Se y σ y σ y n which is an estimator of the standard

SE
(
´
Y
)
=
^
σ
´
Y
=
σ
Y
√
n
which is an estimator of the standard deviation of the sampling distribution of
´
Y .
2. Compute the
t
-statistic
t =
´
Y − μ
Y , 0
SE (
´
Y )
=
´
Y − μ
Y , 0
s
Y
√
n
3. Compute the p -value, which is the smallest significance level at which the null hypothesis could
be rejected, based on the test statistic actually observed; equivalently, the p -value is the
probability of obtaining a statistic, by random sampling variation, at least as different from the
null hypothesis value as is the statistic actually observed, assuming that the null hypothesis is
correct. In a few words,
if p -value
¿ α ,reject H
0
-
Key Concept 5.1: General Form of the t -Statistic
In general, the t -Statistic has the form
t =
estimator − hypothesized value
standard error of the estimator
=
^
β − β
0
SE
(
^
β
)
, ( 5.1 )
Testing hypotheses about the slope
β
1
The null and alternative hypothe s es need to be s tated precisely before they c a n be tested . The angry
taxpayer ' s hypothesis is that
β
R ∧ D
= 0
.

More generally , under the null hypothesis the true population slope
β
1
takes on some specific value ,
β
1,0
. Under the two-sided alternativ e,
β
1
does not equal
β
1,0
. That is , the null hypothesis and
the two-sided alternative hypothesis are
H
0
: β
1
= β
1,0
H
1
: β
1
≠ β
1,0
,
(
5.2
)
 The first step is to compute the standard error of
^
β
1
, SE
( ^
β
1
)
. The standard error of
^
β
1
is an estimator of
^
σ
^
β
1
, the standard deviation of the sampling distribution of
^
β
1
. Specifically,
SE
( ^
β
1
)
=
√
σ
^
β
1
2
, ( 5.3 )
where
σ
^
β
1
2
=
1
n
×
1
n − 2
∑
i = 1
n
(
X
i
−
´
X
)
2
^ u
i
2
[
1
n
∑
i = 1
n
(
X
i
−
´
X
) 2
]
2
, ( 5.4 )
Although the formula for
σ
^
β
1
2
is complicated, in applications the standard error is
computed by regression software so that it is easy to use in practice.
 The second step is to compute the t -statistic,
t =
^
β
1
− β
1,0
SE
( ^
β
1
)
, ( 5.5 )
 The third step is to compute the p -value , the probability of observing a value of
^
β
1
at
least as different from
β
1,0
as the estimate actually computed
^
β
1
act
, assuming that the
null hypothesis is correct. Stated mathematically,
p − value = Pr
H
0
[
|
^
β
1
− β
1,0
|
]
> Pr
H
0
[
|
^
β
1
act
− β
1,0
|
]
¿ Pr
H
0
[
|
^
β
1
− β
1,0
SE
( ^
β
1
)
|
]
> Pr
H
0
[
|
^
β
1
act
− β
1,0
SE
( ^
β
1
)
|
]
= Pr
H
0
(
|
t
|
>
|
t
act
|
)
, ( 5.6 )
where
Pr
H
0
denotes the probability computed under the null hypothesis, the second
equality follows by dividing by
SE
( ^
β
1
)
, and t
act
is the value of the t -statistic actually
computed.