Statistics 512: Problem Set No. 0
Solutions
1. A regression analysis relating test scores (
Y
) to training hours (
X
) produced the following
ﬁtted question: ˆ
y
= 25

0
.
5
x
.
(a) What is the ﬁtted value of the response variable corresponding to
x
= 7?
Solution:
The ﬁtted value at
x
= 7 is
ˆ
y
= 25

(0
.
5)(7) = 25

3
.
5 = 21
.
5
.
(b) What is the residual corresponding to the data point with
x
= 3 and
y
= 30?
Solution:
The ﬁtted value corresponding to the data point with
x
= 3 is
ˆ
y
= 25

(0
.
5)(3) = 25

1
.
5 = 23
.
5
.
The residual corresponding to the data point with
x
= 3 and
y
= 30 is thus
e
i
= 30

23
.
5 = 6
.
5
.
(c) If
x
increases 3 units, how does ˆ
y
change?
Solution:
For increase of 1 in
x
, ˆ
y
changes by the slope. Therefore, if
x
increase
by 3 units, ˆ
y
will decrease by 3 times the slope, i.e. by (3)(

0
.
5) =

1
.
5.
(d) An additional test score is to obtained for a new observation at
x
= 6. Would the test
score for the new observation necessarily be 22? Explain.
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 Spring '08
 Staff
 Statistics, Regression Analysis, fitted value, new observation, ﬁrst equation

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