Stat 312: Lecture 22
Inference on Correlation
Moo K. Chung
[email protected]
April 17, 2003
Concepts
1. Assume that
X
and
Y
are two random variables
whose joint probability density function
f
is the
bivariate normal
such that
X
∼
N
(
μ
X
, σ
2
X
)
,
Y
∼
N
(
μ
Y
, σ
2
Y
)
and
ρ
(
X, Y
) =
ρ
. In this situation, we
can show that
(
Y

X
=
x
) =
β
0
+
β
1
x
, where
β
1
=
ρσ
Y
/σ
X
. This is the regression model we
studied!
2. Testing
H
0
:
ρ
= 0
vs.
H
1
:
ρ
6
= 0
is equiva
lent to
H
0
:
β
1
= 0
vs.
H
1
:
β
1
6
= 0
.
So the
appropriate test statistic for testing
H
0
:
ρ
= 0
is
T
=
ˆ
β
1
/S
ˆ
β
1
∼
t
n

2
(see Lecture 19, Concept 3).
This test statistic is equivalent to
T
=
r
√
n

2
√
1

r
2
.
3. Testing for independence is equivalent to testing
ρ
= 0
in bivariate data.
Inclass problems
Exercise 12.67.
A sample of 10,000
(
x
i
, y
i
)
pairs re
sulted in the sample correlation coefficient
r
= 0
.
022
.
Test
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This note was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Spring '04 term at University of Wisconsin.
 Spring '04
 Chung
 Statistics, Correlation, Probability

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