HP 48GII, Руководство пользователя, Страница: 623 / 864

Page 18-52
Let y
i
= actual data value,
^
y
i
= a + b
⋅
x
i
= least-square prediction of the data.
Then, the prediction error is: e
i
= y
i
-
^
y
i
= y
i
- (a + b
⋅
x
i
).
An estimate of
σ 2
is the, so-called, standard error of the estimate,
) 1 (
2
1
2
/ ) (
)] ( [
2
1
2 2
2
2
1
2
xy y
xx xy yy
i
n
i
i e
r s
n
n
n
S S S
bx a y
n
s − ⋅ ⋅
−
−
=
−
−
= + −
−
=
∑
=
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
): b
−
(t
n-2,
α
/2
)
⋅
s
e
/
√
S
xx
<
Β <
b
+
(t
n-2,
α
/2
)
⋅
s
e
/
√
S
xx
,
For the intercept (
Α
):
a
−
(t
n-2,
α
/2
)
⋅
s
e
⋅
[(1/n)+

x
2
/S
xx
]
1/2
<
Α
< a
+
(t
n-2,
α
/2
)
⋅
s
e
⋅
[(1/n)+

x
2
/S
xx
]
1/2
,
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
0
, tested against the alternative hypothesis, H
1
:
Β ≠ Β
0
. The test statistic is t
0
= (b -
Β
0
)/(s
e
/
√
S
xx
), 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. The test is carried out as
that of a mean value hypothesis testing, i.e., given the level of
significance,
α
, determine the critical value of t, t
α
/2
, then, reject H
0
if t
0
>
t
α
/2
or if t
0
< - t
α
/2
.
If you test for the value
Β
0
= 0, and it turns out that the test suggests that
you do not reject the null hypothesis, H
0
:
Β
= 0, then, the validity of a
linear regression is in doubt. In other words, the sample data does not
support the assertion that
Β ≠
0. Therefore, this is a test of the
significance of the regression model.