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FOUNDATIONAL CONCEPTS Model Specification: What is SLR? y = β₀ + β₁x + ε Response vs. predictor variable What does the line represent? (conditional mean) Slope interpretation. Intercept interpretation.
Typology: Slides
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Professor
School of Industrial and Systems Engineering
Learning Objectives:
properties of the model parameters
in simple linear regression
regression coefficients to a data
example in R
= 0
For the slope parameter β 1
, we can show
$ β 1
=
∑ x i
−
" x Y i
S xx
= c i
Y i
but x i
fixed →
x
i
−"x
S xx
= c i
fixed
Ε
$ β 1
= Ε 5
i=
n
c i
y i
= 5
i=
n
c i
Ε y i
= 5
i=
n
c i
β 0
+β 1
x i
= β 0
5
i=
n
c i
5
i=
n
c i
x i
= β 1
→ Ε
$ β 1
= β 1
= 1
β 1
) = β 1
Var(
β 1
2 / S xx
I’m unbiased! J
Furthermore, is a linear combination of {Y 1
n
}. If we assume that
e i
~ Normal (0, A
!
) , then is also distributed as
"
&
' '
!!
"
i=
m
c i
i
a linear combination of normally distributed random variables
"
~ Normally distributed
"
"
"
$$
%&!
t–interval for C "
"
'
,&'$
Estimate
of 9
t-critical
point
Standard
Deviation/Error of
: 9
Sampling
distribution
of
: 9
is t n−
1 - α
Confidence
interval
V
: 9
=
σ
2
S xx
σ
2 ← MSE
One way we can test statistical significance is to use the t-test for
o
: β 1
= 0 vs. H a
: β 1
We reject H o
if |t-value| is large. If the null hypothesis is rejected, we
being statistically significant.
t−value =
"
! / ''
"
''
relationship between ] and ^ is
significant.
(
"
"
"
"
= 0. It means we cannot
express ] in terms of ^.
B !
: 9 "
= 0
B "
: 9 "
≠ 0
=
$ !
%!
"#$
%&&
=
$ !
se
$ !
~H ('%()
(“se” means “standard error”)
(
('%()
⟹ Reject B !
> H !
< R ⟹ Reject B !
How will the procedure change if we test:
"
= 0 versus `
"
"
= 0 versus `
"
What if we want to test for positive relationship
"
≤ 0 versus `
"
P-value = P(T n- 2
> t-value)
What if we want to test for negative relationship
"
≥ 0 versus `
"
P-value = P(T n- 2
< t-value)
!
!
"
$
$
%
&
= +
!!
"
#
$
%&'
!
!
"
$
%
Confidence interval :
(
!
,%&!
j
!
''
(
!
,%&!
j
!
''
)
"
m
(
= m
n o − m(
"
(
A company, which sells medical supplies to hospitals, clinics, and doctor's
offices, had considered the effectiveness of a new advertising program.
Management wants to know if the advertisement is related to sales. This
company intends to increase the sales with an effective advertising
program.
What inferences can be made on the regression coefficients?
a. What is the estimate of the coefficient C "
and its variance? What is its
sampling distribution?
b. What is the estimate of the coefficient C (
and its variance?
c. Is the coefficient C "
statistically significant? What is the p-value of the test?
Interpret.
d. Is the coefficient C "
statistically positive? What is the p-value of the test?
Interpret.
e. Obtain the 99% confidence interval for C "
f. What is the p-value of a hypothesis testing procedure?
