Recitation #12 and Homework #10: DOX I
Solutions
OR&IE 3120, Spring 2009
Problem 1
The original data are plotted in the top row of the figure below. The alternated
data are in the bottom row. The left column has boxplots, the middle column has normal
plots, and the right column has plots of thickness versus observations number.
Clearly, the outlier distorts the plots so much that it is difficult to see anything else that
might be a problem.
In the bottom, no outliers are seen, which is good. In the bottom, right plot, we see a
pattern with thickness having low values on the left side and having somewhat of a sinusoidal
pattern. Since examination of the data file shows that the data are ordered by run and then
wafers without runs, this indicates there is some betweenrun or betweenwafer variability
that should be investigated.
Note: The
R
specification
par(mfrow=c(2,3))
puts all six plots in one figure.
10
14
18
original data
3
1
0
1
2
3
10
14
18
Normal QQ Plot
Theoretical Quantiles
Sample Quantiles
0
50
100
200
10
14
18
Index
wdata$thickness
10.3
10.5
10.7
altered data
3
1
0
1
2
3
10.3
10.5
10.7
Normal QQ Plot
Theoretical Quantiles
Sample Quantiles
0
50
100
200
10.3
10.5
10.7
Index
wdata$thickness
1
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Problem 2
We see in the output below that
b
μ
= 10
.
5
,
σ
2
u
= 0
.
00146
,
σ
2
v
= 0
.
01057
, and
σ
2
²
= 0
.
00355
.
> VarCorr(chip.lme)
Variance
StdDev
run =
pdLogChol(1)
(Intercept) 0.00146
0.0381
wafer =
pdLogChol(1)
(Intercept) 0.01057
0.1028
Residual
0.00355
0.0596
> fixef(chip.lme)
(Intercept)
10.5
Since
b
σ
v
is nearly twice as large as
b
σ
²
and nearly three times as large as
b
σ
u
, it seems
the variation in thickness is due mainly to between wafer variation within runs. The next
largest variation is between chips within wafers. Between run variation is relatively small.
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 Spring '09
 JACKSON
 Statistics, Normal Distribution, Null hypothesis, 1, Wafer

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