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1
Chapter 3 ANOVA and F-test
H
0
= E(Y|X=x) = β
0
H
a
= E(Y|X=x) = β
0
+ β
1
x
1
+ … + β
p
x
p
Equivalently
H
0
= β
1
= β
2
= … = β
p
= 0
H
a
= At least one out of β
1
,…, β
p
is not equal to zero
F-Statistic
F=
??
???
/
?
±??
/(
²−?−
1)
The coefficient of Determination
±
2
=
??
???
±??
= 1
−
±??
?³³
R
2
is also called multiple correlation coefficient
Two important results on R
2
(1)
±
2
=
´µ??
2
(
³
,
³
)
(2)
±
2
=
¶°·
¸
´µ??
2
(
³
,
¹¸
)
Hypotheses on one coefficient only
H
0
= β
1
= 0
β
0
, β
2
,…, β
p
arbitrary
H
a
= β
1
≠ 0
β
0
, β
2
,…, β
p
arbitrary
t-statistic
º
=
»
¼
1
−
0
½?¾»
¼
1
¿
~
ºÀ² − ? −
1
Á
Equivalently
1. Fit the model to data
³
=
»
0
+
»
1
·
1
+
…
+
»
?
·
?
+
?
Get RSS

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2. Fit the model without x
1
to data
?
=
±
0
+
±
2
?
2
+
…
+
±
?
?
?
+
?
Get RSS
-1
3. RSS
-1
– RSS
F-Statistic =
²³³
−
1
−²³³
/1
²³³
/(
´−?−
1)
~
µ
(1,
´ − ? −
1)
Sequential Analysis of Variance
SS
reg
can be divided into sum of squares “explained” by each variable

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