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Chapter 14
Simple Linear Regression
14.1
Preliminary Remarks
We have only a short time to introduce the ideas of regression
.Tog
iveyousomeideahowla
rge
the topic of regression is, The Department of Statistics offers a onesemester course on it, Statistics
333.
The topics we
could
cover fall into two broad categories: results; and checking assumptions.
We will have time for results only and that is all that will be ontheFnalexam.IfIevergetaround
to writing Chapter 15, it will address the issue of checking assumptions.
14.2
The Simple Linear Regression Model
±or each unit, or case, as they tend to be called in regression,wehavetwonumbers,denotedby
X
and
Y
.Thenumberofgrea
terin
teres
ttousisdeno
tedby
Y
and is called the
response
.
Predictor
is the common label for the
X
variable. Very roughly speaking, we want to study whether there is
an association or relationship between
X
and
Y
,withspecialinterestinthequestionofusing
X
to
predict
Y
.
It is very important to remember (and almost nobody does) thattheideaofexper
imen
ta
land
observational studies introduced in Chapter 9 applies here too, in a way that will be discussed
below.
We have data on
n
cases. When we think of them as random variables we use upper case letters
and when we think of speciFc numerical values we use lower casele
t
ters
. Thus
,wehavethe
n
pairs
(
X
1
,Y
1
)
,
(
X
2
,Y
2
)
,
(
X
3
,Y
3
)
,...
(
X
n
,Y
n
)
,
which take on speciFc numerical values
(
x
1
,y
1
)
,
(
x
2
,y
2
)
,
(
x
3
,y
3
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Experimental
:Theresearcherde
l
ibera
te
lyse
lec
tstheva
luesof
X
1
,X
2
,X
3
,...X
n
.
2.
Observational
:Theresearcherse
lec
tsun
i
ts(usua
l
lyassumedtobea
trandom from a popu
lation or to be i.i.d. trials) and observes the values of two random variables per unit.
Here are two very quick examples.
1.
Experimental
:Theresea
rche
risin
te
res
tedinthey
ie
ld
,pe
rac
re
,o
face
rtain crop. Denote
the yield by
Y
.There
sea
rche
rbe
l
ieve
stha
tthey
ie
ldw
i
l
lbea
f
fec
tedbythe concentration
of, say, a certain fertilizer that will be applied to the plant. The values of
X
1
,X
2
,X
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 Fall '11
 hanlon
 Statistics, Linear Regression

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