Geoscience Reference
In-Depth Information
Box 2.5: Frequency Distributions Derived from the Normal
Frequency distributions derived from the normal distribution include the
ˇ
2
-,
F
- and
t
-distributions. They are defined for
f
,
f
1
and
f
2
“degrees of
freedom” as:
X
f
p
ˇ
Þ
¼
ˇ
1
f ð =
f
1
2
Z
i
;
ˇ
ðÞ
¼
f
Ff
1
;
ð
f
2
f
2
;
tf
ðÞ
¼
Z
=
2
ðÞ=
f
f
ˇ
2
f ð =
i
¼1
“Degrees of freedom” is a useful concept of classical statistics. It repre-
sents the number of values in the calculation of a statistic that are free to vary
independently. The following example illustrates this concept. In general, the
sample variance satisfies
s
2
¼
˃
2
ˇ
2
ðÞ
. If the population mean
μ
is known,
f
a sample of
n
data has
f
¼
n
degrees of freedom. However, if
μ
is not known
X
n
i
¼1
x
i
, Bessel's correction
and estimated by the sample mean
x
¼
2
2
n
1
¼
˃
ˇ
ð
Þ
(
cf.
Box
2.1
) must be applied and
s
2
so that
f
¼
n
1 (Fig.
2.4
).
n
1
2.3.2 Significance Tests and 95 %-Confidence Intervals
Fractiles
(
Z
P
) that are widely used in significance tests and to define 95 %-
confidence intervals are
ʦ
(1.645) ¼0.95 and
ʦ
(1.96) ¼0.975. They correspond
to setting the level of significance
ʦ
0.05 in one-tailed and two-tailed significance
tests, respectively. The so-called
z
-test of significance provides a prototype for all
other tests of significance. It works as follows: Suppose that
n
numbers are dr
aw
n
from a normal population with mean
ʱ
¼
2
, then the sample mean
X
is
μ
and variance
˃
2
/
n
. Consequently,
normally distributed about
μ
with variance
˃
X
μ
˃=
n
Z
¼
p
is normally distributed about 0 with standard deviation one. Suppose that there is
outside informati
on
suggesting that
μ
¼
μ
0
. Then the test- or null-hypothesis H
0
that
the sample mean
X
originates from a normal distribution with mean
μ
0
and standard
˃=
p
is rejected if |
Z
|
deviation
>
1.96 after
μ
is replaced by
μ
0
. The value |
Z
P
|
¼
1.96 represents the significance limit for the test. If |
Z
|
1.96, the hy
p
othesis
μ
¼
μ
0
is accepted. This test is two-tailed because the absolute value of
X
<
is
being considered. If there is additional outside information that would allow taking
either
μ
μ>μ
0
or
μ<μ
0
as the test hypothesis, a one-tailed test with significance limit
Z
P
¼
1.645 should be used.
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