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Suppose that (
x
i
,y
i
)
i
= 1
,...,n
denote the height in inches of adult male students and their fathers. If
we plot (
x
i
,y
i
)
i
= 1
,...,n
we would see a linear pattern with taller fathers having taller sons. This linear
pattern is not perfect, since there are factors other than father’s height that determine the height of a son.
Yet we feel that to some extent, the height of the sons is a least in part determine by the fathers height.
Thus our model for the height of the sons is of the form:
y
i
=
β
o
+
β
1
x
i
+
²
i
where
x
i
is the height of father
i, β
o
and
β
1
are unknown constants,
²
i
is a random error with mean zero,
and
y
i
is the height of son
i.
Given data (
x
i
,y
i
)
i
= 1
,...,n
the problem is to estimate
β
o
and
β
1
and then to test or validate the
linear model. Once we obtain estimates of
β
o
and
β
1
and validate the model, we can use it to estimate the
average height of sons whose fathers are 62 inches tall, or to ﬁnd a conﬁdence interval of the actual height of
a son whose father is 62 inches tall. In this example the height of the father is the predictor, or independent,
variable and the height of the son the response, or independent, variable
As a second example consider a model to predict the performance of college students in ﬁrst year calculus.
The prediction may be based on a placement exam score, high school GPA and SAT scores. In this case the
model may be of the form
y
i
=
β
o
+
β
1
x
i
1
+
β
2
x
i
2
+
β
3
x
i
3
+
²
i
where
x
i
1
, x
i
2
,
and
x
i
3
represent the placement exam score, the high school GPA and the SAT score of
student
i
and
y
i
represents the ﬁrst year calculus grade of student
i.
In this example, the placement exam
score, the GPA, and the SAT score are predictor, or independent variables, and the ﬁrst year calculus grade
is the response, or independent, variable.
As a third example, consider the situation
y
i
=
β
o
+
β
1
x
i
+
β
2
x
2
i
+
...
+
β
k
x
k
i
+
²
i
.
Here the response variable
y
i
depends on the predictor variable
x
i
and its ﬁrst
k
powers.
There are also some nonlinear models that can be transformed into a linear model. For example, if
y
i
=
e
β
o
+
β
1
x
i
+
²
i
then ln
y
=
β
o
+
β
1
x
i
+
²
i
.
All of the above models are of the form
y
i
=
f
(
x
1
i
,...,x
i,p

1
) +
²
i
where
f
(
x
1
,...,x
p

1
) =
β
o
+
β
1
x
1
+
...
+
β
p

1
x
p

1
and
²
i
is a mean zero random variable. For the most part we will assume that the
²
i
s are independent and
have a common variance
σ
2
.
Given data (
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 Spring '04
 GALLEGO

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