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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+βx
0
:
a+b⋅x−(t
n-2,α/2
)⋅s
e
⋅[(1/n)+(x
0
-⎯x)
2
/S
xx
]
1/2
< α+βx
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
<