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Let y
= actual data value,
i
Then, the prediction error is: e
2
An estimate of σ
is the, so-called, standard error of the estimate,
1
n
∑
2
s
[
y
(
a
e
i
n
2
i
=
1
Confidence intervals and hypothesis testing in linear regression
Here are some concepts and equations related to statistical inference for
linear regression:
•
Confidence limits for regression coefficients:
For the slope (Β):
For the intercept (Α):
a − (t
⋅[(1/n)+x
)⋅s
n-2, /2
e
where t follows the Student's t distribution with ν = n – 2, degrees of
freedom, and n represents the number of points in the sample.
•
Hypothesis testing on the slope, Β:
Null hypothesis, H
0
Β ≠ Β
. The test statistic is t
0
Student's t distribution with ν = n – 2, degrees of freedom, and n
represents the number of points in the sample.
that of a mean value hypothesis testing, i.e., given the level of
significance, α, determine the critical value of t, t
t
or if t
< - t
.
/2
0
/2
If you test for the value Β
you do not reject the null hypothesis, H
linear regression is in doubt.
support the assertion that Β ≠ 0.
significance of the regression model.
^
y
= a + b⋅x
= least-square prediction of the data.
i
i
^
= y
-
y
= y
- (a + b⋅x
i
i
i
i
S
(
S
)
yy
xy
2
bx
)]
i
n
2
b − (t
< Β < b + (t
)⋅s
/√S
n-2, /2
e
xx
2
1/2
< Α < a + (t
/S
]
xx
: Β = Β
, tested against the alternative hypothesis, H
0
= (b -Β
)/(s
0
0
= 0, and it turns out that the test suggests that
0
: Β = 0, then, the validity of a
0
In other words, the sample data does not
Therefore, this is a test of the
).
i
2
/
S
n
1
xx
2
s
1 (
y
n
2
)⋅s
/√S
n-2, /2
e
2
⋅[(1/n)+x
)⋅s
/S
n-2, /2
e
/√S
), where t follows the
e
xx
The test is carried out as
, then, reject H
/2
0
Page 18-52
2
r
)
xy
,
xx
1/2
]
,
xx
:
1
if t
>
0
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