Environmental Engineering Reference
In-Depth Information
(
¯
should be smaller
than a specified limit
r
, the required number of sampling locations can be used by
replacing
d
in the numerator of Eq. (
4.31
)by
r
¯
If instead of the absolute error, the relative error
|
y
−¯
y
)
/
¯
y
|
y
, where
¯
y
is a prior estimate of the
mean, and continuing the procedure as before.
For small numbers of sampling locations (
n
20), we must use the Student dis-
tribution, see Eqs. (
4.6
) and (
4.11
), to compute
V
max
. Now we face the problem that
we must know the number of sampling locations for SI, and the numbers of sampling
locations per stratum for STSI (see Eq. (
4.12
)) in order to compute the number of
degrees of freedom
df
, and subsequently
t
(
df
)
1
<
2
. In this situation we recommend cal-
culating, for a series of total number of sampling locations, the cumulative (lower)
probability of
−
α/
d
√
V
(
y
)
/
n
for the Student distribution with
df
degrees of freedom. In this
case, the required number of sampling locations is the smallest number for which
the cumulative probability is smaller than
2.
The number of sampling locations needed to estimate an areal fraction
p
such
that, with a specified large probability (1
α/
), the absolute or relative error in
the estimated fraction is smaller than a particular limit, can be calculated with
the binomial distribution. This requires a prior estimate of the areal fraction only.
For STSI, use of the binomial distribution for computing the required number of
sampling locations is much more complicated, and I recommend using the Student
distribution for this purpose, see Eq. (
4.21
).
−
α
4.2.5.3 Constraints on Error Rates in Testing of a Hypothesis
If the survey results are used in decision-making by means of statistical testing of
a hypothesis on the spatial mean or the areal fraction, then the quality constraint
is best formulated in terms of the probability of false rejection (type I error) and
the probability of false acceptance (type II error) of the hypothesis. An example is
testing the spatial mean concentration against a soil quality standard (compliance
monitoring). Both probabilities (error rates) are set to a maximum. The complement
of the maximum probability of false acceptance, 1
the probability of
type II error, is referred to as the power of the test. This power must be linked to
a “minimum detectable difference”. Note that a hypothesis on a percentile can be
reformulated as a hypothesis on the areal fraction. For instance, testing the null-
hypothesis “The 95th-percentile of Zn
−
β
with
β
140 mg kg
−
1
” is equivalent to testing the
null-hypothesis “The areal fraction with Zn concentrations
≤
140 mg kg
−
1
0.95”.
I refer to de Gruijter et al. (
2006
, pp. 85-89) and to EPA (
2006
) for more details on
this subject matter.
≤
≥
4.3 Estimating Mean Concentrations for Delineated Blocks
This section describes sampling strategies for estimating the mean contaminant
concentrations of several blocks that are delineated before sampling. The decision
on remediation of a block is typically based on estimates of the block-mean
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