Environmental Engineering Reference
In-Depth Information
n
1
n
¯
y
SI
=
y
i
,
(4.3)
i
=
1
where
n
is the number of selected sampling locations, and
y
i
is the contaminant
concentration at sampling location
i
. The subscript SI is added to stress that this
estimator is intended for simple random sampling designs.
The
sampling
variance of the estimated mean is estimated by
V
¯
y
SI
S
2
(
y
)
n
=
,
(4.4)
where
S
2
(
y
) is the estimated
spatial
variance of the concentrations in the study area
y
i
− ¯
y
SI
2
n
1
S
2
(
y
)
=
.
(4.5)
(
n
−
1)
i
=
1
The 100(1
−
α
)% confidence interval for
y
is given by:
¯
V
y
SI
,
¯
t
(
n
−
1)
1
¯
y
SI
±
2
·
(4.6)
−
α/
where
t
(
n
−
1)
1
1) degrees
of freedom. This confidence interval is based on the assumption that
y
, and as a
consequence
¯
2
is the (1
−
α/
2) quantile of the Student distribution with (
n
−
−
α/
y
SI
, is normally distributed. If the distribution deviates clearly from
normality, the data should be first transformed to normality, for instance by tak-
ing the logarithm. The confidence limits thus found are then back-transformed to
the original scale. However, we must be aware that the confidence intervals thus
obtained are not the confidence intervals of the mean on the original scale. For
instance, for a lognormal distribution, the confidence interval is for the median on
the original scale. A simple rule to obtain the confidence limits of the estimated
mean is to multiply the back-transformed limits by the ratio of the untransformed
mean and the back-transformed mean (R.L. Correll, personal communication).
Transformation is not necessary if
n
is large, because
¯
y
SI
is then approximately
normally distributed according to the Central Limit Theorem.
Stratified Simple Random Sampling
For STSI the spatial mean is estimated by
L
¯
a
h
¯
y
STSI
=
y
h
,
(4.7)
h
=
1
Search WWH ::
Custom Search