Geoscience Reference
In-Depth Information
95% confidence, and
z
0.005
= 2.58 for 99% confidence. The meaning of the con-
idence inter
v
al is best
u
nderstood in terms of a specific case. For instance,
the interval ±
1.64SE(
·
will contain the true population mean with a prob-
ability of approximately 0.90. In other words, about 90% of the intervals cal-
culated in this way will contain the true population mean.
The interval (2.6) is only valid for large samples. For small samples (say
with
n
< 20), it will be better to use
y
y
/2 , 1
·
yt
±
⋅
α−
SE()
y
,
(2.7)
n
where
t
α/2,
n
-1
is the value that is exceeded with probability α/2 for the
t
distri-
bution with
n
− 1 degrees of freedom. This requires the assumption that the
variable being measured is approximately normally distributed in the popu-
lation being sampled. If this is not the case, then no simple method exists for
calculating an exact confidence interval.
2.4 Estimation of Totals
In many situations, the ecologist is more interested in the total of all values
in a population rather than the mean per sample unit. For example, the total
weight of new growth of all the plants in a region might be more important
than the mean growth on individual plants. Similarly, the total amount of
forage eaten by a herd of animals might be more important than the average
amount eaten per animal.
The estimation of a population total is straightforward if the population
size
N
is known and an estimate of the mean is available. It is obvious, for
example, that if each of 500 animals is estimated to require an average of
25 kg of forage, then 500 × 25 = 12,500 kg of forage is required for all animals.
The general equation that applies is that the estimated population total for
the variable
Y
is
·
T y
,
(2.8)
y
the mean per unit multiplied by the number of units. The sampling variance
of
T
ˆ
y
is then
·
TN y
2
Var(
) ar()
,
(2.9)
y
and its standard error (i.e., standard deviation) is
·
TNy
SE() SE()
,
(2.10)
y
