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S
xy
=
n
i
=1
x
i
y
i
−
n
i
=1
x
i
n
i
=1
y
i
n
S
xx
=
n
i
=1
x
i
2
−
n
i
=1
x
i
2
n
S
yy
=
n
i
=1
y
i
2
−
n
i
=1
y
i
2
n
ˆ
y
= ˆ
β
0
+ ˆ
β
1
x
where
ˆ
β
1
=
S
xy
S
xx
and
ˆ
β
0
= ¯
y
−
ˆ
β
1
¯
x
Hypothesis testing for intercept,
β
0
Hypothesis testing for slope,
β
1
For
β
0
= 0
,
For
β
1
= 0
,
t
test
=
ˆ
β
0
−
β
0
se
( ˆ
β
0
)
=
ˆ
β
0
MS
Res
(
1
n
+
¯
x
2
S
xx
)
∼
t
α,ν
t
test
=
ˆ
β
1
−
β
1
se
( ˆ
β
1
)
=
ˆ
β
1
MS
Res
(
1
S
xx
)
∼
t
α,ν
where
ν
=
n
−
2
SS
R
= ˆ
β
1
S
xy
MS
Res
=
S
yy
−
ˆ
β
1
S
xy
n
−
2
Goodness of Fit Test
Expected Frequency:
χ
2
test
=
k
i
=1
(
O
i
−
E
i
)
2
E
i
∼
χ
2
α,ν
E
i
=
np
i
free distribution:
ν
=
k
−
1
where
p
i
is a probability for
i
= 1
, . . . , k
hypothesised distribution:
ν
=
k
−
m
−
1
Test of Contingency Tables
E
ij
=
n
i.
n
.j
n
..
χ
2
test
=
r
i
=1
c
j
=
i
(
O
ij
−
E
ij
)
2
E
ij
∼
χ
2
α,ν
where
ν
= (
r
−
1)(
c
−
1)
9
Statistical Tables and Formulae 2.0