Environmental Engineering Reference
In-Depth Information
sample mean and variance are
X
and
S
2
respectively,
then the quantity
T
defined by
N
1
1
50 1
∑
S
=
(
C C
i
−
)
2
=
(
801 2
. )
=
4 04
.
mg/L
c
N
−
1
−
i
=
1
X
S
− µ
/
N
x
1
N
T
=
(10.89)
∑
C
s
=
⋅
(
X C
−
)
3
N
i
S
3
(
N
−
1
)(
N
−
2
)
x
c
i
=
1
1
4 04
50
50 1 50 2
has a Student's
t
distribution with
N
−
1 degrees of
freedom
.
=
⋅
)
(
6097
. )
3
=
1 96
.
3
.
(
−
)(
−
C
50
= 3 52
.
mg/L
This theorem is particularly useful when addressing
the question of whether it is likely that the population
mean is indeed eq
u
al to
μ
x
, given that the sample mean
and variance are
X
and
S
2
, respectively. The Student's
t
-test is commonly used to test the null hypothesis,
H
0
:
μ
= target, versus the alternative hypothesis, such as
H
1
:
μ
≠ target. The
t
-test was originally developed by the
English statistician W.S. Gossett (1876-1937) while
working for an Irish brewery. Gosset published under
the pseudonym “Student,” as secrecy was widely prac-
ticed in industrial circles.
1
S
C
1
4 50
4 04
4 67
.
.
c
COV =
1
+
=
1
+
=
0 870
.
4
N
(
)
The standard errors of the sample statistics are cal-
culated as follows:
S
N
4 04
50
.
σ
C
=
=
=
0 572
.
mg/L
S
4 04
2 50
.
(
σ
S
=
=
=
0 404
.
mg/L
c
2
N
)
Theorem 10.2
If the random samples
X
i
,
i
∈ [1,
N
]
, are
taken from a normally distributed population with a vari-
ance of
σ
2
, then the quantity
C
x
, defined by the following
relation
6
N N
(
−
1
)
σ
C
s
=
(
N
+
1
)(
N
+
2
)(
N
+
3
)
6 50 50 1
50 1 50 2 50 3
(
)(
−
)
=
=
0
.323
(
+
)(
+
)(
+
)
2
(
N
−
1
S
x
C
=
(10.90)
x
1
1
2 50
σ
x
σ
C
=
S
=
( .
4 04
)
=
0 404
.
mg/L
50
2
N
(
)
has a chi-square distribution with
N
−
1 degrees of
freedom
.
2
2
C
1 2
2
+
C
( .
0 870
)
1 2 0 870
2 50
+
( .
)
v
v
σ
COV
=
=
=
0 138
.
N
(
)
This theorem is useful when addressing the question
of whether it is likely that the population variance is
equal to
σ
2
, given that the sample variance is
S
2
.
It is usually instructive to normalize the standard error
by the expected value of the statistic. In this case, the
normalized standard errors of the mean, standard devia-
tion, skewness, median, and coefficient of variation are
12, 10, 16, 11, and 16%, respectively. These values give
measures of the uncertainty of the statistics, which are
still on the order of 10% even after 50 samples have
been collected.
Theorem 10.3
If two samples of size
M
and
N
are
drawn from two normal populations whose variances
are
σ
2
and
σ
2
, and the sample variances are
S
2
and
S
2
respectively, then the quantity
F
defined by the
equation
10.7.6 Useful Theorems
S
S
2
/
/
σ
σ
2
F
=
(10.91)
There are several quantities whose probability distribu-
tions are particularly useful, and these quantities and
their associated probability distributions are contained
in the following theorems:
2
2
has a
F
distribution with (M
−
1,
N
−
1) degrees of
freedom
.
This theorem is particularly useful in determining the
likelihood that the samples were drawn from popula-
tions with equal variances.
Theorem 10.1
If
N
random samples are taken from a
normally distributed population with mean μ
x
and the
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