
0.56
0.75
0.00
0.14
0.53
0.79
0.93
0.00
0.25
0.75
0.95
0.99
0.00
0.40
0.90
0.99
1.00
0.00
0.11
0.42
0.68
0.86
0.00
0.20
0.65
0.89
0.98
0.00
0.35
0.86
0.99
1.00
0.00
0.54
0.96
1.00
1.00
Power calculations assume a 5% chance of incorrectly finding a violation.
-------
Table 9: Chance of finding violations in monitoring with a 50 ppm standard
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Chance of detecting violation*
Actual PCB concentration
60 70
85 1 100
125
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.00
0.02
1 0.08
0.13
0.18
0.00
0.03
2 0.11
0.19
0.26
0.00
0.04
4 0.15
0.26
0.38
0.00
0.05
8 0.21
0.36
0.51
0.00
0.04
0.15
0.25
0.36
0.00
0.05
0.22
0.39
0.55
0.00
0.08
0.34
0.57
0.76
0.00
0.12
0.48
0.74
0.90
0.00
0.06
0.25
0.43
0.60
0.00
0.10
0.39
0.64
0.83
0.00
0.17
0.59
0.85
0.96
0.00
0.27
0.77
0.96
1.00
0.00
0.08
0.33
0.56
0.75
0.00
0.14
0.53
0.79
0.93
0.00
0.25
0.75
0.95
0.99
0.00
0.40
0.90
0.99
1.00
0.00
0.11
0.44
0.70
0.87
0.00
0.21
0.68
0.91
0.98
0.00
0.37
0.88
0.99
1.00
0.00
0.56
0.97
1.00
1.00
"Power calculations assume a 5% chance of incorrectly finding a violation.
-------
Table 10: Chance of finding violations in monitoring with a 100 ppm standard
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Chance of detecting violation*
Actual PCB concentration
125
150
175
200
250
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.00
0.02
1 0.10
0.16
0.22
0.00
0.03
2 0.14
0.24
0.34
0.00
0.05
4 0.20
0.34
0.49
0.00
0.06
8 0.27
0.47
0.65
0.00
0.04
0.18
0.31
0.45
0.00
0.07
0.28
0.49
0.67
0.00
0.11
0.43
0.69
0.86
0.00
0.17
0.59
0.85
0.96
0.00
0.06
0.26
0.45
0.63
0.00
0.11
0.42
0.68
0.85
0.00
0.18
0.62
0.87
0.97
0.00
0.29
0.80
0.97
1.00
0.00
0.08
0.33
0.56
0.75
0.00
0.14
0.53
0.79
0.93
0.00
0.25
0.75
0.95
0.99
0.00
0.40
0.90
0.99
1.00
0.00
0.11
0.44
0.70
0.87
0.00
0.21
0.68
0.91
0.98
0.00
0.37
0.88
0.99
1.00
0.00
0.56
0.97
1.00
1.00
"Power calculations assume a 5% chance of incorrectly finding a violation.
-------
To calculate the values in Table 8, we allowed the alternative PCS level, /ia,
to vary between 30 and 250, depending on the standard used, and the number of
composited samples, q (= mlc\ to vary between 2 and 25. Finding an appropriate
approximation for s was somewhat complicated. First, notice that s2 is computed using q
composite samples, and that, in general, s2/q should be an unbiased estimate of V(0).
Thus, from formula (4),
suggesting that
£(S2} = Ji±^ + r* = Sampling Variability +
Since sampling error tends to increase with the level of the PCB concentration, we
estimated sampling variability as
o > A A r, A n Ao
rW - (cvTOiAO2 = (i.onar = A
(recalling that CVT£y) was determined empirically to be 1.0 for both white goods and
automobiles). Similarly, to estimate measurement error, we used
Combining these terms,
or
For example, if c = 2, m = 18, /^ = 50, and jua = 70, then
35
-------
50 - ^a , , 50-70 Rr -,
+ f ns = 77T7~rK + l-oo = .76.
-05 54.3/V9
Consulting a table of the standard normal distribution shows that
l-4>(.76) = .22
as shown in Table 9.
Alternative Significance Levels. Tables 11 and 12 show the effects
of alternative significance levels, 10% and 1%, respectively, on the power of hypothesis
test for monitoring when the standard is 50 ppm. The effects are analogous to those seen
when changing the level of confidence earlier in Section 4. Specifically, increasing the
significance level from 5% to 10% results in improvements in power, while lowering the
significance level from 5% to 1% results in sharp decreases in power.
6.2 Clean-Up Verification
Power Calculations. The power calculations for clean-up verification
are similar to those for monitoring. Recall, as discussed in Section 5, that the direction of
the hypothesis is reversed for clean-up verification. In addition, the question of power was
phrased in reverse in Tables A-6 through A-8 (reproduced here as Tables 13 through 15),
since one would want the chance of additional clean-up to be low if the site was in fact
clean. Thus, against a specific alternative, n = j/a, the chance of additional clean-up was
calculated as
(18) Chance of additional clean-up = P(x > Cut-off Value
/x =
36
-------
Table 11: Chance of finding violations in monitoring with a 50 ppm standard with 90% confidence
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Chance of detecting violation*
Actual PCB concentration
60 70
85
100
125
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.00
0.09
1 0.18
0.24
0.30
0.00
0.11
2 0.23
0.31
0.40
0.00
0.14
4 0.29
0.41
0.53
0.00
0.17
8 0.36
0.52
0.67
0.00
0.14
0.28
0.40
0.52
0.01
0.18
0.38
0.55
0.70
0.01
0.25
0.52
0.72
0.86
0.01
0.33
0.66
0.85
0.95
0.01
0.20
0.41
0.59
0.74
0.01
0.28
0.58
0.78
0.91
0.02
0.40
0.75
0.92
0.98
0.03
0.54
0.88
0.98
1.00
0.01
0.25
0.51
0.71
0.86
0.02
0.36
0.70
0.89
0.97
0.03
0.52
0.87
0.98
1.00
0.06
0.68
0.96
1.00
1.00
0.01
0.31
0.62
0.83
0.94
0.02
0.46
0.82
0.96
0.99
0.05
0.65
0.95
1.00
1.00
0.10
0.81
0.99
1.00
1.00
"Power calculations assume a 10% chance of incorrectly finding a violation.
-------
Table 12: Chance of finding violations in monitoring with a 50 ppm standard with 99% confidence
u>
oo
Total
samples
collected
m
Number of
composites
analyzed
q = mfc
Subsamples
in each
composite
c
Chance of detecting violation*
Actual PCB concentration
60 70
85
100
125
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.00
0.00
1 0.01
0.02
0.04
0.00
0.00
2 0.01
0.04
0.08
0.00
0.00
4 0.02
0.07
0.14
0.00
0.00
8 0.03
0.11
0.23
0.00
0.00
0.02
0.07
0.13
0.00
0.00
0.04
0.13
0.26
0.00
0.00
0.07
0.25
0.47
0.00
0.00
0.14
0.42
0.69
0.00
0.00
0.04
0.15
0.30
0.00
0.00
0.10
0.32
0.56
0.00
0.00
0.21
0.57
0.84
0.00
0.00
0.38
0.80
0.97
0.00
0.00
0.07
0.24
0.46
0.00
0.00
0.17
0.49
0.77
0.00
0.00
0.36
0.78
0.96
0.00
0.01
0.60
0.94
1.00
0.00
0.00
0.12
0.38
0.64
0.00
0.00
0.28
0.69
0.92
0.00
0.01
0.55
0.93
0.99
0.00
0.02
0.81
0.99
1.00
'Power calculations assume a 1% chance of incorrectly finding a violation.
-------
Table 13: Chance of requiring additional clean-up with a 25 ppm standard
Total
samples
collected
m
Number of
composites
analyzed
q = mJc
Subsamples
in each
composite
c
Chance of requiring more clean-up*
Actual PCB concentration
1 5 10 15
20
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.82 1.00
0.31
1 - - 0.01
_ _ -
_ _
0.16 1.00
0.07
2
_ _
_ _
1.00
_ _
4
_ _ -
_ _
0.97
- - -
8
_ _ -
« ^ ^
1.00
0.86
0.48
0.22
0.07
1.00
0.74
0.24
0.05
1.00
0.54
0.07
-
1.00
0.33
0.01
-
"
1.00
0.97
0.87
0.79
0.70
1.00
0.96
0.81
0.68
0.54
1.00
0.93
0.72
0.53
0.35
1.00
0.90
0.61
0.37
0.18
These calculations assume a 95% (or greater) chance of requiring additional clean-up when the
concentration of PCB's is 25 ppm or greater. A dash (-) indicates that the chance is less than .005.
-------
Table 14: Chance of requiring additional clean-up with a SO ppm standard
Total
samples
collected
m
Number or
composites
analyzed
Q = m/c
Subsamples
in each
composite
c
Chance of requiring more clean-up*
Actual PCB concentration
10 15 20 30
40
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.82 1.00
0.02
1
- -
0.16
2 - -
0.77
4
- -
0.27
8 - -
_
~~ ""
1.00
0.31
0.01
1.00
0.07
-
1.00
-
-
0.97
-
-
^
1.00
0.86
0.48
0.22
0.07
1.00
0.74
0.24
0.05
1.00
0.54
0.07
1.00
0.33
0.01
"
1.00
0.97
0.87
0.79
0.70
1.00
0.96
0.81
0.68
0.54
1.00
0.93
0.72
0.53
0.35
1.00
0.90
0.61
0.37
0.18
*These calculations assume a 95% (or greater) chance of requiring additional clean-up when the
concentration of PCB's is 50 ppm or greater. A dash (-) indicates that the chance is less than .005.
-------
Table IS: Chance of requiring additional clean-up with a 100 ppm standard
Total
samples
collected
m
Number of
composites
analyzed
q = mlc
Subsamples
in each
composite
c
Chance of requiring more clean-up*
Actual PCB concentration
20 30 40 60
80
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.82 1.00
0.02
1
0.16 0.98
2
0.77
-
4
0.27
8 - -
__
1.00
0.31
0.01
1.00
0.07
1.00
-
0.97
-
-
"
1.00
0.86
0.48
0.22
0.07
1.00
0.74
0.24
0.05
1.00
0.54
0.07
1.00
0.33
0.01
"
1.00
0.97
0.87
0.79
0.70
1.00
0.96
0.81
0.68
0.54
1.00
0.93
0.72
0.53
0.35
1.00
0.90
0.61
0.37
0.18
"These calculations assume a 95% (or greater) chance of requiring additional clean-up when the
concentration of PCB's is 100 ppm or greater. A dash (-) indicates that the chance is less than .005.
-------
The sample standard deviation, s, was approximated as described in Section 6.1. For
example, if c = 4, m = 8, /i0 = 50, and /za = 15, then
50-15 _ _.
Consulting a table of the standard normal distribution shows that
1_(-0.75) =
-------
Table 16: Chance of requiring additional clean-up with a 50 ppm standard with 90% confidence
Total
samples
collected
m
Number of
composites
analyzed
q = m/c
Subsamples
in each
composite
c
Chance of requiring more clean-up*
Actual PCB concentration
10 15 20 30
40
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
0.01 0.47 0.85
0.11
1 -
_ _
-
0.12 0.63
0.01
2 -
_ _
_ _
0.01 0.31
_
4 _
_ _
_ _
0.09
_ _
8
_ _ -
_ __
0.99
0.64
0.31
0.12
0.03
0.97
0.47
0.12
0.02
0.93
0.27
0.02
0.86
0.12
-
"
1.00
0.88
0.75
0.65
0.55
1.00
0.84
0.67
0.52
0.38
0.99
0.79
0.55
0.37
0.21
0.99
0.72
0.43
0.22
0.10
*These calculations assume a 90% (or greater) chance of requiring additional clean-up when the
concentration of PCB's is 50 ppm or greater. A dash (-) indicates that the chance is less than .005.
-------
Table 17: Chance of requiring additional clean-up with a 50 ppm standard with 99% confidence
Total
samples
collected
m
Number of
composites
analyzed
q = ntic
Subsatnples
in each
composite
c
Chance of requiring more clean-up*
Actual PCB concentration
10 15 20 30
40
2
4
9
16
25
4
8
18
32
50
8
16
36
64
100
16
32
72
128
200
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
2
4
9
16
25
1.00 1.00
0.54
1
- -
1.00 1.00
0.07
2 - -
- -
1.00 1.00
4 - -
1.00 1.00
8 - -
~~ "
1.00
0.95
0.08
1.00
0.75
1.00
0.30
1.00
0.04
^
1.00
1.00
0.84
0.52
0.25
1.00
1.00
0.63
0.20
0.04
1.00
0.99
0.32
0.03
1.00
0.96
0.10
-
"
1.00
1.00
0.99
0.95
0.90
1.00
1.00
0.97
0.91
0.81
1.00
1.00
0.95
0.82
0.65
1.00
1.00
0.91
0.69
0.45
*These calculations assume a 99% (or greater) chance of requiring additional clean-up when the
concentration of PCB's is 50 ppm or greater. A dash (-) indicates that the chance is less than .005.
-------
50272-101
REPORT DOCUMENTATION
PAGE
1. REPORT NO.
EPA 747-R-93-010
3. Recipient's Accession No.
4. Title and Subtitle
Sampling Guidance for Scrap Metal Shredders
Technical Background
5. Report Date
August 1993
6.
7. Authors)
James Bethel, Westat, Inc.
8. Performing Organization Rept No.
9. Performing Organization Name and Address
Westat, Inc.
1650 Research Blvd.
Rockville, MD 20850
10. Pro|eet/Taak/Work Unit No.
11. Contract (C) or Grant (G) No.
68-02-4293
12. Sponsoring Organization Name and Address
U.S. Environmental Protection Agency
Office of Prevention, Pesticides and Toxic Substances
Washington, D.C 20460
13. Type of Report & Period Covered
Technical Report
14.
15. Supplementary Notes
16. Abstract (Limit: 200 words)
The purpose of this document is to provide basic instructions for collecting and statistically
analyzing samples of materials that are produced as a result of shredding automobiles and other metal
objects, since die by-products of these recycling operations may contain concentrations of polychlorinated
biphenyl's (PCBs). Shredders are large machines that convert light metal objects into fist size or smaller
pieces of scrap metal. PCBs enter the shredder output when materials containing PCB-bearing fluids are
shredded. Large concentrations of PCBs have been identified in some samples that have been collected at
some recycling sites. Thus agencies may wish to collect data at shredder sites in order to study the situation
in their locality. The sampling procedures described in this document are intended to produce representative
samples of fluff that will give reasonably accurate estimates of the overall concentration of PCBs in the
material being sampled. The document discusses sample selection, laboratory testing, and statistical
procedures for analyzing the data. This volume provides additional detail on the assumptions, background
data, and numerical calculations that support the Field Manual (EPA 747-93-009).
17. Document Analysis a. Descriptors
Environmental contaminants, scrap metal recycling
b. Identiflers/Open-Ended Terms
PCB, sampling, statistical analysis
C.COSATI Field/Group
18. Availability Statement
19. Security Class (This Report)
Unclassified
20. Security Class (This Page)
Unclassified
21. No. of Pages
50
22. Price
(SeeANS(-Z39.18)
See Instructions on Reverse
OPTIONAL FORM 272 (4-77)
(Formerly NTIS-3S)
Department of Commerce
------- |