Environmental Engineering Reference
In-Depth Information
the
α
confidence limits of
y
, denoted by
y
α
, are given by
(Berthouex and Brown, 2002)
N
1
∑
y
=
y
i
=
57 16
.
N
i
=
1
1
(
x
−
x
)
2
N
N
−
y
=
(
a bx
+
)
±
t S
1
+
+
∑
∑
S
=
N
x
2
x
=
19 574
,
α
α
2
e
N
N
(10.138)
xx
i
i
∑
2
(
x
−
x
)
i
=
1
i
=
1
i
i
=
1
2
N
N
−
∑
∑
2
S
=
N
y
y
=
3838
yy
i
i
where
t
α 2
is the
t
variate with
N
−
2 degrees of freedom
with a significance level of
α
/2. It is noteworthy that the
prediction interval increases as
x
deviates from the
mean,
x
.
i
=
1
i
=
1
N
N
N
−
=
∑
∑ ∑
S
=
N
x y
x
y
4161
xy
i
i
i
i
i
=
1
i
=
1
i
=
1
S S
−
−
(
S
)
2
EXAMPLE 10.26
xx
yy
xy
S
=
=
2 59
.
e
N N
(
2
)
S
xx
use the data and results in Example 10.25 to estimate
the 95% confidence intervals of predictions of
y
at
x
= 22 and
x
= 32. Compare the width of these confi-
dence intervals.
S
S
4161
19 574
xy
b
=
=
=
0 213
.
,
xx
a
= −
y bx
=
57 16
.
−
( .
0 213 21 54
)(
.
)
=
52 6
.
Solution
Therefore, the parameters of the linear relation are
a
= 52.6 and
b
= 0.213. To determine the 95% confi-
dence interval, use
α
= 0.05, and the applicable
t
-statistic
with
N
−2 = 20 degrees of freedom is
t
a
/2
=
t
0.025
= 2.086.
The confidence intervals for the population parameters,
α
and
β
, corresponding to
a
and
b
, are given by Equa-
tions (10.135) and (10.136) as
From the data and analyses in Example 10.25, a linear
function of the form
y
=
a
+
bx
was fitted to the data,
and the following results were obtained:
a
= 52.6,
b
= 0.213,
S
e
= 2.59,
N
= 22, and
x
= 21 54
.
. Additional
analysis of the data yields
N
∑
(
x
−
x
)
2
=
889 7
.
S
+
Nx
NS
(
)
2
i
xx
α
= ±
a t S
α
/
2
e
i
=
1
xx
For a 95% confidence interval and
N
−
2 = 20 degrees
of freedom,
t
α 2
=
t
0.025
= 2.086. Substituting these data
into Equation (10.138) for
x
= 22 gives
2
19574 [(22)(21.54)]
(22)(19574)
+
=
52 6
.
±
( .
2 086 2 59
)( .
)
= [
48 5 56 6
. ,
. ]
1
(
x
−
x
)
N
S
y
=
[
a bx
+
]
±
t S
1
+
+
α
α
2
e
β
= ±
b t S
N
N
∑
α
/
2
e
(
x
−
x
)
2
xx
i
i
=
1
22
19574
=
0 213
.
±
( .
2 086 2 59
)( .
)
y
0 05
=
[
52 6
.
+
( .
0 213 22
)(
)]
.
=
[ .
0 031 0 394
,
.
]
1
22
(
22 21 51
889
−
.
)
±
( .
2 086 2 59
)( .
)
1
+
+
.
7
Therefore, the 95% confidence limits of
α
and
β
are
[48.5, 56.6] and [0.031, 0.394], respectively.
=
57 3 5 5
.
±
.
10.10.2.1 Confidence Limits of Predictions.
Aside
from specifying the confidence limits of the parameters
in a linear regression, it is sometimes useful to specify the
confidence limits of values predicted by the linear regres-
sion equation. For fitted linear functions of the form
Therefore, the uncertainty (= 5.5) is approximately
9.7% of the predicted value of
y
(= 57.3). These calcula-
tions can be repeated for
x
= 32 and yield
y
0.05
= 59.4 ± 5.6,
in which case the uncertainty (= 5.6) is approximately
9.4% of the predicted value of
y
(= 59.4). Collectively,
these results show that the width of the confidence
interval increases as
x
deviates from
x
; however, the
percentage error in the predicted values decrease (in
this particular case).
= +
(10.137)
where
y
is the expected value of the random variable
y
, and
x
is a deterministic variable, it can be shown that
y
a bx
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