Sol  1
Chapter 4 Solutions
1.
Consider the assessment data with simple model
value
age
size
residual
=+
+
+
β
01
2
. Use the methods in this chapter to assess the
fit of the model and to suggest remedies. Is the prediction of value for a building with
size 13.9 and age 30 sensitive to any particular cases?
We start by fitting the simple model and looking at various plots of the residuals and the
qq plot of the standardized residuals to examine the fit. Note, for this fit, the 95%
prediction interval for age=30, size=13.9 is 46.7 to 21.6, an interval so wide that it is
useless.
The one common feature of all of the plots is the two large residuals both of which
correspond to a large fitted value and relatively small age and size. The
qq plot of the
standardized residuals is not linear but is highly distorted by the two large standardized
residuals. If we delete these two points and repeat the fit we get the following plots.
There is no evidence against the fit in any of these plots. With the two points deleted, the
prediction interval is 17.0 to 25.6, much narrower but still not useful. The bottom line
here is that it is not feasible to use these data to assess a building that is so much larger
than any other in the sample.
Adapted from
Stat 371 course notes
© R.J. MacKay, University of Waterloo
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2.
In an experimental Plan, there were three explanatory variates
x
x
x
123
,,
t
h
a
t
e
a
c
h
were assigned two values, here coded as 1 and +1. There are 8 combinations.
As
well, the investigators looked at the response variate for the socalled center point
x
x
x
00
===
,
,
0
. The data are shown below and can be found in the file
c
h5Exercise2.txt.
x
1
x
2
x
3
y
1
1
1
11.54
1
1
1
5.45
1
1
1
7.34
1
1
1
17.21
1
1
1
17.87
1
1
1
9.40
1
1
1
11.30
0
0
0
11.57
0
0
0
11.97
0
0
0
11.89
0
0
0
12.15
Suppose we fit a model
y
x
x
x
r
=+
+
+
+
β
01
12
23
3
. The summary output from R is
Call:
lm(formula = y ~ x1 + x2 + x3)
Residuals:
Min
1Q
Median
3Q
Max
1.2527 0.1360
0.1465
0.4419
0.7615
Coefficients:
Estimate
Std. Error
t value
Pr(>t)
(Intercept)
11.6181
0.2372
48.985
3.87e10 ***
x1
2.4654
0.3000
8.218
7.68e05 ***
x2
3.1071
0.3000
10.357
1.70e05 ***
x3
0.5329
0.3000
1.776
0.119

Signif. codes:
0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
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 Spring '11
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 Regression Analysis, R.J. MacKay, notes© R.J. MacKay, course notes© R.J.

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