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,
2
SSx1 x1 SSx2 x2 − SSx1 x2 c22 = SSx1 x1
,
2
SSx1 x1 SSx2 x2 − SSx1 x2 Alternative hypothesis is
Onetailed test
Ha = {βi < 0} (or Ha = {βi > 0}) and reject region is
RR = {t < −tα } (or Ha = {t > tα }), where tα are based on (n − k − 1) degree of freedom and we can ﬁnd it
using Table VI (McClave and Benson).
Twotailed test
Ha = {βi = 0}
and reject region is
RR = {t > tα/2 }.
A 100(1 − α)% conﬁdence interval.
The conﬁdence interval for a parameters βi is
ˆ
βi ± tα/2 sβi ,
ˆ
where tα/2 is based on (n − 3) degree of freedom.
Remark. If the test of H0 = {βi = 0} fails to reject for some i, this
does not mean that we should omit the βi . Several conclusions are possible:
1. there is no relationship between y and xi ;
2. a type II error occurred;
3. a relationship between y and xi exists, but is more complex than
straightline relationship.
Multiple coeﬃcient of determination R2 for the straightline model is
SSE
=1−
R =1−
SSyy
2 12 n
¯2
ν =1 (yν − yν )
.
n
(yν − y )2
¯
ν =1 A. Zhensykbaev Regression Testing global usefulness of the model.
Consider the global F test to indicate that the model is useful for prediction. The null hypothesis is
H0 = {β1 = β2 = · · · = βk = 0}.
Alternative hypothesis is
Ha = {at least one βi = 0}.
Test statistics is R2 /k
,
(1 − R2 )/(n − k − 1)
where n  sample size, k  number of variables in the model.
F= RR = {F > Fα }
where Fα is is deﬁned from the Tables VIIIXI (McClave and Benson) with
ν1 = k and ν2 = n − k − 1.
Prediction.
The prediction interval for the mean value of y for speciﬁc values of x1 ,...,
xk of level (P = 1 − α) is
k ˆ
βi (xi − xi ) ± tα/2 σy ,
¯
ˆ y+
¯
i=1 where tα/2 is deﬁned from Table VI with df = n − k − 1,
σy = σ
ˆ
ˆ 1
+
n k k cij (xi − xi )(xj − xj ),
¯
¯
i=1 j =1 and for k = 2 the quantities c11 , c22 was deﬁned early,
c12 = c21 = SSx1 x2
.
2
SSx1 x1 SSx2 x2 − SSx1 x2 Example 5. Supp...
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 Spring '13
 ChristopherStocker
 Math, Linear Regression, Regression Analysis

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