&EPA
United States
Environmental Protection
Agency
Office of Research and
Development
Washington, DC 20460
EPA/600/4-
September 1993
Revision
USEPA Manual of
Methods for Virology
Chapter 13
September 1993
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September 19S3
Chapter 13
Data Analysis and Experimental Design
Prepared by
Larry J. Wymer and Manohari Sivaganesan
Computer Sciences Corporation, Cincinnati, Ohio
1. Introduction
1.1 This chapter provides
descriptions and examples of
methods for the statistical evaluation
of data from virus plaque counting
assays. The primary emphasis is on
techniques for evaluating the
precision of estimates obtained from
such assays, in particular, titer
estimates of total plaque forming units
(PFU), PFU titer by type of virus, and
relative frequencies of virus types.
Also presented are techniques which
should prove useful in the area of
quality assurance, namely, methods
for evaluating dispersion of the
plaques, examining differences
between subsets of plaque counts,
screening the data for outliers and a
discussion of sample design, based
on the concepts of precision
presented in the previous sections.
1.2 The final section of this
chapter includes an extensive
compilation of statistical tables
designed to facilitate these analyses.
With the exception of Table 13-9,
these tables have been newly
compiled using Statistical Analysis
System (SAS) version 5.18 on an IBM
3090 computer. The source for Table
13-9 is ASTM Standard Practice
E-178 on dealing with outlying
observations (1980).
1.3 The following notations are
used uniformly throughout this
chapter:
X the total number of plaques
enumerated in all cell culture
bottles
n the total number of cell culture
bottles used
x, the number of plaques enumer-
ated in the i* cell culture bottle,
where the value of the index, i,
ranges from 1 through n
V the total volume of eluate
inoculated in all cell culture
bottles combined
v, the volume of eluate inoculated
in the i81 cell culture bottle,
i = 1 ,2 n
t the estimated titer (PFUs per
unit volume) of the eluate,
t = X/V
k the number of different types of
viruses identified in the assay
Xh the total number of plaques
enumerated in all cell culture
bottles, identified as being virus
type h, h = 1 ,2 k
xlh the number of plaques enumer-
ated in the i* cell culture bottle,
i = 1 ,2 n, and identified as
being virus type h, h = 1 ,2 k
th the estimated titer (PFUs per
unit volume) of the eluate for
the h* virus type,
The following relationships apply:
X=
1=1
I
h =1
k
h=l
X,. = 2 X!i
V= X Vi=
Volumes are expressed in terms of
eluate volume; however, these may
be converted into equivalent sample
volumes without affecting the validity
of the analyses.
For example, suppose 10 L of
sample is concentrated into 40 mL of
eluate of which 1 mL IS inoculated in
each cell culture bottle. In this case, v.
represents 1 mL, the same for all
values of the index, i; if 20 cell culture
bottles are inoculated in all (n=20),
then V represents 20 mL of eluate.
Also, 1 mL of eluate represents 0.25
liter equivalents (Leg) of sample
(10 L /40 mL = 0.25 Leq per mL);
therefore, for this example, results of
analyses of titer in terms of PFUs per
mL of eluate may be converted to
PFUs per Leq of sample volume by
dividing by a factor of 0.25.
2. Test for Random
Dispersion of Plaques
2.1 Introduction
2.1.1 Many of the statistical
methods presented in this chapter are
based on the assumption that plaques
enumerated in the assay are randomly
dispersed among cell culture bottles.
2.1.2 This section presents
methods for testing the validity of this
assumption. In cases where the
assumption of randomness is shown to
be invalid, alternative methods for
analysis of the data are available and
included in the appropriate section.
Precision of the assay will be greater
when the plaques are randomly
dispersed, rather than occurring in
clusters. In addition, a lack of
randomness may indicate a potential
problem with the assay, such as
contamination of cell lines, or other
non-random sources of error.
2.2 Formulas and test statistic
2.2.1 The index of dispersion
(Fisher, 1950) is used to test for a
random dispersion and is given by
t'V:
where
n
total number of cell culture
bottles used
x, = number of plaques in the P
cell culture bottle;
i = 1 ,2,3 n
v, = eluate volume used in the i*1
cell culture bottle
t = estimated PFU titer =
13-1
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2.2,2 When the same eluate
volume Is used for each cell culture
bottle (v. - v for 1-1,2,3 n)
calculation of the index of dispersion
simplifies to
M
whore F Is the average plaque count
per bottle,
1-1
and s2 is the sample variance, given
by
i-l
n-1
the latter expression for s2 being more
convenient for hand calculation.
2.2.3 The test statistic, D, is
compared to the 0.05 critical value for
chi-square with n-1 degrees of
freedom fcc^, «}. The null hypothesis
that the plaques are randomly
distributed is accepted when
D <%2 , «. Table 13-3 gives 0.05
crittearvaTues for x2 for degrees of
freedom ranging from 1 through 50.
In order for this test to be valid,
none of the expected plaque counts (t
vj should be less than one. In
addition, no more than 20% of the
expected counts should be less than
five (Cochran, 1954). When ihese
conditions are not satisfied in the raw
data, plaque counts from consecu-
tively numbered cell culture bottles,
starting with the highest dilutions,
must be combined until these condi-
tions are satisfied by the resulting
aggregated data. When data are
grouped in this manner, the counts
from the combined data are're-
indexed; ^ and v; now refer to the
combined counts and volumes,
respectively, and n becomes the
number of groups, counting any
combined data as a single group. If
necessary, for example when the
expected plaque count from the
combined cell culture bottles at the
highest dilution level is still lass than
one, data may be combined among
cell culture bottles which have been
inoculated with different volumes of
eluate.
2.3 Sample calculations
2.3.1 Dataset I is summarized as
follows:
bottle eluate
number per volume
(i) bottle (vt)
plaque counts (x,)
1-10
11-20
21-30
0.001 mL
0.01 mL
0.1 mL
(all zero)
0,2,2,0,2,0,0,2,0,0
10,12,10,6,16,13,9,
14,6,17
2.3.2 Steps for performing test for
random dispersion (Dataset I):
(a) Calculate the expected plaque
count for each cell culture
bottle.
E(x,) = t v,, where
t = total plaque count / total
volume of inoculum used
For this experiment, X = 121 and
V=1.11 ml, giving t= 121 71.11 =
109 PFU/mL Expected counts for
each bottle are, therefore,
Expected
bottle count
numbers v, (mL) per bottle (t vt)
1-10
11-20
21-30
0.001
0.01
0.1
109 x. 001= 0.109
109 x. 01
109X.1 =
= 1.09
10.9
(b) If necessary, group the data so
that the expected count for each
group is at least one and no
more than 20% of the groups
have an expected count of less
than five. In this example, the
ten bottles at the highest
dilution (v,-v,0 = 0.001 mL) have
expected counts (0.109) of less
than one, and the expected
count for each of the ten bottles
at the next highest dilution
(vn-v?0 = 0.01 mL) is less than
five (1.090). Combining the data
from cell culture bottles num-
bered 1 through 10, 11 through
15, and 16 through 20 results in
13 groups with expected and
observed counts as given in
columns 4 and 5, respectively,
in Table 13-1.
(c) Calculate the individual terms
of the index of dispersion (D)
from the observed and ex-
pected count for each cell
culture bottle, or group of cell
culture bottles if any data are
combined. Individual terms of
D are calculated from
(x, - t'V^/O'V,), where i ranges
from 1 through 13 for this
example. Results of these
calculations are shown in
column 6 of Table 13-1.
(d) Compare the D statistic, from
the summation of the individual
terms in step 3 to the appropri-
ate critical value for chi-square.
For this example, n = 13;
therefore, the appropriate 0.05
critical value for £2with n-1 =12
degrees of freedom is 21.026,
from Table 13-3. Since the
computed value of D = 15.413
is less than 21.026, we accept
the data as being randomly
distributed.
2.3.3 Sample calculations for
Dataset II are shown in Table 13-2.
Note that only a single dilution of the
eluate was used in this experiment,
and that no grouping of the data was
necessary, since expected plaque
counts were greater than five for each
cell culture bottle. Otherwise, steps in
the calculation of D are identical to
those in Section 2.3.2. Because equal
volumes of eluate were inoculated in
each cell culture bottle, the simplified
formula for D given in Section 2.2.2
may be used. Because the value
calculated for D (32.530) is greater
than the critical value of%s with 13
degrees of freedom (22.362, from
Table 13-3), the hypothesis that the
plaques are randomly dispersed is
rejected.
2.4 Rationale for the test
2.4.1 A "random dispersion of
PFUs" means that the probability of
finding a PFU in some small volume of
the liquid depends only on the volume
itself and is not affected by any known
presence or absence of PFUs in
neighboring regions of the medium. In
other words, knowledge that a given
region of the eluate contains one or
more PFUs would not be useful in
identifying other regions where PFUs
are most likely to be found. This is a
formal statement of what is meant by
saying that the eluate is "well mixed."
2.4.2 When PFUs are randomly
dispersed, the probability of finding
any particular number of PFUs in a
fixed volume of the liquid is given by
the Poisson probability distribution
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function (pdf). The variance of counts
that follow a Poisson pdf is equal to
their mean. The index of dispersion, D,
is simply the ratio of the sample
variance to the sample mean of the
counts, times n-1; therefore, when the
counts follow a Poisson pdf, one would
expect the value of D to be near n-1.
Because the sample mean and
variance used in calculating the index
of dispersion will vary from their true
values, rarely will the value of D be
exactly equal to n-1 even when the
data are randomly dispersed.
2.4.3 In those cases in which the
counts are not Poisson distributed, the
departure from random dispersion is
likely to be a result of factors which
increase the variance of the counts;
thus, when counts are non-Poisson, we
expect the test statistic, D, to be
greater than n-1. The upper 0.05
critical value for chi-square with n-1
degrees of freedom (Table 13-3)
establishes an upper limit for D,
beyond which it is not likely
(probability < 5%) that such a large
value for D could have resulted from
counts which follow a Poisson pdf. The
test is said to have a critical level
(sometimes referred to as the alpha
level) of 0.05.
2.4.4 When it becomes necessary
to combine data because some of the
expected plaque counts are less than
one or because more than 20% of the
expected plaque counts are less than
five, any arbitrary grouping of the data
is permissible. However, in no case
should data be combined on the basis
of observed plaque counts; for
instance, intentionally grouping data
from cell culture bottles which contain
few plaques specifically with data from
cell culture bottles containing a greater
number of plaques may invalidate the
test for randomness. The
recommendation to combine results
from consecutively numbered cell
culture bottles is made in order to
remove, or at least minimize, any
influence that the observed plaque
counts may have one's decision with
regard to any necessary grouping of
the data.
2.5 Implications for virus assays
2.5.7 When the index of dispersion
(D) test leads to the conclusion that the
plaque counts are not randomly
distributed, there are two possibilities to
consider; the PFUs are not randomly
dispersed throughout the medium, or
the process of assaying for PFUs has
introduced extraneous variability to the
results.
In the first case, because the PFUs
themselves are not randomly dis-
persed, one would not expect the
resulting plaques to be randomly
dispersed. The second possibility may
imply that assay conditions were "out
of control."
2.5.2 Figure 13-1 illustrates two
extreme examples of non-random
plaque dispersions (a and b) along
with a simulation of a completely
random dispersion (c). Each of the
three large squares represents the
total volume of inoculum; the smaller
squares represent those portions of
the total volume which are inoculated
into individual cell culture bottles. In
Figure 13-1 (a) the spacing among
PFUs is seen to be too uniform to be
considered random; in fact, this figure
was generated from a uniform lattice of
foci with small, random perturbations
about each lattice point. On the other
hand, plaques are seen to form three
distinct clusters in Figure 13-1 (b),
which was generated from a probability
distribution of points about three
central locations. As a result, plaques
tend to be evenly distributed among all
nine cell culture bottles in case a
(uniform), while most of the plaques
appear in only four of the bottles in
case b (clustered). Indices of
dispersion for the three cases are
calculated to be 1.00 (a: uniform case),
50.75 (b: clustered), and 6.75
(c: random), with eight degrees of
freedom each (n = 9). The 0.05 critical
value for %2 with eight degrees of
freedom is 15.507, from Table 13-3.
2.5.3 A uniform distribution of
PFUs, such as simulated in Figure
13-1 (a), would occur if PFUs exhibited
a tendency to repel one another. While
such a tendency may not be
unreasonable, the notion that PFUs
would interact over a range sufficient
to affect their behavior on a large scale
is not so reasonable. In addition, and
most importantly, a uniform distribution
is so seldom observed that one feels
reasonably assured in discounting any
experiment in which it occurs as simply
representing an unusual result from a
random mixture. Clustering, as
simulated in Figure 13-1(b), however,
may occur when PFUs are mutually
attracted to (or repelled from) specific
locations in the liquid. This is not at all
unreasonable; areas of net positive or
negative ion charge, for instance,
might have the large scale effects
necessary to produce clustering. Most
observed deviations from a random
distribution will be consistent with
clustering effects.
2.5.4 Procedures are presented in
the following sections for determining
precision of the estimated titer for both
Poisson and non-Poisson distributed
plaque counts. If the data are non-
Poisson because of variability
introduced by the assay procedure,
however, the resulting estimate is likely
to be biased. Unfortunately, it is not
usually possible to distinguish between
non-random dispersion of PFUs and
systematic biases resulting from the
assay procedure solely by means of
statistical analysis of the data.
Statistical procedures are presented in
Sections 3 and 4, however, which may
be useful for evaluating non-random
counts under certain circumstances.
Occasionally, it may not be possible to
perform the test for a random
dispersion on a given set of
experimental data. This would be the
case whenever fewer than ten plaques
are observed in total from among all
cell culture bottles used in the assay.
In such cases, while the assumption of
a random dispersion cannot be
verified, the recommendation,
nevertheless, is to proceed with this
assumption in the ensuing statistical
evaluation.
3. Tests for Outliers
3.1 Introduction
3.1.1 An outlier is an observation
that is discordant in relation to the
other sample data. In the context of
virus plaque count assays, an
individual plaque count which appears
to be unusually high or low compared
to the other counts may be considered
an outlier.
3.7.2 Statistical procedures are
presented in this section for evaluating
the hypothesis that an apparent
outlying observation is merely a result
of the inherent variability of the data,
rather than representing a truly
aberrant value as a result of some
discrepancy in the way in which that
observation was obtained. When these
procedures indicate that an
observation is a true outlier, this
should not be construed to imply that
such an observation be automatically
discarded. While any datum found to
be, statistically, an outlier should be
noted in reporting the results of the
assay, the decision to exclude the
observation must be based on, at a
minimum, the researcher's opinion that
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the observation is likely to have been
in error.
3.2 Statistical tests for outliers
3.2.1 The test to be used in
evaluating whether an observation (or
two observations) may be considered
an outlier depends on whether the
remaining observations, excluding the
potential outlier(s), are randomly
distributed. The test for a random
distribution of plaque counts, as
presented in Section 2 may be
performed on plaque counts excluding
those cell culture bottles yielding the
suspected outlier counts. However, if
the test of randomness performed on
all the plaque counts, including the
suspected outlier(s), indicates that the
assumption of a random dispersion is
reasonable, it is not necessary to
retest the subset of counts excluding
the outlior(s).
Because tha tests for outliers from
a random distribution are simple to
perform, involving no computation
other than totalling all plaque counts,
the recommendation is to perform a
test on the assumption of random-
ness for the remaining counts only if
the test for an outlier from a random
distribution indicates that the value in
question is significant. If the observa-
tion k not an outlier based on the
assumption of randomness, neither
will it be a statistical outlier with this
assumption removed.
3.2.2 Tests are presented for four
potential outlier situations:
(a) a single apparent high value
(b) a single apparent low value
(c) a pair of apparent upper
outliers
(d) a pair of apparent lower
outliers
33.3 Tests for outliers: randomly
dispersed data
(a) For a single upper or lower
outlier, the test statistic
consists of the value itself,
conditional on the total plaque
count in all cell culture bottles
at the same dilution of eluate.
For a pair of upper or lower
outliers, the test statistic
becomes the sum of tha two
extreme counts, conditional on
the total count. Descriptions of
these tests may be found on
pages 198-200 in Barnatt and
Lewis (1984).
(b) To test for a single upper
outlier, determine the greatest
number of plaques obtained in a
single bottle from among all cell
culture bottles inoculated with
identical volumes of eluate.
Enter Table 13-4 at the row
corresponding to this number
and the column corresponding
to the number of cell culture
bottles used. If the total of all
plaques enumerated in all cell
culture bottles (including the
value in question) is lower than
the entry in Table 13-4, then the
observation in question may be
considered to be an outlier at
the 0.01 critical level.
(c) Table 13-5 indicates lower
outlier values. Enter this table at
the row corresponding to the
lowest count among all cell
culture bottles inoculated with
the same volume of eluate. If
the total plaques enumerated in
all cell culture bottles (including
the value in question) is greater
than the entry under the column
for the appropriate number of
cell culture bottles, then the
observation in question may be
considered a lower outlier at the
0.01 critical level.
(d) Critical values for upper and
lower outlier pairs are indicated
in Tables 13-6 and 13-7,
respectively.
These tables are used in a
manner similar to their single
outlier counterparts (Tables 13-
4 and 13-5), except that the
rows correspond to the sum of
the two highest or two lowest
counts, respectively.
3.2.4 Tests for outliers: non-
randomly dispersed data
(a) Order the counts from n cell
culture bottles from lowest to
highest, so that
,< x,, -sx,
xn
Compute the variance of the
counts:
=i=l _ * _
n-l
where x is the sample mean of
the counts,
(b) For a single upper outlier,
compute the statistic,
_ (xn -x)
* n
where s, the sample standard
deviation, is the square root of
s2, the sample variance as
given above, in Section 3.2.4
Step (a).
Compare the resulting value of
T with the appropriate entry in
Table 13-8 for the total number,
n, of cell culture bottles used. If
T is greater than or equal to
this entry, then the count, xn,
may be considered an upper
outlying value. The procedure
for testing a lower outlier is
similar, except that the appropri-
ate test statistic is given by,
Table 13-8 gives the minimum
values for T, such that x, may
be considered a lower outlying
value.
(c) Given a pa/rof potential lower
outliers, the following test
statistic may be used,
where SS is the total sum of
squares, SS = (n-1)s2, and SS12
is the sum of squares omitting '
the two lowest observations,
«=3
The sample mean of the
remaining counts, *12> is given
by
«=3
Table 13-9 gives the critical
values for T such that TI 2 < T
indicates a statistically signifi-
cant lower outlier pair. For a
pair of apparent upper outlier
values, use the statistic,
where
"
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September 1993
and -n, n is the sample mean of
the remaining counts,
xn-l,n
n-2
t=l
Again, Table 13-9 gives the
maximum values for Tn, n for the
pair to be considered as"
representing two upper outlying
values.
3.3
Example
3.3.1 Data for this example consist
of the following set of plaque counts
from ten cell culture bottles, all
inoculated with 0.1 mL of sample
eluate:
8, 12,7,4,6, 11, 10,5,5, 19
3.3.2 The procedure for performing
the test for outliers is
(a) Identify apparent outlying
observations.
In this case, the last count in
the series, 19, appears to be
high relative to the other values.
(b) Find the total of all plaques in
the series.
For these data, the total
number of plaques enumerated
is 87.
(c) Compare the total from Step (b)
with the entry from the appropri-
ate table (Table 13-4, 13-5,
13-6 or 13-7), depending on the
direction and number of
suspected outliers.
One potentially high value
was identified; therefore, Table
13-4 is appropriate for this
example. The total count (87)
from step 2 is less than the
tabulated value (89) for a high
count of 19 among 10 cell
culture bottles; therefore, this
observation is considered to be
a statistical outlier. This conclu-
sion is, however, conditional on
whether the remaining data are
randomly dispersed.
(d) If steps 1-3 above indicate that
the observation(s) is an outlier,
recompute the D statistic
(Section 2) to perform the test
for randomness on the remain-
ing observations.
Omitting the count of 19 from
these data, and following the
steps outlined in Section 2.3.2,
Step (d) is recomputed to be
8.765. This is less than the 0.01
critical value for Chi-square with
8 degrees of freedom (= 15.507
from Table 13-3); therefore, we
accept that the remaining data
are randomly distributed. This
validates the finding that the
result of 19 plaques in one cell
culture bottle represents a
statistically significant outlier.
3.4 Rationale for the test
3.4.1 Tests for outliers using Tables
13-4 through 13-7 consider the results
to represent a multinomial distribution
of plaques among cell culture bottles,
where a plaque may appear in any of
the cell culture bottles with equal
probability. The multinomial distribution
function gives the probability for any
given arrangement of n objects
(plaques) among m containers, each
object being randomly and
independently assigned to one
container. Given this distribution, the
probability that the number of objects
assigned to a single container (or two
containers in the case of a pair of
outliers) will exceed, or fall below a
given value can be calculated. The
values in Tables 13-4 through 13-7 are
such that this probability is 1% or less.
The test for outliers, therefore, is said
to have a critical levelof 1%; use of
the 1% critical level is in accordance
with the American Society for Testing
and Materials (ASTM) standard
practice E-178 (1980) for dealing with
outlying observations.
For further discussion of outliers
from Poisson samples, see Barnett
and Lewis (1984).
3.4.2 Tests for outliers based on
Tables 13-4 through 13-7 assume that
the plaques are randomly (Poisson)
distributed among cell culture bottles,
with the possible exception of the
outlying observation(s) itself. The index
of dispersion is recomputed for the
remaining data to test this assumption.
When the remaining data fail the test
for randomness, these outlier tests are
not valid.
3.4.3 Alternative tests for outliers
when the random dispersion
assumption is shown to be invalid,
given in Section 3.2.4 and using
Tables 13-8 and 13-9, assume that
plaque counts are normally distributed
with unknown mean and variance, and
are based on ASTM standard practice
E-178 (1980) for dealing with outlying
observations.
Unless the data are approximately
normally distributed, the actual critical
levels associated with these tests will
differ from the nominal level of 1%.
The requirements for statistical outliers
based on Tables 13-8 and 13-9,
however, are more stringent than the
requirements for statistical outliers
from a random distribution. The
outliers have to be more extreme,
because of the assumption that the
variance of non-randomly dispersed
plaque counts will be greater than that
of randomly dispersed plaque counts.
3.5 Implications for virus assays
3.5.1 Some potential reasons for
outlying observations among viral
plaque count data include
(a) Contamination of cell culture
bottles
(b) Errors in inoculation
(c) Variation in incubation condi-
tions
(d) Counting errors.
3.5.2 The above conditions, among
others, will produce outliers only when
they differentially affect only one or two
observations. If, for example, all cell
culture bottles become contaminated,
this may be evidenced by an increased
variance among plaque counts (non-
random counts) or by a non-detectable
bias, but the observations may still be
concordant among themselves.
3.5.3 When an aberrant condition is
known to have affected the results in
one or more of the cell culture bottles,
these data should be discarded
without regard to whether they appear
to be outliers. Tests for outliers are
inappropriate when there are known
reasons for invalidating data.
3.5.4 When an observation is
shown to be an outlier, an effort should
be made to discover any variance in
the conditions under which that
observation was obtained. In practice,
however, such efforts often prove to be
fruitless; thus, the decision whether to
reject the observation in question
becomes a matter for the researcher's
judgement. At a minimum, the
existence of statistical outliers should
be noted in reporting the assay results.
When outlying observations are
discarded, the researcher's reasons for
doing so, beyond their being
13-5
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September 1993
statistically significant outliers, should
bo stated.
4. Test for Differences Be-
tween Groups of Plaque
Counts
4.1 Introduction
4.1.1 This section presents
methods for comparing results
between two sets of plaque counts.
4.1.2 Groupings may be defined in
any logical manner, such as two
different dilutions in a series, different
cell culture lines, or different methods
of incubation.
One would never expect the results
between two such groupings to agree
perfectly, even when there is known
to be no real difference between the
groups themselves, simply because of
random sampling variation. The
purpose of statistical analysis of the
difference is to determine whether one
might reasonably attribute the
difference to chance.
4.2 Test statistic
4.2.1 Calculate the PFU titer
estimate for each group:
where X, is the total plaque count in
all eel) culture bottles in group i (i=1,2)
and V, is the total volume of eluate
with which group i was inoculated.
Designate the groups as group 1 and
group 2 so that t,l / 0.1
2-0.1 _ V. 27
334 +
1
2-1.0
1.629
(c) Compare the value of R from
step 2 with the 0.05 critical
value of the F distribution with
2 X, + 1 and 2 X + 1 degrees
of freedom (Table f3-10).
In this case, the numerator
and denominator degrees of
freedom are, respectively, 669
(=2'334+1), and 109
(= 2 54 + 1). From Table 13-
10, the 0.05 critical value for the
F distribution is less than 1.378
whenever the degrees of
freedom for the numerator are
greater than 500 and the
degrees of freedom lor the
denominator are greater than
100. Therefore, the calculated
value for R from step 2, 1.629,
clearly exceeds the critical F
value, and we may infer that the
results between the two dilution
levels differ significantly.
4.3.3 Example 2
Data for this example consist of
plaque counts from ten cell culture
bottles of equal volumes of eluate.
bottle eluate volume
numbers per bottle (v,) plaque counts
1 thru 10 0.1 ml
11 thru 20 0.1 mL
5,4,4,6,8,3,5,4,4,7
8,7,4,6,9,5,7,4,5,7
A test for difference between the
two groups is to be performed.
4.3.4 Steps for performing the test
for difference and for constructing a
95% confidence interval are
(a) Identify the group with the lower
PFU titer and label it as group
1.
For this example, the
observed lower count X, is
equal to SO and the observed
higher count Xz is equal to 62.
(b) Compare the value of X2 from
step 1 with the 0.05 two-tailed
critical value M for the larger of
13-6
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September 1993
(c)
the two Poisson counts (Table
13-11).
In this case, the 0.05 critical
value M =72 (for lower count of
50) is greater than the observed
higher count 62. This means
that there is no significant
difference (statistically) between
the two groups.
Construct a 95% confidence
interval for the ratio (higher PFU
titer/lower PFU titer). For
example 2
R = (2X2+1)/(2X,+ 1)
= (2 62+1)7(2 -50 + 1)
= 125/101=1.24
The upper 2.5% critical value
F+ of F with (2 X, +1, 2- X. +
/; = (125,101) degrees of
freedom is 1.459 (Table 3.10).
The lower 2.5% critical value of
Fwith (125,101) degrees of
freedom is the reciprocal of the
upper 2.5% critical value of F
with (101,125) degrees of
freedom and is equal to
1/1.448 = 0.691. A 95%
confidence interval is given by
(R ' F-, R' F+)= (0.856,
1.809).
4.3.5 Occasionally, interpolation of
the values in Table 13-10 may be
necessary in order to determine
whether a calculated value for R is
statistically significant. In such cases,
inverse interpolation on degrees of
freedom is recommended; this requires
taking the inverses of the tabulated
degrees of freedom and desired
degrees of freedom before performing
the actual interpolation. While
interpolation is not necessary in the
example given in this section, because
the calculated value for R clearly
exceeds any possible critical F-value,
the appropriate critical F-value may be
estimated as follows:
(a) The entries in Table 13-10
which bracket the required
numerator and denominator
degrees of freedom (669 and
109, respectively) are
Denominator Numerator degrees
degrees of of freedom (ndf):
freedom: (ddf) 500 1000
100
150
1.378
1.307
1.363
1.290
(b) Interpolate the critical F-values
for numerator degrees of
freedom equal to 669. For this
interpolation, use the multiplier:
ml =
_ 669
i
500
1000
1
500
= 0.505
Perform this interpolation for
denominator degrees of
freedom (ddf) = 100 and 150:
(b.l)Forddf = 100:
669,100
1.378 +
m1 -(1.363-1.378)
1.378 + 0.505
(-0.015)
1.378 - 0.008
1.370
(b.2)Forddf = 150:
F669150= 1.307 +
m1 (1.290 -1.307)
= 1.307+0.505'(-0.017)
= 1.307 - 0.009
= 1.298
(c)
Interpolate the critical F-values
for denominator degrees of
freedom equal to 109. For this
interpolation, use the multiplier:
l
109
l
.100
0.248
1 _ 1
150 100
Use the F-values found in Step
(b):
= .1-370 +
m2 (1 .298 - 1 .370)
= 1.370 + 0.248 '(-0.072)
= 1.370-0.018
= 1.352
4.4 Rationale for the test
4.4. 1 Inverse sampling refers to a
sampling technique in which the
number of events (plaques) to be
observed is fixed in advance; what
varies in inverse sampling is the
volume required for exactly this
number of events to be observed.
Under inverse sampling, if Xt and X2,
as defined in Section 4.2.1, had been
specified in advance, and exactly Vt
and V2 volumes of medium had been
required to attain these predetermined
plaque counts, then the ratio t,/!, can
be shown to follow an F distribution
with 2 X, degrees of freedom in the
numerator and 2 X2 degrees of
freedom in the denominator, where
tj = Xj / Vj, j=1 ,2. In our case, because
sampling was not stopped at the
moment that the X * plaque occurred,
we conclude that V, was most likely to
have been greater than would have
been necessary for exactly X, plaques
to form, but less than was necessary
for X, + 1 plaques to form. Therefore,
X, + 1/2 is taken as the number of
occurrences. The same reasoning
applies to X2, leading to the test
statistic,
4.4.2 The test for differences
between groups of plaque counts, as
presented in this section is a two-tailed
(or two-sided) test because the group
with the higher titer estimate is always
placed in the numerator of the test
statistic, R. Thus, the test is for any
difference between the two groups,
regardless of which group may be
determined to have the higher titer.
4.5 Implications for virus assays
4.5.1 A statistically significant
difference between two groups of
plaque counts indicates that the
magnitude of the difference is large
enough that it is not likely to have
occurred by chance. The "critical level"
for the test specifies the risk assumed
in rejecting the equality between the
two groups, when, in fact, the true
means for the two groups are equal (a
'Type I" or "alpha" error).
A critical level of 0.05 is probably
the most commonly used value in
most fields of application, although
critical levels of 0.01 and 0.10 are
also commonly used, depending on
the level of risk the experimenter is
willing to assume.
4.5.2 The F test may be used
whenever one wishes to compare data
from two groups. Some examples of
comparisons that might be made, in
addition to dilution series as in the
example of Section 4.3, are
susceptibilities of different lines of a
cell culture, and inoculations at
different points in time.
Comparisons of this sort should be
made only between plaques from the
same titration, however, since
differences detected between plaques
obtained from separate titrations may
simply reflect differences in the
relative recoveries of PFUs between
the titrations.
13-7
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September 1993
5. Confidence Intervals for
PFU Titer
5.1 Introduction
5.1.1 Methods presented in this
section deal with the precision of the
PFU titer obtained from a single
titration viral assay.
5.1.2 The titer calculated from a
sample volume of material is a single-
valued, or "point", estimate of the PFU
titer in the population from which the
sample was drawn. An interval
estimate, on the other hand, is a range
of values such that there is a known
probability that the true value of the
population parameter, such as titer, will
be contained within this range.
Commonly, this probability is set at
95%, yielding a 95% confidence
interval.
5.7.3 Two point estimates of titer
may be identical but have different
precisions, as defined by the widths of
their respective 95% confidence
intervals.
5.2 Procedure
5.2.1 Following the procedure given
in Section 2, first determine whether
the assumption that the plaques are
randomly dispersed is valid. The
method used for computing 95%
confidence limits will depend on
whether the plaques are found to be
randomly distributed.
5.2.2 When the data are randomly
dispersed, use Table 13-12 for the
tower and upper 95% confidence limits
for the total number of plaques, based
on the total plaques actually
enumerated in the sample. Let X
represent the total number of plaques
in all cell culture bottles, let XL and \
represent the lower and upper 95%
confidence limits, respectively, from
Table 13-12, and let V represent the
total volume of eluate (in mL) used in
alt cell culture bottles. The estimated
titer Is, then, given by t - X/V, and
lower and upper 95% confidence limits
for titer are given by XU/V and Xy/V,
respectively.
5.2.3 When the data fail the test for
randomness (Section 2), 95%
confidence limits for total PFU titer
may be approximated by
t± 1.96
where
1.96
n-l
is the sample variance of t, and
s, the square root of s2, is the sample
standard deviation of t
n is the number of cell culture bottles
used
x, is the plaque count in the i"1 bottle,
i=1,2 n
v. is the eluate volume used in the i*
bottle
t and V are as defined in Section
5.2.2, above.
When equal volumes of eluate are
inoculated (v,=v, a constant volume),
the sample variance (s2) may be
calculated as
S,? _ I
i, Xj
;=i n
=i
n-l
5.3 Examples
5.3.1 For Dataset I (see
Section 2.3.1), the assumption that the
plaques are randomly dispersed was
previously shown to be valid. Titer is
estimated as 121 + 1.11 =109.0
PFU/mL. Table 13-12 gives lower and
upper confidence limits of 100.4 and
144.6 for the total plaque count, given
the sample count of 121. Therefore,
the 95% confidence interval for the
true titer is
100.4+1.11 = 90.5 PFU/mL to
144.6+1.11 =130.3 PFU/mL
5.3.2 Plaque counts in Dataset II
(Table 2.2) were shown to be non-
randomly dispersed. Calculation of the
95% confidence interval for the PFU
titer is as follows:
(a) Calculate the titer estimate,
t = X/V= 166/14 =11.9 PFU/mL
(b) Compute
Z (Xi-vi - t)2
JT1
385.74
13
= 29.672
1.96
14
1.96 3.742 5.447
14
= 2.9
(c) Compute the quantity
(d) Subtract the result from Step (c)
from the result of Step (a) to
obtain the lower limit, and add
these quantities for the upper
limit. The resulting 95% confi-
dence interval is
11.9 - 2.9 = 9.0 PFU/mL to
11.9 + 2.9 = 14.8 PFU/mL.
5.4 Rationale
5.4.1 When the data are randomly
distributed, the probability of finding a
given number of PFUs in the sample
volume of eluate can be calculated by
means of the Poisson probability
distribution function (pdf). The Poisson
pdf depends only on the mean number
of PFUs per unit volume. The lower
limit for the confidence interval in Table
13-12 is the value of this mean such
that the probability of finding at least
as many plaques as were actually
observed is 2.5%. Likewise, the upper
limit is the value of the mean for which
the probability of finding as many
plaques as were actually observed, or
fewer, is 2.5%.
This procedure was originally
presented by Garwood (1936).
5.4.2 The alternate procedure for
determining a 95% confidence interval
for the PFU titer is based on the
normal probability distribution
approximation to the confidence limits
for a ratio estimate (Cochran, 1977:
pp 150-165). Regardless of the
distribution of the actual data, the
sample ratio of PFUs per unit volume,
t, will approach a normal distribution as
the sample size increases. How large
the sample needs to be in order for the
normal distribution to approximate the
confidence interval reasonably well will
vary. In general, a smaller sample will
be required for this approximation to
be adequate when the number of
PFUs per unit volume is relatively large
than when there are relatively few
PFUs per unit volume.
On pages 39-44, Cochran (1977)
discusses the adequacy of the normal
approximation for computing confi-
dence limits for the sample mean in
general.
13-8
-------
September 1993
5.4.3 Implications for virus assays
(a) The sample volume of material
which is assayed is taken to be
representative of a larger
population of the same material.
Confidence intervals as given
in this section pertain to the true
value of "recoverable" PFUs per
unit volume in the population,
assuming assay conditions
resulting in identical relative
recoveries of PFUs. That is,
only single titration, and not
multi-titration, precision is
considered.
(b) Failure of the titration method to
recover all viruses present in
the original sample is regarded
as a bias.
It is not possible to evaluate
this bias for a single titration
assay without parallel assay of
a known standard.
6. Simultaneous Confidence
Intervals for PFU Titer by
Virus Type
6.1 Introduction
6.1.1 Simultaneous 95%
confidence intervals for titer by each
type of virus identified in the sample
are calculated in such a way that the
probability that all the intervals contain
the true values of their respective
population parameters is 95%.
6.1.2 For this to be true, confidence
intervals for the individual estimates
must be wider than they would be for a
single confidence interval for total titer,
because the probability of error (that a
confidence interval will not contain the
true value of its respective population
parameter) is compounded by the fact
that there are more opportunities for
error.
6.2 Procedure
6.2.1 As in the procedure for the
determination of a single 95%
confidence interval for total PFU titer,
the method used for computing
simultaneous 95% confidence intervals
for PFU by type of virus will depend on
whether the test for random dispersion
of all plaques (see Section 2) has
indicated that the assumption of
randomness is valid. The method for
computing simultaneous confidence
intervals, then, will correspond to the
method used for determining the
confidence interval for overall PFU
titer.
6.2.2 When the data are randomly
dispersed, use Table 13-13 for the
lower and upper 95% confidence limits
for the total number of plaques for
each type of virus identified in the
sample, based on the total plaques of
that virus type actually enumerated in
the sample and the number of different
types identified. Let X h represent the
total number of plaques for the given
virus type in all cell culture bottles, let
X hL and X hu represent the lower and
upper 95% confidence limits,
respectively, from Table 13-13, and let
V represent the total volume of eluate
(in mL) used in all cell culture bottles.
The estimated titer is, then, given by
t h = X h / V, and lower and upper 95%
confidence limits for titer are given by
X hL / V and X hu / V, respectively.
6.2.3 When the data fail the test for
randomness (Section 2), 95%
simultaneous confidence limits for the
PFU titer by virus type may be
approximated by
the sample variance (s2) may be
calculated as
th±-
where
.1 _ j=l
" n-1
is the sample variance of th, and
sh (the square root of s) is the sample
standard deviation
n is the number of cell culture bottles
used
xih is the plaque count in the i* bottle,
i=1,2,3 n
v. is the eluate volume inoculated in
the i* bottle
t h and V are as defined in Section
6.2.2, above.
The value for "z" will depend on the
number of different types of viruses
identified in the sample:
number of
virus types
2
3
4
5
6
7
8
z
2.24
2.39
2.50
2.58
2.64
2.69
2.73
n-1
v2
"if!'
When equal volumes of eluate are
inoculated (v.=v, a constant volume),
n-1
6.3 Examples
6.3.1 For the plaque counts of
Dataset I (Section 2.3.1), the
assumption of random dispersion was
shown to be valid. The 95%
confidence interval for overall titer was
calculated in Section 5, and found to
be 90 to 130 PFU/mL Identification of
the plaques observed in Dataset I
yields the following results:
Type Number found
Coxsackie virus CB2 67
Coxsackie virus CBS 37
Polio virus PV1 17
Total eluate volume used in all cell
culture bottles was 1.11 mL.
6.3.2 Because three virus types
were identified in the sample, 95%
lower and upper confidence intervals
for the number of plaques by type of
virus are found from Table 13-13 for
"number of groups = 3". These limits
are divided by the total eluate volume
(1.11 mL) to obtain the 95%
simultaneous confidence intervals for
titer by type of virus as shown below:
CB2 CBS PV1
Number found 67 37 17
Confidence
Limits from
Table 13-13:
lower 48.99 24.02 8.717
upper 89.31 54.32 29.71
Overall titer
PFU/mL 60 33 15
95%
Confidence
Interval
PFU/mL 44 - 80 22-49 8-27
6.3.3 Plaque counts in Dataset II
(Table 13-2) were shown to be non-
randomly dispersed. The 95%
confidence interval for the overall PFU
titer was calculated in Section 5 to be
from 9.0 to 14.8 PFU/mL. Identification
of the plaques found in Dataset II
yields the following results:
13-9
-------
September 1993
Type
Number
found
in each
bottle
Total
CB2
CBS
PV1
3,6,4,4,9,7,3,
8,4,2,13,5,9,4
2,6,1,3,5,10,6,
6,2,2,4,2,5,2
2,2,2,3,1,4,0,
4,0,2,3,1,3,2
81
56
29
6.3.4 Detailed calculations for CB2
virus (h-1) are shown below. Because
the same eluate volume is used in
each cell culture bottle (1 ml each),
the simplified formula for calculating
the sample variance, given in Section
6.2.3, may be used.
Bottle (i) x. x*, v,(mL)
1
2
3
4
5
6
7
8
9
10
11
12
13
H
Total
3
6
4
4
9
7
3
8
4
2
3
5
9
4
81
9
36
16
16
81
49
9
64
16
4
169
25
81
J£
591
1
1
1
1
1
1
1
1
1
1
1
1
1
1
14
2 ,591-Sl2/ 14 m 591-468.64
1 14-1 13
*3.068
ti -81/14 = 5.8 PFUAnL
95% tower confidence limit =
z
-------
September 1993
be obtained simply by dividing the total
number of plaques found into the
number of plaques testing positive for
the given serotype.
7.1,2 Simultaneous 95%
confidence intervals for the relative
frequencies for all virus types that are
identified in the sample are calculated
in such a way that, as in the case of
titer estimates by type of virus, the
probability that all the intervals contain
the true value of their respective
population parameters is 95%.
7.1.3 Section 7 presents methods
for obtaining 95% simultaneous
confidence intervals for these
proportions.
7.2 Procedure
7.2.1 Arbitrarily number the various
types of viruses identified in the
sample from 1 to k. Let Xh represent
the number of plaques identified as
being type h, h=1,2 k. The
proportion of virus type h is given by
Ph = \M; where X is the total number
of plaques observed in all cell culture
bottles.
7.2.2 If only two types of viruses
have been identified in the sample
(k=2), only one confidence interval
need be calculated; the lower (upper)
95% confidence limit for the one type
corresponds to one minus the upper
(lower) 95% confidence limit for the
other type. When a total of 50 or fewer
viruses have been identified (X < 50),
use Table 13-14 to find the lower and
upper 95% confidence limits; this table
is based on the exact probabilities
associated with a binomial occurrence.
In cases where more than 50 viruses
have been typed and identified, or for
results not found in Table 13-14, the
95% confidence interval for either type
can be calculated using the following
equation (Cochran, 1977: pp 57-59):
^["'^
1
2X
where qh = 1 -ph.
7.2.3 When three or more types of
viruses have been identified in the
sample, the following formula may be
used to determine simultaneous 95%
confidence limits for the respective
proportions of each type (Quesenberry
and Hurst, 1964):
"~
+ ^X2+2Xh+C
"h~ 2(X+X2)
where
ph~ and ph* represent the lower and
upper confidence limits, respectively,
X is the total number of plaques
enumerated in the sample,
Xh is the number of plaques identified
as virus type h,
C is calculated as
+ 4' Xh(X-Xh)]
The value for %2 (chi-square) will
depend on the number of different
types of viruses identified in the
sample:
Number of
Virus Types %2
k = 3
4
5
6
7
8
5.731
6.239
6.635
6.960
7.237
7.477
The formulas forph~ and ph* are valid
as long as the lower confidence limit
(p) is large enough to account for at
least five viruses, given the total
number of viruses identified (that is,
X p > 5).
7.2.4 Table 13-15 gives selected
results for this formula for the lower
and upper simultaneous 95%
confidence limits for the respective
proportions of each type. In cases
where the formula in Section 7.2.3,
above, is not valid (X ph < 5), the
entries in Table 13-15 are based on
exact multinomial probabilities, using
the Bonferroni adjustment described in
Section 6.4.1.
To use Table 13-15, first find the
appropriate subtable based on the
number of different virus types
identified in the sample (k). The lower
(ph-) and upper (p+) confidence limits
are found in the entries corresponding
to the respective sample proportion of
the virus type (pj and the total
number of plaques found (X). Interpo-
lation may be necessary where these
values differ from the tabulated
values. Table 13-15 includes only
sample proportions up to 0.5; in cases
where the proportion exceeds 0.5
(ph > 0.5), simply use the table to find
confidence limits forqh, as defined in
Section 7.2.2.
7.3 Examples
7.3.1 95% simultaneous confidence
intervals for titer by type of virus were
calculated in Section 6.3 for Dataset I.
Identification of the plaques observed
in this dataset are reproduced below:
Type
Coxsackievirus (CB2)
Coxsackievirus (CB5)
Poliovirus (PV1)
Number
found
67
37
17
A total of 121 plaques were
enumerated in all.
7.3.2 Because three types of
viruses were found in the sample, the
appropriate value of %2 to be used is
5.731. Detailed calculations for the
CB2 virus are shown below:
{
"\
5.731 [5.731 + 4 67 (121-67)]
121 -
-_ 5.731 + 2 67-26.186
Pl~ 2 (121+5.731)
113.545
' 253.462 '
:0.45
5.731 + 2 67 + 26.186
2 (121 + 5.731)
165.917
253.462
= 0.66
Calculations for all virus types found
in the sample yield the following
results:
95%
Simultaneous
Sample Confidence
Type Xh Proportion Interval
CB2
CBS
PV1
67
37
17
0.554
0.306
0.140
0.45
0.22
0.08
-0.66
-0.41
-0.23
7.4 Rationale
7.4.1 The formula for confidence
limits when two viruses have been
identified is based on the normal
approximation to a binomial
distribution. The binomial distribution
13-11
-------
September 1993
describes the probability of obtaining a
gh/en number of "successes" (one
particular type of virus) out of a fixed
number of "trials" (total plaques). The
tower confidence limit for the true
proportion of the given type of virus is
such that the probability of obtaining
as many or more plaques from that
type of virus as were actually identified
in the sample is 0.025 (2.5%).
Conversely, the upper limit is such that
the probability of obtaining as few or
fewer plaques of that type as were
Identified is also 0.025. In other words,
the lower (upper) confidence limit
represents a value for the true relative
frequency of that particular vims in the
sample for which the proportion that
was actually observed still may be
regarded as being a "reasonable"
result.
7.4.2 Theterm"1/(2X)"inthe
expression for the normal
approximation to binomial confidence
limits, as given in Section 7.2.2, is a
continuity correction factor. It arises
from the fact that only discrete values
of p., are possible, in increments of 1/X,
while the use of the normal
approximation requires that the p,, be
continuously variable. More familiar
may be the expression "pq/X" under
the radical, rather than "pq/(X-1)". The
latter term is always correct as an
unbiased estimator of the sampling
variance of p, although the difference
between the two is not appreciable for
large values of X.
7.4.3 For cases when three or more
virus types are identified in the sample,
the formula given in Section 7.2.3 is an
approximation to the multinomial
probability of finding a given number of
viruses of each type. The multinomial
probability distribution was used in
Section 3 to test for outliers in plaque
counts among cell culture bottles,
where it was assumed that the
probabilities of plaque formation are
equal among cell culture bottles. Here,
the probabilities, ph, of finding the
various types of viruses are not
necessarily equal; the procedure of
Section 7.2.3 establishes limits on
these joint probabilities such that the
observed set of proportions may be
considered a likely result.
A discussion of this procedure may
ba found In Goodman (1965) or
Quesenbarry and Hurst (1964).
7.5 Implications for virus assays
7.5.1 While the procedure given in
Section 6 for estimating the liter for
each type of virus requires that all
plaques found in each cell culture
bottle, or at least a subset of these
bottles, be identified, relative
frequencies among the virus types can
be estimated from identification of a
subsample of the plaques found in
each cell culture bottle. However,
subsampling of plaques within a cell
culture bottle is not recommended,
because of the possibility that the
operator's selection of plaques to be
serotyped may result in substantial
bias in favor of certain types of virus. If
it is not feasible to identify all the
plaques observed in all cell culture
bottles, then a subsample of the
bottles should be selected and all
plaques in those bottles identified. In
order to avoid bias as a result of the
selection of cell culture bottles for
serotyping, a fixed selection scheme
should be used, such as selecting
consecutively numbered bottles
starting with the lowest number. Note
that there is no requirement to
randomize the selection of bottles, as
long as the selection scheme is fixed
in advance, or at least without
knowledge of the results in the
individual bottles.
7.5.2 As in the case of
simultaneous confidence intervals for
PFU t'rter by virus type (Section 6),
simultaneous confidence intervals for
proportions may depend on which
virus types are considered relevant to
the assay. If only viruses belonging to
a particular class are of interest, then
other classes of viruses may be
grouped together in an "all other"
category. Say that two virus types
belonging to the class of interest were
identified; in this case, the number of
types of viruses is effectively 3 (k=3),
assuming that other types of viruses
were also found in the sample. One
may also group viruses which
represent less than a given percentage
of the sample together into an "all
other" category.
8. Experimental Design
8.1 Precision vs. sample size
8.1.1 Precision of the virus plaque
counting assay is dependent on the
total number of plaques ultimately
enumerated by the assay. Assuming
the PFU be randomly dispersed, the
effect of total count on assay precision
may be illustrated by referring to Table
13-12 for the 95% confidence limits
applicable to various plaque counts.
For example, say that only 10 plaques
have been observed in all cell culture
bottles. Table 13-12 indicates a 95%
confidence interval of 4.795 to 18.39
for the true population plaque count
13-12
when 10 plaques are observed in the
sample; therefore, the true count may
be as much as 5.205 less than or
8.39 more than the observed count.
The "absolute precision" will be
defined as the mean of these two
extremes, which in this case is
(5.205+8.39)72 = 6.80. Notice that this
corresponds to one-half the width of
the 95% confidence interval. When
100 plaques are enumerated in all
cell culture bottles, the
corresponding absolute precision
is seen to be 20.1, once again
taking the half-width of the 95%
confidence interval (81.36 -121.6).
Absolute precision for total plaque
count vs. observed plaque count is
shown in Figure 13-2, and is seen to
increase with total plaque count, but at
a decreasing rate as the count
increases.
8.7.2 Relative precision is defined
as absolute precision divided by the
observed count. From Section 8.1.1,
above, the relative precision is 68%
when 10 PFU have been enumerated
(6.80/10). When 100 PFU are
enumerated in all cell culture bottles,
the corresponding relative precision is
seen to be 20% (20.1/100). Figure
13-3 is a graph of the relative precision
as defined above (half-width of the
95% confidence interval divided by
observed count) vs. the observed
count. Dramatic reductions in relative
precision are seen to occur initially as
the total count increases. Roughly, the
relative precision drops by a factor of
about 3/10 for a doubling of the total
count; for example, relative precision is
100% for a count of 5, 68% for 10,
47% for 20, etc. Increases in the total
plaque count are realized simply by
assaying a larger volume of the
original sample.
8.2 Sample size determination
8.2.7 The question of how much
original sample needs to be assayed
can be answered in terms of the
degree of absolute or relative precision
one desires to achieve. One problem
with this approach is that one needs to
know how many PFUs will be
recovered from a given volume of
sample in order to gauge precision;
this implies that the researcher needs
to know in advance what the results of
the assay will be before deciding how
much sample to assay in the first
place, a common dilemma for sample
design problems in general.
8.2.2 In many cases, a reasonable
estimate of the expected PFU recovery
per unit volume can be made. This is
especially true if the sample is from a
-------
September t993
source that has previously been
assayed for viruses; in this instance,
one simply uses the last result or some
average result over the last n periods
for a previously sampled site and
assume that the current outcome will
be the same. Even when there is no
historical data for a particular site, the
nature of the site may enable one to
make reasonable assumptions about
the expected magnitude of the PFU
density based on experience in
assaying similar types of samples.
Given an assumed value for the
number of PFUs that will be recovered
per unit volume of sample, or even a
range of possible values, one can use
Figures 13-2 and 13-3 and Table
13-12 to arrive at an adequate sample
volume which will yield the desired
degree of precision.
8.2.3 Another solution to sample
size determination is to perform a
screening experiment on the sample.
When historical data from the sampling
site is lacking, a screening experiment
is also desirable from the standpoint of
determining proper dilution levels for
the eluate; this in itself is important to
maximizing the precision of the assay
by avoiding loss of data through cases
where plaques are too numerous to
count (TNTC). However, in many
instances, the extra time and expense
involved in performing a screening
experiment may be prohibitive.
8.2.4 A practical approach to
determining the total volume of sample
to be assayed is to establish some
minimum absolute precision level
when the PFU density in the original
sample falls below some
predetermined low level while targeting
for a minimum relative precision for
densities beyond some other, higher
value of PFU density. For example,
one may set the estimation error to be
no more than 20% (relative precision)
if the PFU density in the original
sample is at least 50 PFU/L, but be
willing to accept estimation error of
5 PFU/L (absolute precision) if the
PFU density is found to be 10 PFU/L
or less. These two criteria lead to two
sample size requirements:
(a) For a relative precision of 20%,
about 100 total plaques need to
be enumerated in the assay
(Figure 13-3). When the density
of PFUs in the sample is 50 per
liter, two liters of sample need
to be assayed to achieve a total
plaque count of 100. If the
density is higher than 50 PFU/L,
then assaying two liters of
sample will yield a plaque count
over 100, and, as can be seen
in Figure 13-3, the relative
precision will be better than
20%.
(b) An absolute precision of
5 PFU/L corresponds to a
relative precision of 50% when
the true density is 10 PFU/mL;
from Table 13-12, the relative
precision is 49.3% when the
sample plaque count totals 18.
Therefore, with a density of
10 PFU/L, 1.8 liters of sample
are required in order to achieve
the second objective. If the
density is less than 10 PFU/L,
then fewer than 18 plaques are
expected to be enumerated,
and, as can be seen from
Figure 13-2, the absolute
sampling error will be lower
than 5 PFU/L (absolute preci-
sion in total plaque count < 9).
(c) The larger of the two sample
requirements, in this case two
liters, is used. The stricter
requirement turns outto be the
limit of 20% relative precision
for a density of 50 PFU/L or
more.
8.2.5 When adsorption, elution,
and/or concentration methods are
used in recovering viruses, choices
must be made as to the volume of
sample to be used in the recovery
procedure, whether to further
concentrate the eluate, the volume and
dilutions of eluate to be inoculated,
and the number of cell culture bottles
to receive these inoculations. As long
as the number of plaques ultimately
enumerated in the assay are sufficient
to meet the sampling requirements, as
described above, and assuming that
PFUs are randomly dispersed, these
details of the experimental design do
not affect sampling precision, and are
not relevant to sample size
considerations.
8.2.6 Other factors, such as cost or
time, may be considered in making
these decisions. As an illustration,
consider the sample size requirement
derived in the previous section.
Assume that the adsorption-elution
process is expected to yield about
40 mL of eluate, and that a 1:10
dilution series will be used so that cell
cultures will be inoculated with 1, 0.1,
and 0.01 mL of eluate. Three cell
culture bottles, one at each dilution
level, will be considered as comprising
a single replicate; therefore, one
replicate accounts for 1.11 mL of
13-13
eluate volume. The following represent
different ways in which two liters of
sample can be subjected to the assay
in terms of the initial sample volume
used in the adsorption-elution
procedure and the number of replicate
observations:
Initial Total
Sample Cell Eluate
Volume Culture Volume
(L) Replicates Bottles Used
(mL)
2
4
8
36
18
9
108
54
27
39.96
19.98
9.99
In general, it is desirable to use a
dilution series, similar to that in this
example, and to use as many
replicates as feasible in order to guard
against potential data losses due to
TNTC situations, toxicity, or other
effects which may lead to the loss of
results in individual cell culture bottles.
8.3 Sampling requirements for
estimation by virus type
8.3.1 The procedure given in
Section 6 for estimating PFU titer by
type of virus is valid only when all
plaques found in a set of cell culture
bottles, if not all cell culture bottles, are
serotyped. The method of Section 7 for
estimating relative frequencies of virus
types, however, can be applied even if
only selected plaques from each cell
culture bottle are typed, with the minor
modification that the total number of
plaques enumerated be replaced by
the total number of plaques which
have been so identified. Nonetheless,
even if only relative frequency, and not
titer by virus type were of interest, the
recommendation is to identify all of the
plaques found in a cell culture bottle
until all plaques enumerated in the
assay have been identified, or until a
number of plaques have been
identified which is sufficient to meet
precision requirements. This eliminates
bias that may result from operator
selection of specific plaques to be
identified.
8.3.2 Estimating sample size
requirements for attaining a target
precision for PFU density or frequency
by type of virus is more uncertain than
estimating sample size requirements
based on overall PFU density;
precision now depends on the number
of different types one expects to find
and their expected relative
-------
September 1993
frequencies, in addition to the total
PFU density. Generally, unless it is
very important to obtain precise
estimates of densities of the various
types of viruses that may be found in
the sample, one designs the assay for
total PFU recovery and accepts
whatever result occurs with respect to
virus type.
9. References
American Society for Testing and
Materials: Committee E-11 on
Statistical Methods. 1980. Stan-
dard practice for dealing with
outlying observations,
ASTM E 178-180.
Barnett, V. and Lewis, T. 1984.
Outliers in Statistical Data, 2nd
ed., John Wiley and Sons, NY.
Cochran, W.G. 1954. Some methods
for strengthening the common %2
tests. Biometrics 10:417-451.
Cochran, W.G. 1977. Sampling
Techniques, 3rd ed., John Wiley
and Sons, NY.
Cox, D.R. 1953. Some-simple
approximate tests for Poisson
variates. Biometrika 40:354-360.
Garwood, F. 1936. Fiducial limits for
the Poisson distribution. Biometrika
43:423-435.
Fisher, R.A. 1950. The significance
of deviations from expectations in
a Poisson series.
Biometrics 6:17-24.
Goodman, LA. 1965. On simulta-
neous confidence intervals for
multinomial proportions.
Technometrics 7:247-254.
Quesenberry, C.P. and Hurst, D.C.
1964. Large sample simultaneous
confidence intervals for multinomial
proportions. Technometrics 6:191-
195.
Sachs, L. 1984. Applied Statistics: A
Handbook of Techniques, 2nd ed.,
Springer-Verlag, NY.
Snedecor, G.W. and Cochran, W.G.
1980. Statistical Methods, 7th ed.,
Iowa State University Press, Ames,
Iowa.
13-14
-------
September 1993
Table 13-1. Test of Dispersion for Dataset I
(D (2) (3) (4)
bottle vf Expected
I number(s) (ml) count (t- v)
1
2
3
4
5
6
7
8
9
10
11
12
13
TOTAL
1-10
11-15
16-20
21
22
23
24
25
26
27
28
29
30
.01
.05
.05
.1
.1
.1
.1
.1
.1
.1
.1
.1
J_
1.11
D= 15.413
1.09
5.45
5.45
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
10.9
121
<*i2,.05=21.026
(5)
Observed
count xi
0
6
2
10
12
10
6
16
13
9
14
6
12
121
(6) i
(xi - 1 v,y
t'V;
1.090
0.056
2.184
0.074
0.111
0.074
2.203
2.386
0.405
0.331
0.882
2.203
3.414
15.413
Table 13-2. Test of Dispersion for Dataset II
(D (2) (3) (4)
bottle v;. Expected
1 number (ml) count (x)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
TOTAL
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
1
1
1
1
1
1
1
1
1
1
1
1
1
14
11.857
11.857
1 1.857
1 1.857
1 1.857
1 1.857
11.857
11.857
11.857
1 1.857
11.857
11.857
11.857
11.857
166
D = 32.530 <%h,.os = 22.362
(5)
Observed
count x,
7
14
7
10
15
21
9
18
6
6
20
8
17
&
166
(6)
(x-.-x)*'*
1.990
0.387
1.990
0.291
0.833
7.050
0.688
3.182
2.893
2.893
5.592
1.255
2.231
1.255
32.530
13-15
-------
September 1993
Table 13-3.
Degrees of
freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
0.05 Critical Values of the Chi-Square (%2) Cumulative Distribution
Chi' Degrees of Chi
square freedom square
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
38.885
40.113
41.337
42.557
43.773
44.985
46. 194
47.400
48.602
49.802
50.998
52. 192
53.384
54.572
55.758
56.942
58. 124
59.304
60.481
61.656
62.830
64.001
65.171
66.339
67.505
'For degrees of freedom (v) > 50, use % - -112 (1.645 + Y2v-l )2
13-16
-------
September 1993
Table 13-4. 0.01 Critical Values for a Single Upper Outlier from a Poisson (Random) Distribution
Number of cell culture bottles
3 4 5 6 7 8 9 10 11 12 13 14 15 16
Highest
count
Is an upper outlier when the total count is less than
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
-
-
-
7
9
10
12
14
16
18
20
22
25
27
29
31
33
36
38
40
42
45
47
49
52
54
57
59
61
64
66
69
71
73
76
78
81
83
86
88
91
93
96
98
101
103
106
108
-
-
6
8
10
12
15
17
20
22
25
28
30
33
36
39
42
44
47
50
53
56
59
62
65
69
72
75
78
81
84
87
90
94
97
100
103
107
110
113
116
120
123
126
129
133
136
139
-
5
7
9
12
14
17
20
23
26
29
33
36
39
43
46
50
53
57
60
64
68
71
75
79
83
87
90
94
98
102
106
110
114
118
122
125
129
133
137
141
145
150
154
158
162
166
170
-
5
7
10
13
16
19
23
26
30
34
38
41
45
49
54
58
62
66
70
75
79
83
88
92
97
101
106
110
115
119
124
129
133
138
143
147
152
157
162
166
171
176
181
186
190
195
200
-
5
8
11
14
18
22
25
29
34
38
42
47
51
56
61
66
70
75
80
85
90
95
100
105
111
116
121
126
132
137
142
148
153
158
164
169
175
180
186
191
197
202
208
213
219
225
230
-
6
9
12
16
20
24
28
33
37
42
47
52
57
63
68
73
79
84
90
95
101
107
113
118
124
130
136
142
148
154
160
166
172
178
185
191
197
203
210
216
222
228
235
241
247
254
260
-
6
9
13
17
21
26
31
36
41
46
52
58
63
69
75
81
87
93
99
106
112
118
125
131
138
145
151
158
164
171
178
185
192
198
205
212
219
226
233
240
247
254
261
268
276
283
290
4
6
10
14
18
23
28
33
39
45
50
57
63
69
75
82
89
95
102
109
116
123
130
137
144
151
159
166
173
181
188
196
203
211
218
226
234
241
249
257
264
272
280
288
296
304
312
319
4
7
10
15
19
25
30
36
42
48
55
61
68
75
82
89
96
104
111
118
126
134
141
149
157
165
173
181
189
197
205
213
222
230
238
246
255
263
272
280
289
297
306
314
323
332
340
349
4
7
11
15
21
26
32
38
45
52
59
66
73
81
88
96
104
112
120
128
136
144
153
161
170
178
187
196
204
213
222
231
240
249
258
267
276
285
294
303
313
322
331
341
350
359
369
378
4
7
11
16
22
28
34
41
48
55
63
70
78
86
94
103
111
120
128
137
146
155
164
173
182
192
201
210
220
229
239
248
258
268
277
287
297
307
317
327
337
347
357
367
377
387
397
407
4
7
12
17
23
29
36
43
51
59
67
75
83
92
101
110
119
128
137
146
156
166
175
185
195
205
215
225
235
245
255
266
276
286
297
307
318
329
339
350
361
371
382
393
404
415
426
436
4
8
12
18
24
31
38
46
54
62
71
79
88
97
107
116
126
136
146
156
166
176
186
197
207
218
229
239
250
261
272
283
294
305
316
327
339
350
361
373
384
396
407
419
430
442
454
465
4
8
13
19
25
32
40
48
57
65
74
84
93
103
113
123
133
144
154
165
176
186
197
209
220
231
242
254
265
277
288
300
312
324
336
348
360
372
384
396
408
420
432
445
457
470
482
494
17
4
8
13
20
26
34
42
51
60
69
78
88
98
109
119
130
141
152
163
174
185
197
209
220
232
244
256
268
280
293
305
317
330
342
355
368
380
393
406
419
432
445
458
471
484
497
510
523
18
4
9
14
20
28
36
44
53
62
72
82
93
103
114
125
136
148
159
171
183
195
207
220
232
244
257
270
282
295
308
321
334
348
361
374
387
401
414
428
442
455
469
483
496
510
524
538
552
19
4
9
14
21
29
37
46
55
65
75
86
97
108
120
131
143
155
167
180
192
205
218
231
244
257
270
283
297
310
324
338
351
365
379
393
407
422
436
450
464
479
493
508
522
537
551
566
581
20
5
9
15
22
30
39
48
58
68
79
90
101
113
125
137
150
162
175
188
201
214
228
241
255
269
283
297
311
325
340
354
368
383
398
412
427
442
457
472
487
502
517
532
548
563
578
594
609
(continued)
13-17
-------
September 1993
Tabla 13-4. (continued)
Number of cell culture bottles
3 4 5 6 7 8 9 10 11 12 13
Highest
count
Is an upper outlier when the total count is less than
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
111
113
116
118
121
124
126
129
131
134
136
139
142
144
147
149
152
155
157
160
162
165
168
170
173
175
178
181
183
186
189
191
194
196
199
202
204
207
210
212
215
218
220
223
226
228
231
234
236
239
143
146
149
153
156
159
163
166
169
173
176
179
183
186
190
193
196
200
203
207
210
214
217
220
224
227
231
234
238
241
245
248
251
255
258
262
265
269
272
276
279
283
286
290
293
297
300
304
307
311
174
178
182
186
190
195
199
203
207
211
215
220
224
228
232
236
241
245
249
253
257
262
266
270
274
279
283
287
292
296
300
304
309
313
317
322
326
330
334
339
343
347
352
356
360
365
369
373
378
382
205
210
215
220
225
230
235
239
244
249
254
259
264
269
274
279
28-4
283
294
293
304
30.9
315
320
325
330
335
340
345
350
355
360
365
371
376
381
386
397
395
401
407
412
417
422
427
432
438
443
448
453
236
242
247
253
259
264
270
276
282
287
293
299
305
310
316
322
328
334
339
345
351
357
363
369
375
381
386
392
398
404
410
416
422
428
434
440
446
452
458
464
470
476
482
488
494
500
506
512
518
524
267
273
279
286
292
299
305
312
318
325
331
338
345
351
358
364
371
378
384
391
398
404
411
418
424
431
438
444
451
458
465
471
478
485
492
498
505
512
519
526
532
539
546
553
560
567
573
580
587
594
297
304
311
319
326
333
341
348
355
362
370
377
384
392
399
407
414
421
429
436
444
451
459
466
474
481
489
496
504
511
519
527
534
542
549
557
564
572
580
587
595
603
610
618
626
633
641
649
656
664
327
335
343
351
359
367
375
384
392
400
408
416
424
432
440
449
457
465
473
482
490
498
506
515
523
531
540
548
556
565
573
581
590
598
607
615
623
632
640
649
657
666
674
683
691
700
708
717
725
734
358
366
375
384
393
401
410
419
428
437
446
455
464
473
482
491
500
509
518
527
536
545
554
563
572
581
590
599
609
618
627
636
645
655
664
673
682
692
701
710
719
729
738
747
757
766
775
785
794
803
388
397
407
416
426
435
445
455
464
474
484
493
503
513
522
532
542
552
562
572
581
591
601
611
621
631
641
651
661
671
681
691
701
711
721
731
741
751
761
771
781
791
801
812
822
832
842
852
862
873
418
428
438
448
459
469
480
490
500
511
521
532
542
553
563
574
584
595
606
616
627
638
648
659
670
680
691
702
713
723
734
745
756
767
778
788
799
810
821
832
843
854
865
876
887
898
909
920
931
942
14
447
458
470
481
492
503
514
525
536
548
559
570
581
593
604
615
627
638
649
661
672
684
695
707
718
730
741
753
765
776
788
799
811
823
834
846
858
869
881
893
905
916
928
940
952
963
975
987
999
1011
15 16 17 18 19 20
477 507 536 566 595 625
489 519 550 580 610 640
501 532 563 594 625 656
513 545 576 608 640 672
524 557 590 622 655 687
536 570 603 637 670 703
548 583 617 651 685 719
560 595 630 665 700 734
572 608 644 679 715 750
584 621 657 694 730 766
596 634 671 708 745 782
608 646 684 722 760 798
620 659 698 737 775 814
632 672 712 751 791 830
645 685 725 766 806 846
657 698 739 780 821 862
669 711 753 795 836 878
681 724 767 809 852 894
693 737 780 824 867 910
705 750 794 838 882 926
718 763 808 853 898 943
730 776 822 868 913 959
742 789 836 882 929 975
754 802 850 897 944 991
767 815 863 912 960 1008
779 828 877 926 975 1024
791 841 891 941 991 1040
804 855 905 956 1006 1057
816 868 919 971 1022 1073
829 881 933 985 1037 1089
841 894 947 1000 1053 1106
853 907 961 1015 1069 1122
866 921 975 1030 1084 1139
878 934 989 1045 1100 1155
891 947 1004 1060 1116 1172
903 961 1018 1075 1132 1188
916 974 1032 1090 1147 1205
928 987 1046 1105 1163 1222
941 1001 1060 1120 1179 1238
953 1014 1074 1135 1195 1255
966 1027 1088 1150 1210 1271
979 1041 1103 1165 1226 1288
991 1054 1117 1180 1242 1305
1004 1068 1131 1195 1258 1321
1016 1081 1145 1210 1274 1338
1029 1094 1160 1225 1290 1355
1042 1108 1174 1240 1306 1372
1054 1121 1188 1255 1322 1388
1067 1135 1203 1270 1338 1405
1080 1148 1217 1285 1354 1422
13-18
-------
September 1993
Table 13-5. 0.01 Critical Values for a Single Lower Outlier from a Poisson (Random) Distribution
Number of cell culture bottles
3 4 5 6 7 8 9 10 11 12 13 14 15 16
Lowest
count
Is a lower outlier when the total count is greater than
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
14
19
25
29
34
38
43
47
51
55
59
63
67
71
75
79
83
86
90
94
98
101
105
109
112
116
120
123
127
131
134
138
141
145
149
152
156
159
163
166
170
173
177
180
184
187
191
194
198
201
205
20
29
36
42
49
55
61
66
72
78
83
89
94
99
105
110
115
120
125
130
136
141
146
151
156
161
166
170
175
180
185
190
195
200
205
209
214
219
224
228
233
238
243
247
252
257
262
266
271
276
280
27
38
47
56
64
71
79
86
94
101
108
114
121
128
135
141
148
155
161
167
174
180
187
193
199
206
212
218
224
230
237
243
249
255
261
267
273
279
285
291
297
303
309
315
321
327
333
339
345
351
357
35
48
59
69
79
88
98
106
115
124
132
141
149
157
165
173
181
189
197
205
213
220
228
236
243
251
259
266
274
281
288
296
303
311
318
325
333
340
347
355
362
369
376
384
391
398
405
412
419
427
434
42
57
71
83
94
106
116
127
137
147
157
167
177
187
196
205
215
224
233
243
252
261
270
279
288
297
306
314
323
332
341
349
358
367
376
384
393
401
410
418
427
435
444
452
461
469
478
486
494
503
511
50
67
83
97
110
123
135
147
159
171
183
194
205
216
227
238
249
259
270
281
291
302
312
322
333
343
353
363
373
383
393
403
413
423
433
443
453
463
473
482
492
502
512
521
531
541
551
560
570
579
589
57
77
95
111
126
140
155
168
182
195
208
221
233
246
258
271
283
295
307
319
331
343
354
366
378
389
401
412
424
435
446
458
469
480
491
503
514
525
536
547
558
569
580
591
602
613
624
635
645
656
667
65
88
107
125
142
158
174
189
204
219
234
248
262
276
290
304
317
331
344
357
371
384
397
410
423
436
449
461
474
487
499
512
525
537
550
562
575
587
599
612
624
636
648
661
673
685
697
709
721
733
746
73
98
119
139
158
176
193
210
227
243
259
275
291
306
321
337
352
367
381
396
411
425
440
454
468
483
497
511
525
539
553
567
581
594
608
622
636
649
663
677
690
704
717
731
744
758
771
784
798
811
824
81
108
132
154
174
194
213
232
250
268
285
303
320
337
353
370
386
403
419
435
451
467
483
498
514
530
545
560
576
591
606
622
637
652
667
682
697
712
727
742
757
771
786
801
816
830
845
859
874
889
903
89
119
145
168
191
212
233
253
273
292
311
330
349
367
385
403
421
439
456
474
491
509
526
543
560
577
594
610
627
644
660
677
693
710
726
742
759
775
791
807
823
839
855
871
887
903
919
935
951
967
982
97
130
157
183
207
230
253
275
296
317
337
358
378
398
417
437
456
475
494
513
532
550
569
587
606
624
642
660
678
696
714
732
750
767
785
803
820
838
855
873
890
907
925
942
959
976
993
1011
1028
1045
1062
105
140
170
198
224
249
273
296
319
342
364
386
407
429
450
470
491
512
532
552
573
593
612
632
652
671
691
710
730
749
768
787
806
825
844
863
882
901
920
938
957
976
994
1013
1031
1050
1068
1086
1105
1123
1141
114
151
183
212
240
267
293
318
342
367
390
414
437
459
482
504
526
548
570
592
613
635
656
677
698
719
740
761
781
802
822
843
863
884
904
924
944
964
984
1004
1024
1044
1064
1084
1103
1123
1143
1162
1182
1202
1221
17
122
162
196
227
257
285
313
340
366
391
417
442
466
490
514
538
562
585
608
631
654
677
700
722
745
767
789
811
833
855
877
899
920
942
963
985
1006
1028
1049
1070
1091
1113
1134
1155
1176
1197
1218
1239
1259
1280
1301
18
131
173
209
242
274
304
333
362
389
417
443
470
496
521
547
572
597
622
647
671
695
719
743
767
791
815
838
862
885
908
931
954
977
1000
1023
1046
1069
1091
1114
1136
1159
1181
1204
1226
1248
1271
1293
1315
1337
1359
1381
19
139
184
222
257
291
323
353
384
413
442
470
498
525
553
580
606
633
659
685
711
736
762
787
813
838
863
888
912
937
962
986
1010
1035
1059
1083
1107
1131
1155
1'179
1203
1227
1250
1274
1297
1321
1344
1368
1391
1415
1438
1461
20
148
195
235
273
308
341
374
406
437
467
497
526
555
584
612
640
668
696
723
751
778
805
831
858
884
911
937
963 '
989
1015
1041
1066
1092
1118
1143
1168
1194
1219
1244
1269
1294
1319
1344
1369
1394
1418
1443
1468
1492
1517
1541
13-19
-------
September 1993
Table 13-6. 0.01 Critical Values for a Pair of Upper Outlier from a Poisson (Random) Distribution
Number of cell culture bottles
5 6 7,8 9 10 11 12 13 14 15 16 17 18
Sumo/2
Highest
counts Are an upper outlier pair when the total count is less than
A
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
-
-
7
8
10
11
13
15
17
18
20
22
24
26
28
30
32
33
35
37
39
41
43
45
47
49
52
54
56
58
60
62
64
66
68
70
72
74
77
79
81
83
85
87
89
91
94
-
-
7
9
11
12
14
16
18
20
22
25
27
29
31
33
36
38
40
42
45
47
49
52
54
57
59
61
64
66
69
71
74
76
78
81
83
86
88
91
93
96
98
101
103
106
108
-
6
7
9
11
13
15
18
20
22
25
27
30
32
35
37
40
42
45
47
50
53
55
58
61
64
66
69
72
75
77
80
83
86
89
91
94
97
100
103
106
109
111
114
117
120
123
-
6
8
10
12
14
17
19
22
24
27
30
32
35
38
41
44
46
49
52
55
58
61
64
67
70
73
77
80
83
86
89
92
95
99
102
105
108
111
115
118
121
124
128
131
134
137
-
6
8
10
13
15
18
20
23
26
29
32
35
38
41
44
47
51
54
57
60
64
67
70
74
77
81
84
87
91
94
98
101
105
108
112
116
119
123
126
130
133
137
141
144
148
152
-
6
8
11
13
16
19
22
25
28
31
34
38
41
44
48
51
55
58
62
65
69
73
76
80
84
88
91
95
99
103
107
110
114
118
122
126
130
134
138
142
146
150
154
158
162
166
-
6
9
11
14
17
20
23
26
30
33
37
40
44
47
51
55
59
63
66
70
74
78
82
86
90
94
99
103
107
111
115
119
124
128
132
136
141
145
149
154
158
162
167
171
175
180
5
7
9
12
15
18
21
24
28
31
35
39
43
47
51
55
59
63
67
71
75
79
84
88
92
97
101
106
110
115
119
124
128
133
137
142
147
151
156
161
165
170
175
179
184
189
194
5
7
9
12
15
19
22
26
29
33
37
41
45
49
54
58
62
67
71
76
80
85
89
94
99
103
108
113
118
122
127
132
137
142
147
152
157
162
167
172
177
182
187
192
197
202
208
5
7
10
13
16
19
23
27
31
35
39
43
48
52
57
61
66
70
75
80
85
90
95
100
105
110
115
120
125
130
135
141
146
151
156
162
167
172
178
183
188
194
199
205
210
216
221
5
7
10
13
17
20
24
28
32
37
41
45
50
55
60
64
69
74
79
84
90
95
100
105
111
116
121
127
132
138
143
149
154
160
166
171
177
183
188
194
200
206
211
217
223
229
235
5
7
10
14
17
21
25
29
34
38
43
48
53
57
63
68
73
78
83
89
94
100
105
111
117
122
128
134
140
145
151
157
163
169
175
181
187
193
199
205
211
217
224
230
236
242
248
5
7
11
14
18
22
26
30
35
40
45
50
55
60
65
71
76
82
88
93
99
105
111
117
122
128
135
141
147
153
159
165
172
178
184
190
197
203
210
216
223
229
236
242
249
255
262
5
a
11
14
18
23
27
32
36
41
47
52
57
63
68
74
80
86
92
98
104
110
116
122
128
135
141
147
154
160
167
173
180
187
193
200
207
213
220
227
234
241
248
254
261
268
275
19
5
8
11
15
19
23
28
33
38
43
48
54
60
65
71
77
83
89
96
102
108
115
121
128
134
141
147
154
161
168
175
182
188
195
202
209
217
224
231
238
245
252
259
267
274
281
289
20
5
8
11
15
19
24
29
34
39
45
50
56
62
68
74
80
87
93
100
106
113
119
126
133
140
147
154
161
168
175
182
190
197
204
212
219
226
234
241
249
256
264
271
279
287
294
302
(continued)
13-20
-------
September 1993
Table 13-6. (continued)
Number of cell culture bottles
5 6 7 8 9 10 11 12 13 14
Sum of 2
Highest
counts Are an upper outlier pair when the total count is less than
51 96
52 98
53 100
54 102
55 104
56 107
57 109
58 111
59 113
60 115
61 117
62 120
63 122
64 124
65 126
66 128
67 131
68 133
69 135
70 137
71 139
72 142
73 144
74 146
75 148
111
113
116
119
121
124
126
129
131
134
137
139
142
144
147
149
152
155
157
160
162
165
168
170
173
126
129
132
135
138
141
144
146
149
152
155
158
161
164
167
170
173
176
179
182
185
188
191
194
197
141
144
147
151
154
157
161
164
167
171
174
177
181
184
187
191
194
198
201
204
208
211
215
218
221
155
159
163
166
170
174
178
181
185
189
192
196
200
204
207
211
215
219
223
226
230
234
238
242
245
170
174
178
182
186
190
194
198
203
207
211
215
219
223
227
231
236
240
244
248
252
257
261
265
269
184
189
193
198
202
206
211
215
220
224
229
233
238
242
247
252
256
261
265
270
274
279
284
288
293
199
203
208
213
218
223
227
232
237
242
247
252
257
262
267
272
276
281
286
291
296
301
306
311
316
213
218
223
228
233
239
244
249
254
260
265
270
275
281
286
291
297
302
307
313
318
323
329
334
340
227
232
238
243
249
255
260
266
271
277
283
288
294
300
305
311
317
322
328
334
340
345
351
357
363
15
241
247
253
258
264
270
276
282
288
294
300
306
312
319
325
331
337
343
349
355
361
367
374
380
386
16
255
261
267
273
280
286
292
299
305
312
318
324
331
337
344
350
357
363
370
376
383
389
396
402
409
17
268
275
282
288
295
302
309
315
322
329
336
342
349
356
363
370
376
383
390
397
404
411
418
425
432
18
282
289
296
303
310
317
324
332
339
346
353
360
367
375
382
389
396
403
411
418
425
433
440
447
455
19
296
303
311
318
325
333
340
348
355
363
370
378
385
393
401
408
416
423
431
439
446
454
462
470
477
20
310
317
325
333
341
348
356
364
372
380
388
396
404
411
419
427
435
443
451
459
468
476
484
492
500
(continued)
13-21
-------
September 1993
Table 13-6. (continued)
Number of cell culture bottles
5 6 7 8 9 10 11 12 13 14
Sum of 2
Highest
counts Are an upper outlier pair when the total count is less than
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
150
153
155
157
159
161
164
166
168
170
173
175
177
179
182
184
186
188
191
193
195
197
200
202
204
176
178
181
183
186
189
191
194
197
199
202
204
207
210
212
215
218
2SO
223
226
228
231
234
236
239
200
203
206
209
212
215
219
222
225
228
231
234
237
240
243
246
249
252
255
258
261
264
267
271
274
225
228
232
235
239
242
245
249
252
256
259
263
266
270
273
277
280
284
287
291
294
298
301
304
308
249
253
257
261
265
268
272
276
280
284
288
292
295
299
303
307
311
315
319
323
326
330
334
338
342
273
278
282
286
290
295
299
303
307
312
316
320
324
329
333
337
341
346
350
354
359
363
367
372
376
297
302
307
311
316
321
325
330
334
339
344
348
353
358
363
367
372
377
381
386
391
395
400
405
410
321
326
331
336
341
346
351
356
361
367
372
377
382
387
392
397
402
407
412
417
423
428
433
438
443
345
350
356
361
367
372
377
383
388
394
399
405
410
416
421
427
432
438
443
449
454
460
465
471
476
369
374
380
386
392
398
403
409
415
421
427
433
439
444
450
456
462
468
474
480
486
492
498
504
510
15
392
398
404
411
417
423
429
435
442
448
454
461
467
473
479
486
492
498
505
511
517
524
530
536
543
16
415
422
429
435
442
448
455
462
468
475
482
488
495
502
508
515
522
528
535
542
548
555
562
569
575
17
439
446,
453
460
467
474
481
488
495
502
509
516
523
530
537
544
551
558
565
573
580
587
594
601
608
18
462
469
477
484
491
499
506
514
521
528
536
543
551
558
566
573
581
588
596
603
611
618
626
633
641
19
485
493
500
508
516
524
532
539
547
555
563
571
579
586
594
602
610
618
626
634
642
650
658
666
673
20
508
516
524
532
541
549
557
565
573
582
590
598
606
615
623
631
639
648
656
664
673
681
689
698
706
13-22
-------
September 1993
Table 13-7. 0.01 Critical Values fora Pair of Lower Outliers from a Poisson (Random)
Number of cell culture bottles
5 6 7 8 9 10 11 12 13 14 15 16
Sum of 2
Lowest
counts Are a lower outlier pair when the total count is greater than
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
10
15
19
23
26
30
33
37
40
43
47
50
53
56
59
63
66
69
72
75
78
81
84
87
90
93
96
99
102
105
108
111
114
117
120
123
125
128
131
134
137
140
143
146
149
151
154
157
160
163
166
14
19
25
29
34
38
43
47
51
55
59
63
67
71
75
79
83
86
90
94
98
101
105
109
112
116
120
123
127
131
134
138
141
145
148
152
156
159
163
166
170
173
177
180
184
187
191
194
198
201
205
17
25
31
36
42
47
52
57
62
67
72
77
81
86
91
95
100
104
109
113
118
122
126
131
135
139
144
148
152
157
161
165
169
173
178
182
186
190
194
199
203
207
211
215
219
223
228
232
236
240
244
21
30
37
44
50
56
62
68
74
80
85
91
96
101
107
112
117
122
128
133
138
143
148
153
158
163
168
173
178
183
188
193
198
202
207
212
217
222
227
231
236
241
246
250
255
260
265
269
274
279
284
26
35
44
51
59
66
72
79
86
92
98
105
111
117
123
129
135
141
147
153
158
164
170
176
181
187
193
198
204
209
215
221
226
232
237
243
248
254
259
264
270
275
281
286
292
297
302
308
313
318
324
30
41
50
59
67
75
83
90
98
105
112
119
126
133
139
146
153
159
166
173
179
186
192
198
205
211
217
224
230
236
243
249
255
261
267
273
280
286
292
298
304
310
316
322
328
334
340
346
352
358
364
34
46
57
67
76
85
93
101
110
118
125
133
141
149
156
164
171
178
186
193
200
207
214
221
228
236
243
249
256
263
270
277
284
291
298
304
311
318
325
331
338
345
352
358
365
372
378
385
391
398
405
39
52
64
74
85
94
104
113
122
131
139
148
156
165
173
181
189
197
205
213
221
229
237
245
252
260
268
275
283
291
298
306
313
321
328
336
343
351
358
365
373
380
387
395
402
409
417
424
431
438
445
43
58
71
82
93
104
114
124
134
144
153
162
172
181
190
199
208
216
225
234
242
251
260
268
276
285
293
302
310
318
326
335
343
351
359
367
375
383
391
399
407
415
423
431
439
447
455
463
471
479
487
48
64
78
90
102
114
125
136
147
157
167
177
187
197
207
217
226
236
245
255
264
273
282
292
301
310
319
328
337
346
355
363
372
381
390
399
407
416
425
434
442
451
459
468
477
485
494
502
511
519
528
53
70
85
99
112
124
136
148
159
170
181
192
203
214
224
235
245
255
265
275
285
295
305
315
325
335
344
354
364
373
383
393
402
412
421
430
440
449
459
468
477
487
496
505
514
523
533
542
551
560
569
58
76
92
107
121
134
147
159
172
184
196
207
219
230
241
253
264
275
286
296
307
318
328
339
349
360
370
381
391
401
412
422
432
442
452
462
472
482
492
502
512
522
532
542
552
562
572
581
591
601
611
Distribution
17 18
62
82
99
115
130
144
158
171
184
197
210
222
235
247
259
271
283
294
306
317
329
340
352
363
374
385
396
407
418
429
440
451
462
473
484
494
505
516
526
537
548
558
569
579
590
600
611
621
632
642
653
67
89
107
124
139
154
169
183
197
211
224
238
251
264
276
289
302
314
326
339
351
363
375
387
399
411
422
434
446
458
469
481
492
504
515
527
538
549
561
572
583
594
606
617
628
639
650
661
672
683
694
19
72
95
114
132
149
165
180
195
210
225
239
253
267
280
294
307
321
334
347
360
373
386
398
411
424
436
449
461
473
486
498
510
523
535
547
559
571
583
595
607
619
631
643
654
666
678
690
701
713
725
737
20
77
101
122
141
158
175
192
208
223
238
254
268
283
297
312
326
340
354
368
381
395
408
422
435
449
462
475
488
501
514
527
540
553
566
579
591
604
617
629
642
655
667
680
692
704
717
729
742
754
766
779
13-23
-------
September 1993
Table 13-8. 0.01 Critical Values for the Test for a Single Upper ("Tn") or Lower ("T,") Outlier from
a Normal Distribution
Number of
cett culture
bottles
3
4
5
6
7
8
&
10
11
Critical
T value
1.155
1.492
1.749
1.944
2.097
2.221
2.323
2.410
2.484
Number of
cell culture
bottles
17
18
19
20
21
22
23
24
25
Critical
T value
2.785
2.821
2.854
2.884
2.912
2.939
2.963
2.987
3.009
Table 13-9. 0.01 Critical Values for the Test for an Upper ("Tn_tJ,") or Lower ("T12") Outlier Pair
from a Normal Distribution
Number of
cell culture
bottles
4
5
15
7
8
9
10
11
12
13
14
15
W
17
Critical
T value
0.0000
0.0035
0.0186
0.0440
0.0750
0.1082
0.1414
0.1736
0.2043
0.2333
0.2605
0.2859
0.3098
0.3321
Number of
cell culture
bottles
18
19
20
21
22
23
24
25
26
27
28
29
30
Critical
T value
0.3530
0.3725
0.3909
0.4082
0.4245
0.4398
0.4543
0.4680
0.4810
0.4933
0.5050
0.5162
0.5268
Upper or lower outlier pairs are statistically significant outliers if Tn_,,n or T, r
is less than the tabulated values
13-24
-------
September 1993
Table13-10
DF* 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
32
34
36
38
40
42
44
46
48
50
55
60
65
70
80
100
150
200
250
300
350
400
450
500
1000
00
*nc
G4U
38.51
17.44
12.22
10.01
8.813
8.073
7.571
7.209
6.937
6.724
6.554
6.414
6.298
6.200
6.115
6.042
' 5.978
5.922
5.871
5.827
5.786
5.750
5.717
5.686
5.659
5.633
5.610
5.588
5.568
5.531
5.499
5.471
5.446
5.424
5.404
5.386
5.369
5.354
5.340
5.310
5.286
5.265
5.247
5.218
5.179
5.126
5.100
5.085
5.075
5.067
5.062
5.058
5.054
5.039
5.024
0.05 Two-Tailed Critical Values for the Cumulative F Distribution
Numerator degrees of freedom
2 5 10 15 20 25 30 40 50
800 322 '§69 985 593
39.00 39.30 39.40 39.43 39.45
16.04 14.88 14.42 14.25 14.17
10.65 9.364 8.844 .8.657 8.560
8.434 7.146 6.619 6.428 6.329
7.260 5.988 5.461 5.269 5.168
6.542 5.285 4.761 4.568 4.467
6.059 4.817 4.295 4.101 3.999
5.715 4.484 3.964 3.769 3.667
5.456 4.236 3.717 3.522 3.419
5.256 4.044 3.526 3.330 3.226
5.096 3.891 3.374 3.177 3.073
4.965 3.767 3.250 3.053 2.948
4.857 3.663 3.147 2.949 2.844
4.765 3.576 3.060 2.862 2.756
4.687 3.502 2.986 2.788 2.681
4.619 3.438 2.922 2.723 2.616
4.560 3.382 2.866 2.667 2.559
4.508 3.333 2.817 2.617 2.509
4.461 3.289 2.774 2.573 2.464
4.420 3.250 2.735 2.534 2.425
4.383 3.215 2.700 2.498 2.389
4.349 3.183 2.668 2.466. 2.357
4.319 3.155 2.640 2.437 2.327
4.291 3.129 2.613 2.411 2.300
4.265 3.105 2.590 2.387 2.276
4.242 3.083 2.568 2.364 2.253
4.221 3.063 2.547 2.344 2.232
4.201 3.044 2.529 2.325 2.213
4.182 3.026 2.511 2.307 2.195
4.149 2.995 2.480 2.275 2.163
4.120 2.968 2.453 2.248 2.135
4.094 2.944 2.429 2.223 2.110
4.071 2.923 2.407 2.201 2.088
4.051 2.904 2.388 2.182 2.068
4.033 2.887 2.371 2.164 2.050
4.016 2.871 2.356 2.149 2.034
4.001 2.857 2.341 2.134 2.019
3.987 2.844 2.329 2.121 2.006
3.975 2.833 2.317 2.109 1.993
3.948 2.807 2.291 2.083 1.967
3.925 2.786 2.270 2.061 1.944
3.906 2.769 2.252 2.043 1.926
3.890 2.754 2.237 2.028 1.910
3.864 2.730 2.213 2.003 1.884
3.828 2.696 2.179 1.968 1.849
3.781 2.652 2.135 1.922 1.801
3.758 2.630 2.113 1.900 1.778
3.744 2.618 2.100 1.886 1.764
3.735 2.609 2.091 1.877 1.755
3.728 2.603 2.085 1.871 1.748
3.723 2.598 2.080 1.866 1.743
3.719 2.595 2.077 1.862 1.739
3.716 2.592 2.074 1.859 1.736
3.703 2.579 2.061 1.846 1.722
3.689 2.567 2.048 1.833 1.708
998
39.46
14.12
8.501
6.268
5.107
4.405
3.937
3.604
3.355
3.162
3.008
2.882
2.778
2.689
2.614
2.548
2.491
2.441
2.396
2.356
2.320
2.287
2.257
2.230
2.205
2.183
2.161
2.142
2.124
2.091
2.062
2.037
2.015
1.994
1.976
1.960
1.945
1.931
1.919
1.891
1.869
1.850
1.833
1.807
1.770
1.722
1.698
1.683
1.674
1.667
1.662
1.658
1.655
1.640
1.626
1001
39.46
14.08
8.461
6.227
5.065
4.362
3.894
3.560
3.311
3.118
2.963
2.837
2.732
2.644
2.568
2.502
2.445
2.394
2.349
2.308
2.272
2.239
2.209
2.182
2.157
2.133
2.112
2.092
2.074
2.041
2.012
1.986
1.963
1.943
1.924
1.908
1.893
1.879
1.866
1.838
1.815
1.796
1.779
1.752
1.715
1.665
1.640
1.625
1.616
1.608
1.603
1.599
1.596
1.581
1.566
1006
39.47
14.04
8.411
6.175
5.012
4.309
3.840
3.505
3.255
3.061
2.906
2.780
2.674
2.585
2.509
2.442
2.384
2.333
2.287
2.246
2.210
2.176
2.146
2.118
2.093
2.069
2.048
2.028
2.009
1.975
1.946
1.919
1.896
1.875
1.856
1.839
1.824
1.809
1.796
1.768
1.744
1.724
1.707
1.679
1.640
1.588
1.562
1.546
1.536
1.529
1.523
1.519
1.515
1.499
1.484
1008
39.48
14.01
8.381
6.144
4.980
4.276
3.807
3.472
3.221
3.027
2.871
2.744
2.638
2.549
2.472
2.405
2.347
2.295
2.249
2.208
2.171
2.137
2.107
2.079
2.053
2.029
2.007
1.987
1.968
1.934
1.904
1.877
1.854
1.832
1.813
1.796
1.780
1.765
1.752
1.723
1.699
1.678
1.660
1.632
1.592
1.538
1.511
1.495
1.484
1.476
1.47Q
1.465
1.462
1.445
1.428
100
1013
39.49
13.96
8.319
6.080
4.915
4.210
3.739
3.403
3.152
2.956
2.800
2.671
2.565
2.474
2.396
2.329
2.269
2.217
2.170
2.128
2.090
2.056
2.024
1.996
1.969
1.945
1.922
1.901
1.882
1.846
1.815
1.787
1.763
1.741
1.720
1.702
1.685
1.670
1.656
1.625
1.599
1.577
1.558
1.527
1.483
1.423
1.393
1.374
1.361
1.352
1.345
1.340
1.336
1.316
1.296
200
1016
39.49
13.93
8.289
6.048
4.882
4.176
3.705
3.368
3.116
2.920
2.763
2.634
2.526
2.435
2.357
2.289
2.229
2.176
2.128
2.086
2.047
2.013
1.981
1.952
1.925
1.900
1.877
1.855
1.835
1.799
1.767
1.739
1.713
1.691
1.670
1.651
1.634
1.618
1.603
1.571
1.543
1.520
1.500
1.467
1.420
1.355
1.320
1.299
1.285
1.274
1.266
1.259
1.254
1.230
1.205
500
1017
39.50
13.91
8.270
6.028
4.862
4.156
3.684
3.347
3.094
2.898
2.740
2.611
2.503
2.411
2.333
2.264
2.204
2.150
2.103
2.060
2.021
1.986
1.954
1.924
1.897
1.872
1.848
1.827
1.806
1.770
1.737
1.708
1.682
1.659
1.638
1.618
1.600
1.584
1.569
1.536
1.507
1.483
1.463
1.428
1.378
1.307
1.269
1.245
1.228
1.215
1.206
1.198
1.192
1.162
1.128
1000
1018
39.50
13.91
8.264
6.022
4.856
4.149
3.677
3.340
3.087
2.890
2.733
2.603
2.495
2.403
2.324
2.256
2.195
2.142
2.094
2.051
2.012
1.977
1.945
1.915
1.888
1.862
1.839
1.817
1.797
1.760
1.727
1.698
1.672
1.648
1.627
1.607
1.589
1.573
1.557
1.523
1.495
1.471
1.449
1.414
1.363
1.290
1.250
1.224
1.206
1.192
1.182
1.173
1.166
1.132
1.090
oo
7078
39.50
13.90
8.257
6.015
4.849
4.142
3.670
3.333
3.080
2.883
2.725
2.595
2.487
2.395
2.316
2.247
2.187
2.133
2.085
2.042
2.003
1.968
1.935
1.906
1.878
1.853
1.829
1.807
1.787
1.750
1.717
1.687
1.661
1.637
1.615
1.596
1.578
1.561
1.545
1.511
1.482
1.457
1.436
1.400
1.347
1.271
1.229
1.201
1.182
1.166
1. 154
1.145
1.137
1.094
1.000
13-25
-------
September 1993
Tabla 13-11.
Lower M
Count
1
2
3
4
5
6
7
a
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
7
9
10
12
14
15
17
18
20
21
23
24
26
27
28
30
31
32
34
35
36
37
39
40
41
43
44
45
46
48
49
SO
51
53
54
55
56
58
59
60
61
62
64
65
66
67
68
70
71
72
0.05 Critical Values for the Larger of Two Poisson Counts (M) Based on
R.(M+1/2)/(N+1/2),M>N
Lower M Lower M Lower M Lower M
Count Count Count Count
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
73
74
76-
77
78
79
80
82
83
84
85
86
87
89
90
91
92
93
94
96
97
98
99
100
101
103
104
105
106
107
108
110
111
112
113
114
115
116
118
119
120
121
122
123
124
126
127
128
129
130
W1
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
131
132
134
135
136
137
138
139
140
142
143
144
145
146
147
148
149
151
152
153
154
155
156
157
158
160
161
162
163
164
165
166
167
169
170
171
172
173
174
175
176
177
179
180
181
182
183
184
185
186
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
188
189
190
191
192
193
194
195
196
198
199
200
201
202
203
204
205
206
207
209
210
211
212
213
214
215
216
217
219
220
221
222
223
224
225
226-
227
228
230
231
232
233
234
235
236
237
238
239
241
242
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
243
244
245
246
247
248
249
250
252
253
254
255
256
257
258
259
260
261
262
264
265
266
267
268
269
270
271
272
273
274
276
277
278
279
280
281
282
283
284
285
286
288
289
290
291
292
293
294
295
296
the Ratio
Lower
Count
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
Test,
M
297
298
300
301
302
303
304
305
306
307
308
309
310
311
313
314
315
316
317
318
319
320
321
322
323
324
326
327
328
329
330
331
332
333
334
335
336
337
339
340
341
342
343
344
345
346
347
348
349
350
Lower
Count
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
M
352
353
354
355
356
357
358
359
360
361
362
363
364
366
367
368
369
370
371
372
373
374
375
376
377
378
380
381
382
383
384
385
386
387
388
389
390 '
391
392
394
395
396
397
398
399
400
401
402
403
404
13-26
-------
September 1993
Table 13-12. Lower and Upper 95% Confidence Limits for the Mean of a Poisson Variate
Lower Upper Lower Upper Lower Upper Lower Upper
Count Limit Limit Count Limit Limit Count Limit Limit Count Limit Limit
jj ~ * - - - -
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
u 3.o«y
.0253 5.572
.2422 7.225
.6187 8.767
1.090 10.24
1.623 11.67
2.202 13.06
2.814 14.42
3.454 15.76
4.115 17.08
4.795 18.39
5.491 19.68
6.201 20.96
6.922 22.23
7.654 23.49
8.395 24.74
9.145 25.98
9.903 27.22
10.67 28.45
11.44 29.67
12.22 30.89
13.00 32.10
13.79 33.31
14.58 34.51
15.38 35.71
16.18 36.90
16.98 38.10
17.79 39.28
18.61 40.47
19.42 41.65
20.24 42.83
21.06 44.00
21.89 45.17
22.72 46.34
23.55 47.51
24.38 48.68
25.21 49.84
26.05 51.00
26.89 52.16
27.73 53.31
28.58 54.47
29.42 55.62
30.27 56.77
31.12 57.92
31.97 59.07
32.82 60.21
33.68 61.36
34.53 62.50
35.39 63.64
36.25 64.78
bU
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
J/. 1 1 65.92
37.97 67.06
38.84 68.19
39.70 69.33
40.57 70.46
41.43 71.59
42.30 72.72
43.17 73.85
44.04 74.98
44.91 76.11
45.79 77.23
46.66 78.36
47.54 79.48
48.41 80.60
49.29 81.73
50.17 82.85
5'1.04 83.97
51.92 85.09
52.80 86.21
53.69 87.32
54.57 88.44
55.45 89.56
56.34 90.67
57.22 91.79
58.11 92.90
58.99 94.01
59.88 95.13
60.77 96.24
61.66 97.35
62.55 98.46
63.44 99.57
64.33 100.7
65.22 101.8
66.11 102.9
67.00 104.0
67.89 105. 1
68.79 106.2
69.68 107.3
70.58 108.4
71.47 109.5
72.37 110.6
73.27 111.7
74.16 112.8
75.06 1 13.9
75.96 115.0
76.86 116.1
77.76 117.2
78.66 118.3
79.56 119.4
80.46 120.5
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
81.36 121.6
82.27 122.7
83.17 123.8
84.07 124.9
84.98 126.0
85.88 127.1
86.78 128.2
87.69 129.3
88.59 130.4
89.50 131.5
90.41 132:6
91.31 133.7
92.22 134.8
93.13 135.9
94.04 136.9
94.94 138.0
95.85 139.1
96.76 140.2
97.67 141.3
98.58 142.4
99.49 143.5
100.4 144.6
101.3 145.7
102.2 146.8
103.1 147.8
104.0 148.9
105.0 150.0
105.9 151. 1
106.8 152.2
107.7 153.3
108.6 154.4
109.5 155.4
110.4 156.5
111.4 157.6
112.3 158.7
113.2 159.8
114.1 160.9
1 15.0 162.0
115.9 163.0
116.9 154.1
117.8 165.2
118.7 166.3
119.6 167.4
120.5 168.5
121.4 169.5
122.4 170.6
123.3 171.7
124.2 172.8
125.1 173.9
126.0 174.9
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
127.0 176.0
127.9 177.1
128.8 178.2
129.7 179.3
130.6 180.3
131.6 181.4
132.5 182.5
133.4 183.6
134.3 184.6
135.2 185.7
136.2 186.8
137.1 187.9
138.0 189.0
138.9 190.0
139.9 191.1
140.8 192.2
141.7 193.3
142.6 194.3
143.6 195.4
144.5 196.5
145.4 197.6
146.3 198.6
147.3 199.7
148.2 200.8
149.1 201.9
150.0 202.9
151.0 204.0
151.9 205.1
152.8 206.2
153.7 207.2
154.7 208.3
155.6 209.4
156.5 210.4
157.4 211.5
158.4 212.6
159.3 213.7
160.2 214.7
161.2 215.8
162.1 216.9
163.0 218.0
163.9 219.0
164.9 220.1
165.8 221.2
166.7 222.2
167.7 223.3
168.6 224.4
169.5 225.4
170.4 226.5
171.4 227.6
172.3 228.7
Lower Upper
Count Limit Limit
~~2W
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
173.2 229.7
174.2 230.8
175.1 231.9
176.0 232.9
177.0 234.0
177.9 235.1
178.8 236.1
179.8 237.2
180.7 238.3
181.6 239.3
182.6 240.4
183.5 241.5
184.4 242.5
185.4 243.6
186.3 244.7
187.2 245.7
188.2 246.8
189. 1 247.9
190.0 248.9
191.0 250.0
191.9 251.1
192.8 252.1
193.8 253.2
194.7 254.3
195.6 255.3
196.6 256.4
197.5 257.5
198.4 258.5
199.4 259.6
200.3 260.7
201.2 261.7
202.2 262.8
203. 1 263.9
204.0 264.9
205.0 266.0
205.9 267.0
206.8 268.1
207.8 269.2
208.7 270.2
209.7 271.3
210.6 272.4
211.5 273.4
212.5 274.5
213.4 275.6
214.3 276.6
215.3 277.7
216.2 278.7
217.2 279.8
218.1 280.9
219.0 281.9
Lower Upper
Count Limit Limit
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
220.0 283.0
220.9 284.0
221.8 285.1
222.8 286.2
223.7 287.2
224.7 288.3
225.6 289.4
226.5 290.4
227.5 291.5
228.4 292.5
229.4 293.6
230.3 294.7
231.2 295.7
232.2 296.8
233.1 297.8
234.1 298.9
235.0 300.0
235.9 301.0
236.9 302.1
237.8 303.1
238.8 304.2
239.7 305.3
240.6 306.3
241.6 307.4
242.5 308.4
243.5 309.5
244.4 310.6
245.3 311.6
246.3 312.7
247.2 313.7
248.2 314.8
249.1 315.9
250.0 316.9
251.0 318.0
251.9 319.0
252.9 320.1
253.8 321.1
254.8 322.2
255.7 323.3
256.6 324.3
257.6 325.4
258.5 326.4
259.5 327.5
260.4 328.5
261.4 329.6
262.3 330.7
263.2 331.7
264.2 332.8
265. 1 333.8
266. 1 334.9
13-27
-------
September 1993
Tsbls 13-13, Lower and Upper Simultaneous 95% Confidence Limits for the Means of Several Poisson Variates
# Groups-2
Lower Upper Lower Upper Lower Upper Lower Upper Lower Upper Lower Upper
Count Limit Limit Count Umii Limit Count Limit Limit Count Limit Limit Count Limit Limit Count Limit Limit
0 0 4.382 50 35.51 68.33
1 .0/26 6.381 51 36.36' 69.49
2 .1671 8.122 52 37.19 70.64
3 .4741 9.739 53 38.04 71.79
4 .8801 1128 54 38.88 72.95
5 1.354 12.77 55 39.73 74.10
6 1.876 1431 56 40.58 7525
7 2.437 15.63 57 41.43 76.39
8 3.027 17.01 58 4229 77.54
9 3.642 18.38 59 43.14 78.69
10 4278 19.73 60 43.99 79.83
11 4.932 21.06 61 44.85 80.97
12 5.601 22.38 62 45.71 82.11
13 6283 23.69 63 46.55 8326
14 6.978 24.98 64 47.42 84.40
15 7.683 2627 65 4828 85.53
16 8.398 27.55 66 49.15 86.67
17 9.122 28.82 67 50.01 87.81
18 9.853 30.08 68 50.87 88.94
19 10.59 31.33 69 51.74 90.08
20 11.34 32.58 70 52.60 9121
21 12.09 33.82 71 53.47 92.34
22 12.85 35.06 72 54.34 93.48
23 13.61 3629 73 5520 94.61
24 14.38 37.52 74 56.07 95.74
25 15.15 38.74 75 56.94 96.87
26 15.93 39.96 76 57.81 97.99
27 16.71 41.17 77 58.69 99.12
28 17.50 42.39 78 59.56 100.2
29 1829 43.59 79 60.43 101.4
30 19.08 44.80 80 61.31 102.5
31 19.88 46.00 81 62.18 103.6
32 20.68 47.19 82 63.06 104.7
33 21.48 48.39 83 63.93 105.9
34 2229 49.58 84 64.81 107.0
35 23.10 50.77 85 65.69 108.1
36 23.91 51.95 86 66.57 1092
37 24.72 53.14 87 67.45 110.4
38 25.54 54.32 88 68.33 111.5
39 26.36 55.50 89 6921 112.6
40 27.18 56.67 90 70.09 113.7
41 28.01 57.85 91 70.97 114.8
42 28.83 59.02 92 71.85 115.9
43 29.66 60.19 93 72.74 117.1
44 30.49 61.36 94 73.62 1182
45 31.32 62.52 95 74.51 119.3
46 32.15 63.69 96 75.39 120.4
47 32.99 64.85 97 7628 121.5
48 33.83 66.01 98 77.16 122.6
49 34.67 67.17 99 78.05 123.7
100 78.94 124.9 150
101 79.83 126.0 151
102 80.72 127.1 152
103 81.60 128.2 153
104 82.49 129.3 154
105 83.38 130.4 155
106 8428 131.5 156
107 85.17 132.6 157
108 86.06 133.7 158
109 86.95 134.8 159
110 87.84 135.9 160
111 88.74 137.0 161
112 89.63 1382 162
113 90.53 139.3 163
114 91.42 140.4 164
115 92.32 141.5 165
116 93.21 142.6 166
117 94.11 143.7 167
118 95.00 144.8 168
119 95.90 145.9 169
120 96.80 147.0 170
121 97.70 148.1 171
122 98.59 149.2 172
123 99.49 150.3 173
124 100.4 151.4 174
125 101.3 152.5 175
126 102.2 153.6 176
127 103.1 154.7 177
128 104.0 155.8 178
129 104.9 156.9 179
130 105.8 158.0 180
131 106.7 159.1 181
132 107.6 160.2 182
133 108.5 161.3 183
134 109.4 162.4 184
135 110.3 163.5 185
136 1112 164.6 186
137 112.1 165.7 187
138 113.0 166.8 188
139 113.9 167.9 189
140 114.8 168.9 190
141 115.7 170.0 191
142 116.6 171.1 192
143 117.5 1722 193
144 118.5 173.3 194
145 119.4 174.4 195
146 120.3 175.5 196
147 1212 176.6 197
148 122.1 177.7 198
149 123.0 178.8 199
123.9 179.9
124.8 181.0
125.7 182.1
126.6 183.1
127.5 184.2
128.4 185.3
186.4
187.5
188.6
189.7
190.8
191.9
129.4
130.3
1312
132.1
133.0
133.9
134.8 192.9
135.7 194.0
136.6 195.1
137.6 1962
138.5 197.3
139.4 198.4
140.3 199.5
1412 200.6
142.1 201.6
143.0 202.7
144.0 203.8
144.9 204.9
145.8 206.0
146.7 207.1
147.6 2082
148.5 209.2
149.4 210.3
150.4 211.4
151.3 212.5
1522 213.6
153.1 214.7
154.0 215.7
154.9 216.8
155.9 217.9
156.8 219.0
157.7 220.1
158.6 221.1
159.5 222.2
160.5 223.3
161.4 224.4
162.3 225.5
163.2 226.6
164.1 227.6
165.0 228.7
166.0 229.8
166.9 230.9
167.8 232.0
168.7 233.0
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
169.7 234.1
170.6 235.2
171.5 236.3
172.4 237.3
173.3 238.4
174.3 239.5
175.2 240.6
176.1 241.7
177.0 242.7
177.9 243.8
178.9 244.9
179.8 246.0
180.7 247.0
181.6 248.1
182.6 249.2
183.5 250.3
184.4 251.4
185.3 252.4
186.3 253.5
187.2 254.6
188.1 255.7
189.0 256.7
190.0 257.8
190.9 258.9
191.8 260.0
192.7 261.0
193.7 262.1
194.6 263.2
195.5 264.3
196.4 265.3
197.4 266.4
198.3 267.5
199.2 268.5
200.1 269.6
201.1 270.7
202.0 271.8
202.9 272.8
203.8 273.9
204.8 275.0
205.7 276.1
206.6 277.1
207.6 2782
208.5 279.3
209.4 280.3
210.3 281.4
211.3 282.5
2122 283.6
213.1 284.6
214.1 285.7
215.0 286.8
250 215.9 287.8
251 216.8 288.9
252 217.8 290.0
253 218.7 29.1.1
254 219.6 292.1
255 220.6 2932
256 221.5 294.3
257 222.4 295.3
258 223.3 296.4
259 224.3 297.5
260 225.2 298.5
261 226.1 299.6
262 227.1 300.7
263 228.0 301.8
264 228.9 302.8
265 229.9 303.9
266 230.8 305.0
267 231.7 306.0
268 232.7 307.1
269 233.6 308.2
270 234.5 309.2
271 235.5 310.3
272 236.4 311.4
273 237.3 312.4
274 238.2 313.5
275 239.2 314.6
276 240.1 315.6
277 241.0 316.7
278 242.0 317.8
279 242.9 318.8
280 243.8 319.9
281 244.8 321.0
282 245.7 322.0
'283 246.6 323.1
284 247.6 324.2
285 248.5 325.2
286 249.4 326.3
287 250.4 327.4
288 251.3 328.4
289 2522 329.5
290 253.2 330.6
291 254.1 331.6
292 255.0 332.7
293 256.0 333.8
294 256.9 334.8
295 257.9 335.9
296 258.8 337.0
297 259.7 338.0
298 260.7 339.1
299 261.6 340.2
13-28
(continued)
-------
September 1993
Table 13-13.
# Groups = 3
Lower
Count Limit
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
0
.0084
.1350
.4075
.7800
1222
1.715
2.248
2.812
3.403
4.015
4.646
5.293
5.955
6.630
7.315
8.011
8.717
9.430
10.15
10.88
11.62
12.36
13.11
13.86
14.62
15.38
16.15
16.92
17.69
18.47
1926
20.04
20.83
21.63
22.42
23.22
24.02
24.83
25.64
26.45
27.26
28.07
28.89
29.71
30.53
31.35
32.17
33.00
33.83
(continued)
Upper
Limit Count
4.787
6.848
8.636
10.29
11.87
13.39
14.86
16.30
17.72
19.11
20.48
21.83
23.18
24.50
25.82
27.12
28.42
29.71
30.99
3236
33.52
34.78
36.03
37.28
38.52
39.76
41.00
42.22
43.45
44.67
45.89
47.10
48.31
49.52
50.72
51.92
53.12
54.32
55.51
56.70
57.89
59.08
60.26
61.44
62.62
63.80
64.97
66.15
67.32
68.49
bU
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
Lower
Limit
34.66
35.49
36.32
37.16
37.99
38.83
39.67
40.51
41.35
42.20
43.04
43.89
44.73
45.58
46.43
47.28
48.14
48.99
49.84
50.70
51.56
52.41
53.27
54.13
54.99
55.85
56.71
57.58
58.44
59.31
60,17
61.04
61.91
62.77
63.64
64.51
65.38
66.25
67.13
68.00
68.87
69.75
70.62
71.50
72.37
73.25
74.13
75.01
75.88
76.76
Upper
Limit
69.66
70.83
71.99
73.16
74.32
75.48
76.64
77.80
78.95
80.11
81.26
82.41
83.57
84.72
85.87
87.01
88.16
89.31
90.45
91.59
92.74
93.88
95.02
96.16
97.30
98.44
99.57
100.7
101.8
103.0
104.1
1053
106.4
107.5
108.6
109.8
110.9
112.0.
113.2
114.3
115.4
116.5
117.7
118.8
119.9
121.0
122.1
123.3
124.4
125.5
Count
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
Lower
Limit
77.64
78.52
79.41
80.29
81.17
82.05
82.94
83.82
84.70
85.59
86.48
87.36
88.25
89.14
90.02
90.91
91.80
92.69
93.58
94.47
95.36
96.25
97.14
98.03
98.93
99.82
100.7
101.6
102.5
103.4
104.3
105.2
106.1
107.0
107.9
108.8
109.7
110.6
111.5
112.4
113.3
114.2
115.1
116.0
116.9
117.8
118.7
119.6
120.5
121.4
Upper
Limit
126.6
127.7
128.9
130.0
131.1
132.2
133.3
134,4
135.6
136.7
137.8
138.9
140.0
141.1
1423
143.4
144.5
145.6
146.7
147.8
148.9
150.0
151.1
152.2
1S3.3
154.4
155.5
156.7
157.8
158.9
160.0
161.1
162.2
163.3
164.4
165.5
166.6
167.7
168.8
169.9
171.0
172.1
173.2
174.3
175.4
176.5
177.6
178.7
179.8
180.9
Count
150
151
152
153
154
155
156
157
158
159
160
'161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
Lower
Limit
122.3
123.2
124.1
125.0
125.9
126.8
127.7
128.6
129.5
130.4
131.3
1323
133.1
134.0
134.9
135.8
136.7
137.6
138.6
139.5
140.4
141.3
142.2
143.1
144.O
144.9
145.8
146.7
147.6
148.6
149.5
150.4
151.3
1523
153.1
154.0
154.9
155.8
156.8
157.7
158.6
159.5
160.4
161.3
1622
163.2
164.1
165.0
165.9
166.8
Upper
Limit
182.0
183.1
184.2
185.3
186.4
187.5
188.6
189.7
190.8
191.9
192.9
194.0
195.1
196.2
197.3
198.4
199.5
200.6
201.7
202.8
203.9
205.0
206.1
207.1
2082
209.3
210.4
211.5
212.6
213.7
214.8
215.9
217.0
218.0
219.1
220.2
221.3
222.4
223.5
224.6
225.7
226.7
227.8
228.9
230.0
231.1
232.2
233.3
234.3
235.4
Count
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
Lower
Limit
167.7
168.6
169.6
170.5
171.4
172.3
173.2
174.1
175.1
176.0
176.9
177.8
178.7
179.6
180.6
181.5
182.4
183.3
184.2
185.2
186.1
187.0
187.9
188.8
189.8
190.7
191.6
192.5
193.4
194.4
195.3
196.2
197.1
198.0
199.0
199.9
200.8
201.7
202.6
203.6
204.5
205.4
206.3
207.3
208.2
209.1
210.0
211.0
211.9
212.8
Upper
Limit
236.5
237.6
238.7
239.8
240.8
241.9
243.0
244.1
245.2
246.3
247.3
248.4
249.5
250.6
251.7
252.8
253.8
254.9
256.0
257.1
258.2
259.2
260.3
261.4
262.5
263.6
264.6
265.7
266.8
267.9
269.0
270.0
271.1
2722
273.3
274.3
275.4
276.5
277.6
278.7
279.7
280.8
281.9
283.0
284.0
285.1
286.2
287.3
288.3
289.4
Count
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276'
277
278
279
280
28?
282
283
284
285
286
287
288
289
290
29 1
292
293
294
295
296
297
298
299
Lower
Limit
213.7
214.7
215.6
216.5
217.4
218.4
219.3
220.2
221.1
222.1
223.0
223.9
224.8
225.8
226.7
227.6
228.5
229.5
230.4
231.3
232.2
233.2
234.1
235.0
236.0
236.9
237.8
238.7
239.7
240.6
241.5
242.5
243.4
244.3
245.2
246.2
247.1
248.0
249.0
249.9
250.8
251.7
252.7
253.6
254.5
255.5
256.4
257.3
258.3
259.2
Upper
Limit
290.5
291.6
292.6
293,7
294.8
295.9
296.9
298.0
299.1
300.2
301.2
302.3
303.4
304.5
305.5
306.6
307.7
308.8
309.8
310.9
312.0
313.1
314.1
3152
316.3
317.3
318.4
319.5
320.6
321.6
322.7
323.8
324.8
325.9
327.0
328.1
329.1
330.2
331.3
332.3
333.4
334.5
335.5
336.6
337.7
338.8
339.8
340.9
342.0
343.0
13-29.
(continued)
-------
September 1993
Tabto 13-13. (continued)
# Groups -4
Lower Upper
Count Limit Limit Count
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
0 5.075
.0063 7.176
.1162 8.996
.3665 10.68
.7170 1238
1.138 13.82
1.612 15.31
2.126 16.77
2.673 1820
3247 19.61
3.843 21.00
4.459 22.37
5.092 23.73
5.739 25.07
6.400 26.40
7.072 27.71
7.756 29.02
8.449 30.32
9.150 31.61
9.860 32.90
10.58 34.17
11.30 35.44
12.03 36.71
12.77 37.96
13.51 3922
1426 40.47
15.01 41.71
15.77 42.95
16.53 44.18
17.30 45.41
18.07 46.64
18.84 47.86
19.62 49.08
20.40 50.30
21.19 51.51
21.97 52.72
22.76 53.93
23.56 55.13
24.35 56.33
25.15 57.53
25.95 58.73
26.76 59.92
27.56 61.11
28.37 62.30
29.T8 63.49
29.99 64.68
30.81 65.86
31.63 67.04
32.45 68.22
3327 69.40
SO
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66'
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
Lower Upper Lower Upper Lower Upper Lower Upper
Limit Limit Count Limit Limit Count Limit Limit Count Limit Limitt Count
34.09 70.57
34.91 71.75
35.74 72.92
36.57 74.09
37.40 7526
3823 76.43
39.06 77.59
39.89 78.76
40.73 79.92
41.56 81.08
42.40 8224
4324 83.40
44.08 84.56
44.92 85.72
45.77 86.87
46.61 88.03
47.46 89.18
48.31 90.33
49.15 91.48
50.00 92.63
50.85 93.78
51.70 94.93
52.56 96.08
53.41 97.22
5426 98.37
55.12 99.51
55.98 100.7
56.83 101.8
57.69 102.9
58.55 104.1
59.41 1052
6027 106.4
61.13 107.5
61.99 108.6
62,86 109.8
63.72 110.9
64.59 112.0
65.45 1132
66.32 114.3
67.19 115.4
68.05 116.6
68,92 117.7
69.79 118.8
70.66 120.0
71.53 121.1
72.40 1222
7328 123.3
74.15 124.5
75.02 125.6
75.90 126.7
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
.146
147
148
149
76.77 127.8
77.65 129.0
78.52 130.1
79.40 1312
80.28 132.3
81.16 133.5
82.03 134.6
82.91 135.7
83.79 136.8
84.67 137.9
85.55 139.1
86.43 1402
87.32 141.3
8820 142.4
89.08 143.5
89.96 144.6
90.85 145.8
91.73 146.9
92.62 148.0
93.50 149.1
94.39 150.2
9527 151.3
96.16 152.4
97.05 153.6
97.94 154.7
98.82 155.8
99.71 156.9
100.6 158.0
101.5 159.1
102.4 1602
103.3 161.3
1042 162.4
105.1 163.5
105.9 164.7
106.8 165.8
107.7 166.9
108.6 168.0
109.5 169.1
110.4 1702
111.3 171.3
1122 172.4
113.1 173.5
114.0 174.6
114.9 175.7
115.8 176.8
116.7 177.9
117.6 179.0
118.5 180.1
119.4 1812
120.3 182.3
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
1212 183.4
122.1 184.5
123.0 185.6
123.9 186.7
124.8 187.8
125.7 188.9
126.6 190.0
127.5 191.1
128.4 192.2
129.3 193.3
1302 194.4
131.1 195.5
132.0 196.6
132.9 197.7
133.8 198.8
134.7 199.9
135.6 201.0
136.5 202.1
137.4 2032
138.3 204.3
1392 205.4
140.1 206.5
141.0 207.6
141.9 208.7
142.8 209.8
143.7 210.9
144.6 212.0
145.5 213.1
146.4 214.2
147.3 215.3
1482 216.3
149.1 217.4
150.1 218.5
151.0 219.6
151.9 220.7
152.8 221.8
153.7 222.9
154.6 224.0
155.5 225.1
156.4 226.2
157.3 227.3
158.2 228.4
159.1 229.4
160.0 230.5
161.0 231.6
161.9 232.7
162.8 233.8
163.7 234.9
164.6 236.0
165.5 237.1
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
166.4 2382
167.3 2392
1682 240,3
169.2 241.4
170.1 242.5
171.0 243.6
171.9 244.7
172.8 245.8
173.7 246.9
174.6 247.9
175.6 249.0
176.5 250.1
177.4 251.2
178.3 252.3
1792 253.4
180.1 254.5
181.0 255.5
182.0 256.6
182.9 257.7
183.8 258.8
184.7 259.9
185.6 261.0
186.5 262.0
187.5 263.1
188.4 2642
189.3 265.3
190.2 266.4
191.1 267.5
192.0 268.5
193.0 269.6
193.9 270.7
194.8 271.8
195.7 272.9
196.6 273.9
197.5 275.0
198.5 276.1
199.4 2772
200.3 278.3
201.2 279.4
202.1 280.4
203.1 281.5
204.0 282.6
204.9 283.7
205.8 284.8
206.7 285.8
207.7 286.9
208.6 288.0
209.5 289.1
210.4 290.2
211.3 2912
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
'283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
Lower Upper
Limit Limit
212.3 292.3
2132 293.4
214.1 294.5
215.0 295.5
215.9 296.6
216.9 297.7
217.8 298.8
218.7 299.9
219.6 300.9
220.6 302.0
221.5 303.1
222.4 3042
223.3 305.2
2242 306.3
225.2 307.4
226.1 308.5
227.0 309.6
227.9 310.6
228.9 311.7
229.8 312.8
230.7 313.9
231.6 314.9
232.6 316.0
233.5 317.1
234.4 318.2
235.3 319.2
236.3 320.3
237.2 321.4
238.1 322.5
239.0 323.5
240.0 324.6
240.9 325.7
241.8 326.8
242.7 327.8
243.7 328.9
244.6 330.0
245.5 331.1
246.4 332.1
247.4 333.2
248.3 334.3
2492 335.4
250.1 336.4
251.1 337.5
252.0 338.6
252.9 339.6
253.8 340.7
254.8 341.8
255.7 342.9
256.6 343.9
257.6 345.0
13-30
(continued)
-------
September 1993
Table 13-13.
# Groups = 5
Lower
Count Limit
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
0
.0050
.1035
.3379
.6722
1.078
1.537
2.037
2.571
3.132
3.717
4.321
4.943
5.580
6.231
6.893
7.567
8.251
8.943
9.644
10.35
11.07
11.79
12.52
13.26
14.00
14.74
15.49
16.25
17.00
17.77
18.53
19.30
20.08
20.86
21.64
22.42
23.21
24.00
24.79
25.59
26.38
27.18
27.99
28.79
29.60
30.41
31.22
32.03
32.85
(continued)
Upper
Limit Count
5.298
7.430
9.274
10.98
12.59
14.15
15.66
17.13
18.58
20.00
21.40
22.78
24.14
25.50
26.84
28.16
29.48
30.79
32.09
33.38
34.67
35.95
37.22
38.48
39.74
41.00
42.25
43.50
44.74
45.98
4721
48.44
49.67
50.89
52.11
53.32
54.54
55.75
56.96
58.16
59.36
60.56
61.76
62.96
64.15
65.34
66.53
67.72
68.90
70.08
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65-
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
Lower
Limit
33.66
34.48
35.30
36.13
36.95
37.78
38.60
39.43
40.26
41.09
41.93
42.76
43.60
44.43
4527
46.11
46.95
47.79
48.64
49.48
50.33
51.17
52.02
52.87
53.72
54.57
55.42
56.28
57.13
57.98
58.84
59.70
60.55
61.41
62.27
63.13
63.99
64.85
65.72
66.58
67.44
68.31
69.17
70.04
70.91
71.77
72.64
73.51
74.38
75.25
Upper
Limit
71.27
72.45
73.62
74.80
75.97
77.15
78.32
79.49
80.66
81.82
82.99
84.15
85.32
86.48
87.64
88.80
89.96
91.11
92.27
93.42
94.58
95.73
96.88
98.03
99.18
100.3
101.5
102.6
103.8
104.9
106.1
107.2
108.3
109.5
110.6
111.8
112.9
114.0
115.2
116.3
117.4
118.6
119.7
120.8
122.0
123.1
124.2
125.4
126.5
127.6
Count
100
101
102
103
104
105
106
wr
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
Lower
Limit
76.12
76.99
77.86
78.74
79.61
80.48
81.36
82.23
83.11
83.99
84.86
85.74
86.62
87.50
88.38
89.26
90.14
'91.02
91.90
92.78
93.66
94.54
95.43
96.31
9720
98.08
98.97
99.85
100.7
101.6
102.5
103.4
104.3
105.2
106.1
107.0
107.8
108.7
109.6
110.5
111.4
112.3
113.2
114.1
115.0
115.9
116.8
117.6
118.5
119.4
Upper
Limit
128.8
129.9
131.0
132.1
133.3
134.4
135.5
136.6
137.8
138.9
140.0
141.1
142.3
143.4
144.5
145.6
146.7
147.9
149.0
150.1
151.2
152.3
153.4
154.6
155.7
156.8
157.9
159.0
160.1
161.2
162.4
163.5
164.6
165.7
166.8
167.9
169.0
170.1
171.2
172.4
173.5
174.6
175.7
176.8
177.9
179.0
180.1
181.2
182.3
133.4
Count
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
Lower
Limit
120.3
121.2
122.1
123.0
123.9
124.8
125.7
126.6
127.5
128.4
129.3
130.2
131.1
132.0
132.9
133.8
134.7
135.6
136.5
137.4
138.3
139.2
140.1
141.0
141.9
142.8
143.7
144.6
145.5
146.4
147.3
148.2
149.1
150.0
150.9
151.8
152.7
153.7
154.6
155.5
156.4
157.3
158.2
159.1
160.0
160.9
161.8
162.7
163.6
164.5
Upper
Limit
184.5
185.6
186.7
187.8
188.9
190.0
191.2
192.3
193.4
194.5
195.6
196.7
197.8
198.9
200.0
201.1
202.2
203.3
204.4
205.5
206.6
207.7
208.8
209.9
211.0
212.0
213.1
214.2
215.3
216.4
217.5
218.6
219.7
220.8
221.9
223.0
224.1
225.2
226.3
227.4
228.5
229.6
230.7
231.8
232.8
233.9
235.0
236.1
237.2
238.3
Count
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
Lower
Limit
165.5
166.4
167.3
168.2
169.1
170.0
170.9
171.8
172.7
173.6
174.6
175.5
176.4
177.3
178.2
179.1
180.0
180.9
181.8
182.8
183.7
184.6
185.5
186.4
1Q7.3
188.2
189.2
190.1
191.0
191.9
192.8
193.7
194.6
195.6
196.5
197.4
198.3
199.2
200.1
201.1
202.0
202.9
203.8
204.7
205.6
206.6
207.5
208.4
209.3
210.2
Upper
Limit
239.4
240.5
241.6
242.7
243.8
244.8
245.9
247.0
248.1
249.2
250.3
251.4
252.5
253.6
254.6
255.7
256.8
257.9
259.0
260.1
261.2
262.3
263.3
264.4
265.5
266.6
267.7
268.8
269.9
270.9
272.0
273.1
274.2
275.3
276.4
277.4
278.5
279.6
280.7
281.8
282.9
283.9
285.0
286.1
287.2
288.3
289.4
290.4
291.5
292.6
Count
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
Lower
Limit
211.2
212.1
213.0
213.9
214.8
215.7
216.7
217.6
218.5
219.4
220.3
221.3
2222
223.1
224.0
224.9
225.9
226.8
227.7
228.6
229.6
230.5
231,4
232.3
233.2
234.2
235.1
236.0
236.9
237.9
238.8
239.7
240.6
241.5
242.5
243.4
244.3
245.2
246.2
247.1
248.0
248.9
249.9
250.8
251.7
252.6
253.6
254.5
255.4
256.3
Upper
Limit
293.7
294.8
295.8
296:9
298.0
299.1
300.2
301.2
302.3
303.4
304.5
305.6
306.6
307.7
308.8
309.9
311.0
312.0
313.1
314.2
315.3
316.4
317.4
318.5
319.6
320.7
321.7
322.8
323.9
325.0
326.1
327.1
3282
329.3
330.4
331.4
332.5
333.6
334.7
335.7
336.8
337.9
339.0
340.0
341.1
342.2
343.3
344.3
345.4
346.5
-------
September 1993
Tab/o «
X.C
N P-
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
10 0
11 0
12 0
13 0
14 0
15 0
16 0
17 0
18 0
19 0
20 0
21 0
22 Q
23 0
24 0
25 0
26 0
27 0
28 0
29 0
30 0
31 0
32 0
33 0
34 0
35 0
36 0
37 0
38 0
39 0
40 0
41 0
42 0
43 0
44 0
45 0
46 0
47 0
48 0
49 0
50 0
51 0
52 0
53 0
54 0
55 0
56 0
57 0
58 0
59 0
60 0
t-14.
»
Pf
.842
.708
.602
.522
.459
.410
.369
.336
.308
.285
.265
.247
.232
.218
2QB
.195
.185
.176
.168
.161
.154
.148 '
.142
.137
.132
.128
.123
.119
.116
.112
.109
.106
.103
.100
.097
.095
.093
.090
.088
t08S
.084
.082
.080
.079
.077
.075
.074
.073
.071
.070
.068
.067
.066
.065
.064
.063
.062
.061
.060
Lower (P ) and Upper (P +) 95% Confidence Limits for a Binomial Pro;
x-r x-2 x=3 x=4 x=s x=e
013 .987
.008 .906
.008 .806
.005 .716
.004 .641
.004 .579
.003 .527
.003 .482
.003 .445
.002 .413
.002 .385
.002 .360
.002 .339
.002 .319
.002 .302
.001 .287
.001 .273
.001 .260
.001 £49
.001 .238
t001 .228
.001 .219
.001 .211
.001 .204
.001 .196
.001 .190
.001 .183
.001 .178
.001 .172
.001 .167
.001 .162
.001 .158
.001 .153
.001 .149
.001 .145
.001 .142
.001 .138
.001 .135
.001 .132
-QQ1 .129
.001 .126
.001 .123
.001 .120
.001 .118
.001 .115
.001 .113
.001 .111
.001 .109
.001 .106
.000 .104
.000 .103
.000 .101
.000 .099
.000 .097
.000 .096
.000 .094
.000 .092
.000 .091
.000 .089
rm r*
.158 1.00
.094 .992
.068 .932
.053 .853
.043 .777
.037 .710
.032 .651
.028 .600
.025 .556
.023 .518
.021 .484
.019 .454
.018 .428
.017 .405
.016 .383
.015 .364
.014 .347
.013 .331
.012 .317
.01.2 .304
,011 .292
.011 .280
.010 .270
.010 .260
.009 .251
.009 .243
.009 .235
.008 .228
.008 .221
.008 .214
.008 .208
.007 .202
.007 .197
.007 .192
.007 .187
.007 .182
.006 .177
.006 .173
.006 .169
,008 .165
.006 .162
.006 .158
.006 .155
.005 .151
.005 .148
.005 .145
.005 .143
.005 .140
.005 .137
.005 .135
.005 .132
.005 .130
.005 .127
.004 .125
.004 .123
.004 .121
.004 .119
.004 .117
.004 .115
.292 1.00
.194 .994
.147 .947
.118 .882
.099 .816
.085 .755
.075 .701
.067 .652
.060 ,610
.055 .572
.050 .538
.047 .508
.043 .481
.040 .456
.038 .434
.036 .414
.034 .396
.032 .379
.030 .363
.029 .349
.028 .336
.027 .324
.025 .312
.024 .302
.024 .292
.023 .282
.022 .274
.021 .265
.020 .258
.020 .250
.019 .243
.019 .237
.018 .231
.018 .225
.017 .219
.017 .214
.016 .209
.016 .204
.015 .199
.015 .195
.015 .191
.014 .187
.014 .183
.014 .179
.013 .175
.013 .172
.013 .169
.013 .165
.012 .162
.012 .159
.012 .157
.012 .154
.011 .151
.011 .149
.011 .146
.011 .144
.011 .141
.010 .139
.398 1.00
.284 .995
.223 .957
. 184 .901
.157 .843
.137 .788
.122 .738
.109 .692
.099 .651
.091 .614
.084 .581
.078 .551
.073 .524
.068 .499
.064 .476
.061 .456
.057 .437
.054 .419
.052 .403
.050 .388
.047 .374
.045 .361
.044 .349
.042 .337
.040 .327
.039 .317
.038 .307
.036 .298
.035 .290
.034 .282
.033 .275
.032 .267
.031 .261
.030 .254
.029 .248
.029 .242
.028 .237
.027 .231
.027 .226
.026 .221
.025 .217
.025 .212
.024 .208
.024 .204
.023 .200
.023 .196
.022 .192
.022 .189
.021 .185
.021 .182
.021 .179
.020 .176
.020 .173
.019 .170
.019 .167
.019 .165
'.018 .162
.478
.359
.290
.245
.212
.187
.167
.152
.139
.128
.118
.110
.103
.097
.091
.087
.082
.078
.075
.071
.068
.066
.063
.061
.058
.056
.055
.053
.051
.050
.048
.047
.045
.044
.043
.042
.041
.040
.039
.038
.037
.036
.035
.035
.034
.033
.033
.032
.031
.031
.030
.030
.029
.029
.028
.028
1.00
.996 .541
'.963 .421
.915 .349
.863 .299
.813 .262
.766 .234
.723 .211
.684 .192
.649 .177
.616 .163
.587 .152
.560 .142
.535 .133
.512 .126
.491 .119
.472 .113
.454 .107
.437 .102
.422 .098
.407 .094
.394 .090
.381 .086
.369 .083
.358 .080
.347 .077
.337 .075
.328 .072
.319 .070
.311 .068
.303 .066
.295 .064
.288 .062
.281 .060
.274 .059
.268 .057
.262 ,056
.256 .054
.251 .053
.246 .052
.241 .051
.236 .049
.231 .048
.227 .047
.222 .046
.218 .045
.214 .044
.210 .044
.207 .043
.203 .042
.200 .041
.196 .040
.193 .040
.190 .039
.187 .038
. 184 .038
oortion, P (P=X/N)
X=7 X=8
P* P- p + p- p +
1.00
.996 .590
.968 .473
.925 .400
.878 .348
.833 .308
.789 .277
.749 .251
.711 .230
.677 .213
.646 .198
.617 .184
.590 .173
.566 .163
.543 .154
.522 .146
.502 .139
.484 .132
.467 .126
.451 .121
.436 .116
.423 .111
.410 .107
.397 .103
.386 .099
.375 .096
.364 .093
.355 .090
.345 .087
.336 .084
.328 .082
.320 .080
.313 .077
.305 .075
.298 .073
,292 ,072
.285 .070
.279 .068
.274 .066
.268 .065
.263 .063
.257 .062
.252 .061
.248 .059
.243 .058
.239 .057
.234 .056
.230 .055
.226 .054
.222 .053
.219 .052
.215 .051
.212 .050
.208 .049
.205 .048
1.00
.997 .631
.972 .518
.933 .444
.891 .390
.848 .349
.808 .316
.770 .289
.734 .266
.701 .247
.671 .230
.643 .215
.616 .203
.592 .191
.570 .181
.549 .172
.529 .164
.511 .156
.494 . 149
.478 .143
.463 .138
.449 .132
.435 .127
.423 .123
.411 .119
.400 .115
.389 .111
'.379 .107
.369 .104
.360 .101
.352 .098
.343 .096
.335 .093
.328 .091
.321 .088
.314 .086
.307 .084
.301 .082
.295 .080
.289 .078
.283 .076
.278 .075
.272 .073
.267 .072
.263 .070
.258 .069
.253 .067
.249 .066
.245 .065
.241 .064
.237 .063
.233 .061
.229 .060
.226 .059
1.00
.997
.975
.940
.901
.861
.823
.787
.753
.722
.692
.665
.639
.616
.593
.573
.553
.535
.518
.502
.487
.472
.459
.446
.434
.423
.412
.401
.392
.382
.373
.365
.356
.349
.341
'.334
.327
.321
.314
.308
.302
.297
.291
.286
.281
.276
.271
.267
.262
.258
.254
.250
.246
X=9
P- P*
664 1.00
.555 .997
.482 .977
.428 .945
.386 .909
.351 .872
.323 .837
.299 .802
.278 .770
.260 .740
.244 .711
.231 .685
.218 .660
.207 .636
.197 .615
.188 .594
.180 .575
.172 .557
.165 .540
.159 .524
.153 .508
.147 .494
.142 .480
.137 .467
.133 .455
.129 .444
.125 .433
.121 .422
.118 .412
.114 .402
.111 .393
.108 .385
.106 .376
.103 .368
.100 .360
.098 .353
.096 .346
.094 .339
.091 .333
.089 .326
.088 .320
.086 .314
.084 .309
.082 .303
.081 .298
.079 .293
.078 .288
.076 .283
.075 .279
.073 .274
.072 .270
.071 .266
(continued)
13-32
-------
September 1993
Table 13-14. (continued)
70
77
72
73
74
75
76
X=0
692
.587
.516
.462
.419
.384
.354
1.00
.998
.979
.950
.916
.882
.848
X=1
.715
.615
.546
.492
.449
.413
1.00
.998
.981
.953
.922
.890
X=2
.735
.640
.572
.519
.476
1.00
.998
.982
.957
.927
X=3
.753
.667
.595
.544
7.00
.998
.983
.960
X=4
P_
.768
.687
.677
Pj.
*
7.00
.998
.984
X=5
P.
.782
.698
P +
*
7.00
.998
X=6 X=7 X=8
p- p + p- p+ p- p
.794 7.00
X=9
i j. D ' D +
+ F F*
17 .329 .816 .383 .858 .440 .897 .501 .932 .566 .962 .636 .985 .713 .999 .805 7.00
78 .308 .785 .357 .827 .470 .867 .465 .903 .524 .936 .586 .964 .653 .986 .727 .999 875 7.00
79 .289 .756 .335 .797 .384 .837 .434 .874 .488 .909 .544 .939 .604 .966 .669 .987 .740 .999 .824 7.00
20 .272 .728 .375 .769 .367 .809 .408 :846 .457 .887 .509 .973 .563 .943 .627 .968 .683 .988 .757 .999
27 .257 .702 .298 .743 .340 .782 .384 .879 .430 .854 .478 .887 .528 .978 .587 .946 .637 .970 .696 .988
22 .244 .678 .282 .778 .322 .756 .364 .793 .407 .828 .457 .867 .498 .893 .546 .922 .597 .948 .657 .977
23 .232 .655 .268 .694 .306 .732 .345 .768 .385 .803 .427 .836 .477 .868 .576 .898 :563 .925 .672 .950
24 .227 .634 .256 .672 .297 .709 .328 .744 .366 .779 .406 .872 .447 .844 .489 .874 .533 .902 .578 .929
25 .277 .673 .244 .657 .278 .687 .373 .722 .349 .756 .387 .789 .425 .820 .465 .857 .506 .879 .549 .906
26 .202 .594 .234 .637 .266 .666 .299 .707 .334 .734 .369 .766 .406 .798 .443 .828 .482 .857 .522 .884
27 .794 .576 .224 .672 .255 .647 .287 .687 .379 .773 .353 .745 .388 .776 .424 .806 .460 .835 .498 .862
28 .786 .559 .275 .594 .245 .628 .275 .667 .306 .694 .339 .725 .372 .755 .406 .785 .447 .874 .476 .847
29 .779 .543 .207 .577 .235 .677 .264 .643 .294 .675 .325 .706 .357 .736 .389 .765 .423 .793 .457 .827
30 .773 .528 .799 .56.7 .227 .594 .255 .626 .283 .657 .373 .687 .343 .777 .374 .745 .406 .773 .439 .807
37 .767 .574 .792 .546 .278 .578 .245 .609 .273 .640 .302 .669 .337 .698 .360 .727 .397 .755 .422 .782
32 .767 .500 .786 .532 .277 .563 .237 .594 .264 .623 .297 .653 .379 .687 .347 .709 .377 .736 .406 .763
33 .756 .487 .780 .578 .204 .549 .229 .579 .255 .608 .287 .636 .308 .665 .335 .692 .364 .779 .392 .745
34 .757 .475 .774 .505 .797 .535 .222 .564 .246 .593 .272 .627 .298 .649 .324 .676 .357 .702 .379 .728
35 .746 .463 .769 .493 .797 .522 .275 .557 .239 .579 .263 .606 .288 .634 .374 .660 .340 .686 .366 .772
36 .742 .452 .763 .487 .786 .570 .208 .538 .237 .565 .255 .592 .279 .679 .304 .645 .329 .677 .355 .696
37 .738 .447 .759 .470 .780 .498 .202 .525 .225 .552 .248 .579 .277 .605 .295 .637 .379 .656 .344 .687
38 .734 .437 .754 .459 .775 .487 .796 .574 .278 .540 .240 .566 .263 .592 .286 .677 .370 .642 .334 .666
39 .730 .427 .750 .449 .770 .476 .797 .502 .272 .528 .234 .554 .256 .579 .278 .604 .307 '.628 .324 .652
40 .727 .472 .746 .439 .766 .465 .786 .497 .206 .577 .227 .542 .249 .567 .270 .597 .293 .675 .375 .639
47 .724 .403 .742 .429 .767 .455 .181 .481 .201 .506 .221 .531 .242 .555 .263 .579 .285 .603 .307 .626
42 .727 .395 .739 .420 .757 .446 .776 .477 .796 .495 .276 .520 .236 .544 .256 '.567 .277 .590 .298 .673
43 .778 .386 .735 .472 .753 .437 .772 .467 .797 .485 .270 .509 .230 .533 .250 .556 .270 .579 .297 .607
44 .775 .378 .732 .403 .750 .428 .768 .452 .786 .476 .205 .499 .224 .522 .244 .545 .263 .568 .283 .590
45 .772 .377 .729 .395 .746 .479 .764 .443 .782 .466 .200 .490 .279 .572 .238 .535 .257 .557 .277 .578
46 .709 .364 .726 .388 .743 .477 .760 .435 .777 .458 .795 .480 .274 .502 .232 .525 .257 .546 .270 .568
47 .707 .357 .723 .380 .739 .403 .756 .426 .773 .449 .797 .477 .209 .493 .227 .575 .245 .536 .264 .557
48 .705 .350 .720 .373 .736 .396 .753 .478 .770 .447 .787 .463 .204 .484 .222 .505 .240 .526 .258 .547
49 .702 .343 .778 .366 .733 .389 .749 .477 .766 .433 .783 .454 .799 .475 .277 .496 .234 .577 .252 .538
50 .700 .337 .775 .360 .737 .382 .746 .403 .762 .425 .779 .446 .795 .467 .272 .488 .229 .508 .247 .528
57 .098 .337 .773 .353 .728 .375 .743 .396 .759 .477 .775 .438 .797 .459 .208 .479 .224 .499 .247 .579
52 .096 .325 .777 .347 .725 .368 .740 .389 .756 .470 .777 .437 .787 .457 .203 .477 .220 .497 .236 .570
53 .094 .320 .708 .347 .723 .362 .738 .383 .753 .403 .768 .423 .783 .443 .799 .463 .275 .483 .237 .502
54 .093 .374 .706 .335 .720 .356 .735 .376 .750 .397 .765 .476 .780 .436 .795 .456 .277 .475 .227 .494
55 .097 .309 .704 .330 .778 .350 .732 .370 .747 .390 .767 .470 .776 .429 .797 .448 .207 .467 .222 .486
56 .089 .304 .702 .324 .776 .344 .730 .364 .744 .384 .758 .403 .773 .422 .788 .447 .203 .460 .278 .478
57 .087 .299 .700 .379 .774 .339 .727 .358 .747 .378 .755 .397 .770 .475 .784 .434 .799 .452 .274 .477
58 .086 .294 .099 .374 .772 .334 .725 .353 .739 .372 .753 .390 .767 .409 .787 .427 .795 .445 .270 .463
59 .084 .290 .097 .309 .770 .328 .723 .347 .736 .366 .750 .384 .764 .403 .778 .427 .792 .439 .206 .456
60 .083 .285 .095 .304 .708 .323 .727 .342 .734 .360 .747 .379 .767 .397 .775 .474 .788 .432 .203 .450
N = number of trials, X = number of "successes."
For P>0.5, use confidence limits based on 1-P.
13-33
-------
September 1993
Table 13-15.
#Gfoups-3
P-0.05
W P- P'
20 .000 £96
21 .001 £88
22 .001 £80
23 .001 £73
24 .001 £66
25 .001 £59
26 .001 £53
27 .001 £48
28 .001 £43
29 .002 £38
30 .002 £33
31 .002 £29
32 .002 £25
33 .002 £21
34 .002 £17
35 .003 £14
36 .003 £11
37 .003 £07
38 .003 £04
39 .003 £02
40 .003 .199
41 .004 .196
42 .004 .194
43 .004 .191
44 .004 .189
45 .004 .187
46 .004 .185
47 .005 .182
48 .005 .181
49 .005 .179
SO .005 .177
60 .007 .161
70 .008 .150
80 .010 .141
90 .011 .134
100 .012 .128
110 .013 .123
120 .014 .119
130 .015 .116
140 .016 .113
150 .017 .110
160 .018 .107
170 .019 .105
180 .019 .103
190 .020 .101
200 .020 .100
250 .026 .094
300 .027 .039
350 .029 .086
400 .030 .083
450 .031 .081
500 .031 .079
1000 .036 .069
Lowi
P-.7i
P-
.007
.003
.003
.009
.070
.011
.011
.012
.013
.013
.014
.015
.015
.016
.017
.017
.018
.018
.019
.020
.020
.021
.021
.022
.022
.023
.023
.024
.024
.025
.025
.029
.033
.036
.039
.041
.050
.052
.053
.054
.055
.056
.057
.058
.059
.060
.063
.066
.068
.070
.071
.072
.080
*r (P ) and Upper (P *) Simultaneous 95% Confidence Limits for Multinor
0 P=0.75 P=0.20 P=0.25 P=0.30 P=0.35
§
.366
.358
.350
.343
.336
.330
.324
.373
.373
.303
.304
£99
.295
.291
£88
£84
£81
'£78
.274
£72
.269
.266
.263
.267
.259
.256
.254
.252
.250
.243
.246
.230
.273
.203
.207
.794
.789
.785
.787
.177
.174
.171
.169
.166
.764
.762
.755
.749
.745
.742
.739
.737
.725
r
.027
.023
.024
.026
.027
.028
.030
.031
.032
.033
,035
,036
.037
.038
.039
.040
.047
.042
.043
.044
.045
.046
.047
.047
.048
.049
.050
.050
.057
.052
.053
.059
.075
.078
.087
.084
.086
.088
.090
.092
.093
.095
.096
.097
.098
.099
.704
.707
.770
.772
.774
.776
.725
rT
.429
.420
.473
.406
.399
.393
.387
.382
.376
.372
-.367
.363
.359
.355
.357
.347
.344
.347
.338
.335
.332
.329
.327
.324
.322
.379
.377
.375
.373
.377
.309
.292
.278
.269
.267
.254
.249
.244
.240
.236
.233
.230
.227
.224
.222
.220
.272
.206
.207
.798
.795
.792
.779
r
.041
.044
.046
.048
.050
.052
.054
.056.
.057
.059
.067
.062
.064
.065
.067
.068
.069
.077
.072
.073
.074
.076
.077
.078
.079
.080
.087
.082
.083
.084
.085
.705
.770
.775
.778
.722
.725
.727
.730
.732
.734
.735
.737
.738
.740
.747
.746
.757
.754
.757
.759
.767
.777
A"
.486
.478
.477
.464
.457
.457
.445
.440
.435
.430
.426
.422
.477
.474
.470
.406
.403
.400
.397
.394
.397
.388
.385
.383
.387
.378
.376
.374
.372
.370
.368
.347
.335
.325
.377
.377
.305
.300
.296
.292
.289
.285
.283
.280
.278
.276
.267
.267
.256
.252
.249
.246
.232
r
066
.069
.072
.074
.077
.079
.082
.084
.086
.088
.090
.092
.094
.096
.098
.700
.707
.703
.704
.706
.707
.726
.727
.728
.729
.730
.737
.732
.733
.734
.734
.742
.748
.754
.758
.762
.765
.768
.777
.773
.776
.778
.780
.787
.783
.784
.797
.795
.799
.202
.204
.207
.279
C"
.540
.532
.525
.578
.572
.506
.500
.495
.490
.486
.487
.477
.473
.469
.465
.462
.459
.455
.452
.449
.447
.435
.433
.437
.429
.426
.424
.422
.427
.479
.477
.402
.390
.380
.372
.365
.360
.355
.350
.346
.343
.340
.337
.334
.332
.330
.327
.374
.309
.305
.302
.299
.284
r
.094
.098
.101
.704'
.707
.770
.773
.776
.779
.727
.723
.726
.728
.730
.752
.754
.755
.757
.758
.760
.767
.762
.763
.765
.766
.767
.768
.769
.770
.777
.772
.787
.788
.794
.799
.204
.207
.277
.274
.277
.279
.227
.224
.225
.227
.229
.236
.247
.245
.248
.257
.253
.267
A"
.590
.583
.576
.569
.563
.558
.552
.547
.542
.538
.534
.529
.526
.522
.505
.502
.499
.497
.494
.492
.489
.487
.485
.482
.480
.478
.476
.474
.473
.477
.469
.454
.442
.433
.425
.478
.472
.407
.403
.399
.396
.392
.390
.387
.384
.382
.373
.367
.367
.357
.354
.357
.336
r
.725
.729
.733
.737
.747
.744
.747
.750
.753
.779
.782
.784
.786
.787
.789
.797
.793
.794
.796
.797
.799
.200
.202
.203
.204
.206
.207
.208
.209
.277
.272
.222
.230
.236
.242
.247
.257
.255
.258
.267
.264
.266
.269
.277
.273
.274
.282
.287
.292
.295
.298
.307
.375
r
.638
.631
.624
.678
.672
.607
.602
.597
.592
.570
.567
.563
.560
.557
.554
.557
.549
.546
.543
.547
.539
.536
.534
.532
.530
.528
.526
.524
.523
.527
.579
.505
.493
.484
.476
.469
.464
.459
.455
.457
.447
.444
.447
.439
.436
.434
.425
.478
.473
.409
.405
.403
.387
nial Pi
P=0.4
P-
.769
.763
.768
.772
.776
.206
.209
.272
.274
.277
.279
.227
.224
.226
.223
.230
.232
.233
.235
.237
.239
.240
.242
.243
.245
.246
.248
.249
.250
.252
.253
.264
.273
.280
.286
.297
.296
.300
.303
.307
.370
.372
.375
.377
.379
.327
.329
.335
.339
.343
.346
.349
.364
oportions (P)
0 P=0.45 P=0.50
P* P- P* P- P*
.683
.677
.670
.664
.659
.637
.627
.623
.620
.676
.673
.670
.607
.604
.607
.598
.596
.593
.597
.589
.586
.584
.582
.580
.578
.576
.575
.573
'.577
.569
.568
.554
.543
.534
.526
.520
.574
.509
.505
.507
.498
.495
.492
..489
.487
.485
.476
.469
.464
.460
.456
.453
.438
.795
.200
.234
.238
.247
.244
.247
.250
.253
.256
.259
.267
.264
.266
.268
.270
.272
.274
.276
.278
.280
.282
.284
.235
.287
.288
.290
.297
.293
.294
.296
.307
.377
.325
.337
.337
.342
.346
.350
.353
.356
.359
.362
.364
.366
.368
.377
.383
.388
.392
.395
.398
.473
.726
.720
.687
.682
.678
.674
.677
.667
.664
.667
.657
.654
.652
.649
.646
.644
.647
.639
.637
.635
.632
.630
.628
.627
.625
.623
.627
.620
.678
.676
.675
.607
.597
.582
.575
.569
..563
.559
.554
.557
.547
.544
.547
.539
.537
.534
.526
.579
.574
.570
.506
.504
.488
.264
.268
.273
.277
.280
.284
.288
.297
.294
.297
.300
.302
.305
.308
.370
.372
.375
.377
.379
.327
.323
.325
.327
.329
.330
.332
.334
.335
.337
.338
.340
.352
.362
.377
.378
.384
.389
.393
.397
.407
.404
.407
.470
.472
.474
.477
.425
.432
.437
.447
.444
.447
.462
.736
.732
.727
.723
.720
.776
.772
.709
.706
.703
.700
.698
.695
.692
.690
.688
.685
.683
.687
.679
.677
.675
.673
.677
.670
.668
.666
.665
.663
.662
.660
.648
.638
.629
.622
.676
.677
.607
.603
.599
.596
.593
.590
.588
.586
.583
.575
.568
.563
.559
.556
.553
.538
(continued)
13-34
-------
September 1993
Table
#Gro
N
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
250
300
350
400
450
500
1000
t 13-15.
ops = 4
P=0.05
.000 .308
.000 .299
.000 .291
.001 .283
.001 .276
.001 .270
.001 .263
.001 .258
.001 .252
.001 .247
.001 .242
.002 .238
.002 .233
.002 .229
.002 .226
.002 .222
.002 .218
.002 .215
.003 .212
.003 .209
.003 .206
.003 .203
.003 .201
.003 .198
.004 .196
.004 .193
.004 .191
.004 .189
.004 .187
.004 .185
.005 .183
.006 .167
.008 .155
.009 .146
.010 .138
.012 .132
.013 .127
.014 .123
.015 .119
.015 .116
.016 .113
.017 .110
.018 .108
.018 .106
.019 .104
.019 .102
.025 .097
.027 .092
.028 .088
.029 .085
.030 .082
.031 .080
.035 .070
(continued)
P=.10 P=0.15
.006
.007
.007
.008
.009
.009
.010
.011
.011
.012
.013
.013
.014
.014
.015
.016
.016
.017
.017
.018
.018
.019
.020
.020
.021
.021
.022
.022
.023
.023
.023
.028
.031
.034
.037
.039
.049
.050
.052
.053
.054
.055
.056
.057
.058
.059
.062
.065
.067
.068
.070
.071
.079
.378
.369
.361
.354
.347
.340
.334
.328
.323
.318
.313
.309
.304
.300
.296
.293
.289
'.286
.283
.280
.277
.274
.271
.269
.266
.264
.261
.259
.257
.255
.253
.236
.223
.213
.205
.199
.194
.189
.185
.181
.178
.175
.172
.170
.168
.166
.158
.152
.147
.144
.141
.139
.126
.UW
.021
.022
.023
.025
.026
.027
.029
.030
.031
.032
.033
.034
.035
.036
.037
.038
.039
.040
.041
.042
.043
.044
.045
.045
.046
.047
.048
.049
.049
.050
.056
.073
.076
.079
.082
.084
.086
.088
.090
.091
.093
.094
.095
.096
.098
.102
.106
.108
.111
.113
.114
.124
.440
.432
.424
.417
.410
.403
.397
.392
.386
.381
.377
.372
.368
.364
.360
.356
.353
.349
.346
.343
.340
.337
.335
.332
.329
.327
.325
.322
.320
.318
.316
.299
.285
.275
.266
.259
.254
.249
.244
.240
.237
.234
.231
.228
.226
.224
.215
.209
.204
.200
.197
.194
.180
P=0.20
.038
.040
.042
.044
.046
.048
.050
.052.
.054
.055
.057
.059
.060
.062
.063
.064
.066
.067
.068
.070
.071
.072
.073
.074
.075
.076
.077
.078
.079
.080
.081
.102
.108
.112
.116
.119
.122
.125
.127
.129
.131
.133
.135
.136
.138
.139
.144
.149
.152
.155
.157
.159
.170
.498
.490
.482
.475
.468
.462
.456
.450
.445
.440
.435
.431
.427
.423
.419
.415
.412
.408
.405
.402
.399
.396
.394
.391
.388
.386
.384
.381
.379
.377
.375
.354
.342
.331
.323
.316
.310
.305
.300
.296
.293
.290
.287
.284
.281
.279
.270
.264
.258
.254
.251
.248
.233
P=0.25
.061
.064
.067
.070
.072
.075
.077
.080
.082
.084
.086
.088
.090
.092
.093
.095
.097
.098
.100
.102
.103
.122
.123
.124
.125
.126
.127
.128
.129
.130
.131
.139
.145
.150
.155
.159
.162
.165
.168
.171
.173
.175
.177
.179
.180
.182
.188
.193
.197
.200
.203
.205
.217
.551
.543
.536
.529
.522
.516
.511
.505
.500
.495
.491
.486
.482
.478
.474
.471
.467
.464
.461
.458
.455
.444
.441
.439
.437
.435
.432
.430
.428
.426
.425
.409
.396
.386
.378
.371
.365
.359
.355
.351
.347
.344
.341
.338
.336
.333
.324
.317
.312
.308
.304
.301
.286
P=0.30
.088
.092
.096
.099'
.102
.105
.108
.111
.113
.116
.118
.120
.123
.125
.148
.149
.151
.152
.154
.155
.157
.158
.159
.160
.161
.163
.164
.165
.166
.167
.168
.177
.184
.190
.196
.200
.204
.208
.211
.213
.216
.218
.221
.223
.224
.226
.233
.239
.243
.246
.249
.252
.265
.6U1
.594
.587
.580
.574
.568
.562
.557
.552
.547
.543
.539
.535
.531
.514
.511
.508
.505
.503
.500
.497
.495
.493
.490
.488
.486
.484
.482
.480
.478
.476
.461
.448
.439
.430
.423
.417
.412
.408
.404
.400
.397
.394
.391
.388
.386
.377
.370
.364
.360
.356
.353
.337
.119
.123
.127
.131
.134
.138
.141
.144
.147
.174
.176
.178
.180
.182
.184
.186
.188
.189
.191
.192
.194
.195
.197
.198
.200
.201
.202
.203
.205
.206
.207
.217
.225
.232
.238
.243
.247
.251
.255
.258
.260
.263
.265
.268
.270
.271
.279
.285
.289
.293
.296
.299
.313
5 P=0.40
.649
.641
.635
.628
.622
.617
.611
.606
.602
.579
.575
.572
.569
.566
.562
.560
.557
.554
.552
.549
.547
.544
.542
.540
.538
.536
.534
.532
.530
.528
.526
.511
.499
.490
.482
.475
.469
.464
.459
.455
.451
.448
.445
.442
.440
.438
.428
.421
.416
.411
.408
.405
.388
.151
.156
.161
.165
.169
.200
.203
.206
.208
.211
.213
.216
.218
.220
.222
.224
.226
.228
.230
.231
.233
.235
.236
.238
.239
.241
.242
.244
.245
.246
.248
.259
.268
.275
.282
.287
.292
.296
.300
.303
.306
.309
.311
.314
.316
.318
.326
.332
.337
.341
.344
.347
.362
.693
.687
.680
.674
.669
.640
.636
.632
.628
.625
.621
.618
.615
.612
.609
.606
.604
.601
.599
.596
.594
.592
.590
.588
.585
.584
.582
.580
.578
.576
.575
.560
.549
.539
.531
.525
.519
.514
.510
.506
.502
.499
.496
.493
.491
.488
.479
.472
.467
.462
.459
.456
.439
P=0.45
.187
.192
.197
.231
.234
.237
.241
.244
.247
.249
.252
.255
.257
.259
.262
.264
.266
.268
.270
.272
.274
.276
.277
.279
.281
.282
.284
.285
.287
.288
.290
.302
.312
.320
.327
.332
.337
.342
.346
.349
.353
.355
.358
.361
.363
.365
.374
.380
.385
.389
.393
.395
.411
.736
.729
.723
.691
.687
.683
.679
.675
.672
.668
.665
.662
.659
.656
.654
.651
.649
.646
.644
.642
.640
.638
.635
.634
.632
.630
.628
.626
.625
.623
.621
.607
.596
.587
.580
.574
..568
.563
.559
.555
.551
.548
.545
.543
.540
.538
.529
.522
.517
.512
.509
.506
.489
P=0.50
.256
.261
.265
.269
.273
.277
.280
.283
.287
.290
.293
.295
.298
.301
.303
.306
.308
.310
.312
.314
.316
.318
.320
.322
.324
.326
.327
.329
.330
.332
.333
.347
.357
.366
.373
.379
.384
.389
.393
.397
.400
.403
.406
.408
.411
.413
.422
.429
.434
.438
.442
.444
.461
.744
.739
.735
.731
.727
.723
.720
.717
.713
.710
.707
.705
.702
.699
.697
.694
.692
.690
.688
.686
.684
.682
.680
.678
.676
.674
.673
.671
.670
.668
.667
.653
.643
.634
.627
.621
.616
.611
.607
.603
.600
.597
.594
.592
.589
.587
.578
.571
.566
.562
.558
.556
.539
(continued)
13-35
-------
September 1993
TabJa 13-15.
ff Groups - 5
PmO.05
N P- P*
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
250
300
350
400
450
500
1000
.UOU
.000
.000
.000
.001
.001
.001
.001
.001
.001
.001
.001
.001
.002
.002
.002
.002
.002
.002
.002
.003
.003
.003
.003
.003
.003
.004
.004
.004
.004
.004
.006
.007
.009
.010
.011
.012
.013
.014
.015
.016
.016
.017
.018
.018
.019
.025
.026
.027
.029
.029
.030
.035
.ai/
.308
.299
£91
.284
.277
.271
.265
£59
£54
£49
.244
£40
£36
£32
.228
£25
£21
£18
£15
£12
£09
£06
£04
.201
.199
.196
.194
.192
.190
.188
.171
.159
.149
.141
.135
.130
.125
.121
.118
.115
.112
.110
.108
.106
.104
.099
.093
.089
.086
.084
.082
.071
(continued)
P».rO P»0.75 P=0.20 P=0.25
p. p, p. p, p, p, p. p +
.005
.006
.007
.007
.003
.008
.009
.0)0
.0/0
.011
.012
.012
.013
.013
.014
.015
.015
.016
.016
.017
.017
.018
.018
.019
\.019
.020
.020
.021
.021
.022
.022
.026
.030
.033
.036
.038
.048
.049
.051
.052
.053
.054
.055
.056
.057
.058
.061
.064
.066
.068
.069
.071
.078
.387
.378
.370
.362
.355
.348
.342
.336
.330
.325
.320
.316
.311
.307
.303
.299
.296
.292
£89
£86
£83
£80
.277
£74
.272
.269
.267
.264
.262
£60
.258
.241
.227
£17
.209
£02
.198
.193
.188
.184
.181
.178
.175
.173
.170
.168
.160
.154
.149
.145
.142
.140
.127
.018
.019
.020
.022
.023
.024
.026
.027
.028
.029
.030
.031
.032
.034
.035
.036
.037
.037
.038
.039
.040
.041
.042
.043
.044
.044
.045
.046
.047
.047
.048
.054
.060
.074
.077
.080
.082
.085
.086
.088
.090
.091
.093
.094
.095
.096
.101
.105
.107
.110
.112
.113
.123
.449
.441
.433
.425
.418
.412
.405
.400
.394
.389
.384
.379
.375
.371
.367
.363
.359
.356
.353
.349
.346
.343
.341
.338
.335
.333
.330
.328
.326
.323
.321
.304
.290
.279
.271
.263
.257
.252
.247
.243
.240
£37
£34
.231
.228
.226
.217
.211
.206
.202
.198
.196
.181
.UXti
.038
.040
.042
.044
.046
.048
.049
.051'
.053
.054
.056
.058
.059
.060
.062
.063
.064
.066
.067
.068
.069
.070
.072
.073
.074
.075
.076
.077
.078
.079
.100
.105
.110
.114
.117
.120
.123
.125
.127
.129
.131
.133
.134
.136
.137
.143
.147
.151
.154
.156
.158
.169
.bU/
.498
.490
.483
.476
.470
.464
.458
.453
.448
.443
.438
.434
.430
.426
.422
.418
.415
.412
.408
.405
.402
.400
.397
.394
.392
.389
.387
.385
.383
.380
.360
.347
.336
.327
.320
.314
.309
.304
.300
.296
.293
.290
.287
.284
.282
.273
.266
.260
.256
.253
.250
.235
.058
.061
.064
.067
.069
.072
.074
.076
.079
.081
.083
.085
.087
.088
.090
.092
.094
.095
.097
.098
.100
.101
.121
.122
.123
.124
.125
.126
.126
.127
.128
.136
.142
.148
.152
.156
.160
.163
.166
.169
.171
.173
.175
.177
.178
.180
.187
.191
.195
.199
.201
.204
.216
.560
.552
.544
.537
.530
.524
.518
.513
.508
.503
.498
.493
.489
.485
.481
.478
.474
.471
.467
.464
.461
.458
.448
.445
.443
.441
.438
.436
.434
.432
.430
.414
.401
.390
.382
.375
.368
.363
.358
.354
.350
.347
.344
.341
.338
.336
.326
.319
.314
.310
.306
.303
.287
P=0.30 P=0.35 P=0.40 P=0.45
P- P + P- P* P- P* P-
.085
.088
.092
.095.
.098
.101
.104
.107
.109
.112
.114
.117
.119
.121
.123
.146
.148
.149
.151
.152
.153
.155
.156
.157
.158
.159
.161
.162
.163
.164
.165
.174
.181
.188
.193
.197
.201
.205
.208
.211
.214
.216
.218
.220
.222
.224
.231
.237
.241
.245
.248
.250
.264
.610
.602
.595
.588
.581
.575
.570
.564
.559
.555
.550
.546
.542
.538
.534
.518
.515
.512
.509
.506
.504
.501
.499
.496
.494
.492
.490
.488
.486
.484
.482
.466
.453
.443
.435
.427
.421
.416
.411
.407
.403
.400
.397
.394
.391
.389
.379
.372
.366
.362
.358
.355
.339
.114
.118
.122
.126
.130
.133
.136
.140
.143
.146
.172
.174
.176
.178
.180
.182
.184
.185
.187
.189
.190
.192
.193
.195
.196
.197
.199
.200
.201
.202
.203
.214
.222
.229
.235
.240
.244
.248
.252
.255
.258
.261
.263
.265
.267
.269
.277
.283
.288
.291
.295
.297
.312
.bb/
.649
.642
.636
.630
.624
.619
.614
.609
.604
.582
.578
.575
.572
.569
.566
.563
.560
.558
.555
.553
.550
.548
.546
.543
.541
.539
.537
.535
.534
.532
.516
.504
.494
.486
.479
.473
.467
.463
.458
.455
.451
.448
.445
.443
.440
.431
.423
.418
.413
.410
.407
.390
.146
.151
.155
.160
.164
.168
.199
.201
.204
.206
.209
.211
.213
.216
.218
.220
.222
.224
.225
.227
.229
.231
.232
.234
.235
.237
.238
.240
.241
.242
.244
.255
.264
.272
.278
.284
.289
.293
.297
.300
.303
.306
.309
.311
.313
.315
.324
.330
.335
.339
.342
.345
.361
.701
.694
.687
.681
.676
.670
.642
.638
.634
.631
.627
.624
.621
.618
.615
.612
.609
.607
.604
.602
.600
.597
.595
.593
.591
.589
.587
.585
..583
.582
.580
.565
.553
.543
.535
.529
.523
.517
.513
.509
.505
.502
.499
.496
.493
.491
.481
.474
.469
.464
.461
.457
.440
.181
.186
.191
.226
.229
.232
.236
.239
.242
.244
.247
.250
.252
.255
.257
.259
.261
.263
.265
.267
.269
.271
.273
.275
.276
.278
.280
.281
.283
.284
.285
.298
.308
.316
.323
.329
.334
.339
.343
.346
.350
.353
.355
.358
.360
.362
.371
.378
.383
.387
.391
.394
.410
P=0.50
P* P-
.743
:736
.730
.697
.693
.689
.685
.681
.678
.674
.671
.668
.665
.662
.659
.657
.654
.652
.649
.647
.645
.643
.641
.639
.637
.635
.633
.631
.630
.628
.626
.612
.601
.592
.584
.577
.572
.567
.562
.558
.555
.551
.548
.546
.543
.541
.531
.524
.519
.514
.511
.508
.491
.250
.255
.259
.263
.267
.271
.275
.278
.281
.284
.287
.290
.293
.295
.298
.300
.303
.305
.307
.309
.311
.313
.315
.317
.319
.321
.322
.324
.326
.327
.329
.342
.353
.362
.369
.375
.381
.386
.390
.394
.397
.400
.403
.406
.408
.410
.420
.426
.432
.436
.440
.443
.459
P*
.750
.745
.741
.737
.733
.729
.725
.722
.719
.716
.713
.710
.707
.705
.702
.700
.697
.695
.693
.691
.689
.687
.685
.683
.681
.679
.678
.676
.674
.673
.671
.658
.647
.638
.631
.625
.619
.614
.610
.606
.603
.600
.597
.594
.592
.590
.580
.574
.568
.564
.560
.557
.541
13-36
-------
September 1993
o o
o
0 °
0
o
0 0.
o
o o
o
o o
0 0
_
0
o
o
o
0 0
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(c)
Figure 13-1. Dispersion of PFU in suspension. Dashed lines represent separate cell culture bottles. Types of dispersion represented are (a) uniform,
(b) clustered, and (c) random (Poisson).
13-37
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September 1993
0 25 50 75 100 125 150 175 200 225 250 275 300
Total'Count
Figure 13-2. Maximum sampling error of total plaque count half-length of 95% confidence interval, assuming randomly dispersed (Poisson) counts.
13-38
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September 1993
3.0 n
2.8-
2.6-
0.0
0 25 50 75 100 125 150 175 200 225 250 275 300
Total Count
Figure 13-3. Relative sampling error of total plaque count half-length of 95% confidence interval divided by total plaque count, assuming randomly
dispersed (Poisson) counts.
13-39
'U.S. Government Printing Office: 1993 750-002/80269
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