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Week 7 Practice Problems
1.
In the production of a certain chemical, the yield (y) of the process (measured as %)
is approximately a linear function of the amount of catalyst (x) used. In a laboratory
experiment, 10 levels of catalyst were used with two replicates at each level. The data
are
Catalyst (x)
Yield (y)
Catalyst (x)
Yield (y)
0.3
46.9
0.8
53.9
0.3
50.6
0.8
52.8
0.4
50
0.9
55.3
0.4
48.6
0.9
53.8
0.5
50.1
1
58.9
0.5
50.1
1
60.9
0.6
52.2
1.2
58.7
0.6
50
1.2
61.8
0.7
52.4
1.4
63.4
0.7
57.9
1.4
62.1
To save you calculation time, we provide the following summaries:
22
ˆ
0.78,
54.42,
(
)
2.238,
(
)
30.458,
65.220
ii
i
i
x
y
xx
y
r
==
−
=
−
=
=
∑∑
∑
catalyst
yield
1.4
1.2
1.0
0.8
0.6
0.4
0.2
65
60
55
50
45
Yield vs Catalyst Level
Consider the Gaussian response model to describe the experiment.
(
)
,
~
(0, ),
1,.
..,20independent
i
i
Yx
x
R
R
G
i
α
βσ
=+
− +
=
a)
What do the parameters
and
β
represent?
b)
Find the MLEs of
and
(you can do this from scratch for practice if you
wish – otherwise use the formulas given in class). Add the fitted line to the
scatterplot.
c)
Find the estimate of
σ
. What are the degrees of freedom in this case?
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View Full Documentd)
Find a 95% confidence interval for
β
. How would you interpret this interval
formally?
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 Spring '10
 McKay

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