•
Example:
H
0
:
β
=
c
, where
c
is a value of interest in the
context of a regression model.
•
A common value for
c
is 0
Components 2 : Alternative Hypothesis
2
Alternative Hypothesis
H
1
:
•
Logical alternative hypothesis paired with the null
H
0
(that we will “accept” if the null hypothesis is rejected)
•
Three possible types of
H
1
for
H
0
:
β
=
c
:
1
H
1
:
β
6
=
c
2
H
1
:
β >
c
3
H
1
:
β <
c
•
Testing
H
0
:
β
≥
c
versus
H
0
:
β <
c
is equivalent to testing
H
0
:
β
=
c
versus
H
0
:
β <
c
.
•
Testing
H
0
:
β
≤
c
versus
H
0
:
β >
c
is equivalent to testing
H
0
:
β
=
c
versus
H
0
:
β >
c
.
Summary
Interval Estimation
Hypothesis Testing
Examples of Hypothesis Tests
Key Words
Components 3 : The Test Statistic
3
Test statistic:
statistic (realized random variable) whose probability
distribution is completely
known
when the null hypothesis is
true
!
•
We know that
t
=
b

β
b
se
(
b
)
∼
t
(
N

2)
has a tdistribution based on our model assumptions.
•
Thus,
if
H
0
:
β
=
c
is true
, it follows that:
t
=
b

c
b
se
(
b
)
∼
t
(
N

2)
•
If the null hypothesis is
not true
, then the
t
statistic above
does not have a
t
distribution.
Rodrigo Pinto
Hypothesis Testing
Component 4: Critical Value and Rejection Region
4
Critical Value and Rejection Region:
•
The alternative hypothesis (and the significance level
α
) are
used to define a critical value
t
c
•
This critical value defines a Rejection Region that depends on
the Alternative Hypothesis
•
H
1
:
β
k
6
=
c
implies a
double
sided region (twotail)
•
H
1
:
β
k
>
c
implies a
right
sided region (righttail)
•
H
1
:
β
k
<
c
implies a
left
sided region (lefttail)
:
:
Onetail Test for
H
0
:
β
=
c
versus
H
1
:
β <
c
1
To test the Null
H
0
:
β
k
=
c
,
against Alternative
H
1
:
β
k
<
c
,
2
Rejection Rule:
ˆ
t
=
ˆ
b

c
b
se
(
b
)

{z
}
Test Statistic
<
t
c
(
α,
N

2)

{z
}
Critical Value
t
(
m
)
Reject
H
0
:
β
k
c
Do not reject
H
0
:
β
k
c
0
t
c
t
(
α
,
N
2)
α
3.3
The rejection region for a onetail test of
H
:
b
¼
c
against
H
:
b
<
c
.
Twotail Test for
H
0
:
β
=
c
against
H
1
:
β
6
c
c
c
c
2)
Interpreting the Rejection Region and Significance Level
•
The rejection region consists of values that are unlikely and that
have low probability of occurring when the null hypothesis is true
•
The chain of logic is:
“
If a value of the test statistic is obtained that falls in a region of
low probability, then it is unlikely that the test statistic has the
assumed distribution, and thus it is unlikely that the null hypothesis
is true
”
•
If the alternative hypothesis is true, then values of the test statistic
will tend to be unusually large (or unusually small)
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 Winter '07
 SandraBlack
 Statistical hypothesis testing