HP 50G, User Manual

Page: 620 / 887

Page 18-53
Θ
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.
Θ
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.
Θ
Hypothesis testing on the intercept ,
Α:
Null hypothesis, H
0
:
Α
=
Α
0
, tested against the alternative hypothesis, H
1
:
Α ≠ Α
0
. The test statistic is t
0
= (a-
Α
0
)/[(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. 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
.
Θ
Confidence interval for the mean value of Y at x = x
0
, i.e.,
α
+
β
x
0
:
a+b
⋅
x
−
(t
n-2,
α
/2
)
⋅
s
e
⋅
[(1/n)+(x
0
-
⎯
x)
2
/S
xx
]
1/2
<
α
+
β
x
0
<
a+b
⋅
x+(t
n-2,
α
/2
)
⋅
s
e
⋅
[(1/n)+(x
0
-
⎯
x)
2
/S
xx
]
1/2
.
Θ
Limits of prediction: confidence interval for the predicted value Y
0
=Y(x
0
):
a+b
⋅
x
−
(t
n-2,
α
/2
)
⋅
s
e
⋅
[1+(1/n)+(x
0
-
⎯
x)
2
/S
xx
]
1/2
< Y
0
<