Where c is a constant and is an important value in the context of specific regression
model
Step-by-step procedure for testing hypothesis
1.
Determine the null and alternative hypothesis
2.
Specify the test statistic and its distribution if the null hypothesis is true
3.
Select
α
and determine the
rejection region
4.
Calculate the sample value of the test statistic
5.
State your conclusion
Testing the Significance of a linear relationship
o
We can test the significance of a linear relationship between X and Y by testing:
H
0
: β
2
= 0
H
1
: β
2
≠ 0
The distribution of the slope coefficient estimator, b
2
, and the test statistic
o
Based on assumptions SR1-SR6 we can show that
b
2
N
(
β
2
,
[
σ
2
∑
(
x
i
−
x
)
2
]
)
o
A standardised normal random variable is obtained from b
2
by subtracting its mean and dividing by its
standard deviation
Z
=
b
2
−
β
2
√
σ
2
∑
(
x
i
−
x
)
2
N
(
0,1
)
o
The standardized random variable Z is normally distributed with mean 0 and variance 1
o
Usually σ
2
is unknown and replaced by an estimate of σ
2
,
^
σ
2
The Test Statistic
o
Replacing σ
2
with
^
σ
2
creates a random variable t:
t
=
b
2
−
β
2
√
^
σ
2
∑
(
x
i
−
x
)
2
t
(
n
−
2
)
Where β
2
= H
0
The ratio has a t-distribution with (n-2) degrees of freedom
y = β
1
+ β
2
x + e (so
β
1
& β
2
means n-2 degrees of freedom)
Hypothesis Tests
o
The test statistic for testing the significance of β
2
H
0
: β
k
= 0
t
=
b
k
−
β
k
se
(
b
k
)
for k
=
1,2
Revisiting the Food Expenditure Example – Testing Significance of the Relationship

o
Earlier we estimated a linear relationship b/w food expenditure and income in form
y = β
1
+ β
2
x + e
where y = food expenditure, and x = income
o
Estimated regression model is given by
^
y
=
83.42
+
10.21
x
o
We want to test the claim that income has a significantly positive influence on food expenditure
In terms of our regression model this can be tested by testing
H
0
: β
2
= 0
H
1
: β
2
> 0
The test statistic to perform this hypothesis test is
t
=
b
2
−
0
se
(
b
k
)
for k
=
1,2
If the null hypothesis is true
N = 40
DF = N-2 = 40 – 2 = 38
Let us select α = 0.05
Critical Value: t
(1-α,N-2)
= t
(0.95,38)
= 1.686
Rejection Region: Reject the null hypothesis
H
0
if t ≥ 1.686
Using the food example data, b2 = 10.21; se(b2) = 2.09. The value of the test statistic is
t
=
b
2
−
0
se
(
b
k
)
=
10.21
2.09
=
4.88
Since t = 4.88 > 1.686
we reject the null H
0
: β
2
= 0 hypothesis and accept the alternative H
1
: β
2
>
0.
That is we reject the null hypothesis that there is no relationship b/w income and food
expenditure at the 5% level of significance
Therefore there is enough evidence to conclude that there is statistically significant positive
linear relationship b/w household income and food expenditure
p-Value
o
Reject
the
null hypothesis
when
p-value is less than or equal to, level of significance α
I.e. if p-value ≤ α, then reject H
0
I.e. if p-value > α, then do not reject H
0
Testing for the significance: p-value for a right-tail test
o
Null hypothesis is
H
0
: β
2
= 0
o

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- Fall '19
- Statistics, Normal Distribution, Regression Analysis, Variance, Statistical hypothesis testing, linear regression model