STAT 420
Fall 2007
Homework #2
(due Friday, September 7, by 3:00 p.m.)
1.
Sometimes it is known in advance that the leastsquares regression line must go
through the origin, i.e., the regression model is of the form
Y
i
=
β
x
i
+
ε
i
,
i
= 1, 2, … ,
n
, where
ε
i
’s are i.i.d.
N
(
0,
σ
2
)
,
and the equation of the regression line is
y
ˆ
=
°
ˆ
x
. In this case, finding the
leastsquares line reduces to finding the value
°
ˆ
that minimizes the expression
(
)
[
]
°
=

=
n
i
i
i
x
y
f
1
2
°
°
.
Use the derivative of
f
with respect to
β
to derive the formula for the slope of
the leastsquares regression line in this case.
2.
It has been proposed that the brightness measured in some unit of color or a
commercial product is proportional to the time it is in a certain chemical reaction
during the production process, or
Y
i
=
β
x
i
+
ε
i
,
i
= 1, 2, … ,
n
, where
ε
i
’s are i.i.d.
N
(
0,
σ
2
)
,
where
Y
i
measures brightness,
x
i
measures time, and
β
is a parameter. The
following data on
x
and
Y
are available:
x
1.0
1.2
1.4
1.6
1.8
2.0
Y
3.4
7.0
6.0
9.0
8.0
11.0
a)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 STEPANOV
 Statistics, Least Squares, Linear Regression, Regression Analysis, Variance, Errors and residuals in statistics

Click to edit the document details