STAT 404- Assignment 5
Due Friday, November 27 at 11:59pm. Bonus 0.5 marks for presentation.
[4 marks] 1. Repeat the in-class example on ”fabricating integrated circuit”. An
initial step of this process is to grow an epitaxial layer on polished
silicon wafers. It is desired to have the actual thickness as close to 14.5
µm, and within the limits [14, 15].
run D C B A y¯ s2
1 - - - - 14.59 .270
2 + - - - 13.59 .291
3 - + - - 14.24 .268
4 + + - - 14.05 .197
5 - - + - 14.65 .221
6 + - + - 13.94 .205
7 - + + - 14.40 .222
8 + + + - 14.14 .215
9 - - - + 14.67 .269
10 + - - + 13.72 .272
11 - + - + 13.84 .220
12 + + - + 13.90 .229
13 - - + + 14.56 .227
14 + - + + 13.88 .253
15 - + + + 14.30 .250
16 + + + + 14.11 .192
Replace the numbers in columns y¯ and s2 with
ybar = c(14.58, 13.59, 14.29, 14.05, 14.59, 13.97, 14.43, 14.11,
14.81, 13.81, 13.73, 13.90, 14.44, 13.94, 14.36, 14.18)
ss = c(0.272, 0.289, 0.263, 0.196, 0.221, 0.207,, 0.226 0.212, 0.269,
0.268, 0.217, 0.228, 0.224,, 0.258 0.255, 0.195)
(a) [1] Construct the ANOVA table including all effects for the mean
response (the class example omitted some for the ease of presentation).
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(b) [1] Identify the significant effects based on F-test.
(c) [1] Plot all main effects on the half-normal plot.
(d) [1] Construct a 95% two-sided CI for the main effect of D.
(e) (Not to be handed in) Go over all analyses as you please.
[6 marks] 2. An experimenter obtained eight yields (yield = response) for the design
given in the following table. Note that the factors are names as 1, 2,
3, 4, and 5 instead of A, B, C, D, and E. There are no replicates.
1 2 3 4 5 yield
− − − − − 26.9
+ + − − − 28.4
− + + − + 21.4
+ − + − + 20.4
− + + + − 18.3
+ − + + − 17.2
− − − + + 31.8
+ + − + + 28.6
(a) [1] Create two interaction plots for factors 3 and 4: one for 3 against
4, and the other for 4 against 3.
(b) [2] Obtain estimates for the main effects of 3 and 4, and the 2-factor
interaction effect 3 × 4.
(c) [1] If the standard deviation of each observation (i.e., each run) is
1.2, what is the standard deviation of the estimators in (b)?
(d) [1] Based on the results in (a)-(c), which of the three effects in (b)
are significant?
(e) [1] Identify a main or 2-factor interaction that is aliased with the
2-factor interaction 4 × 5.
Hint: the terms high and low only have symbolical meaning. Pay
attention to the high and low of factor 3.
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[4 marks] 3. An experimenter considers running a 27−3 design with two possibilities:
design A with generators
I = 1235 = 1246 = 12347
and design B with generators
I = 1235 = 1246 = 1347.
(a) [1] Find all members in the defining contrast subgroup in each
design. Note each design contains 7 words in addition to the identity I.
That is, work out the other 4 members (called words) in each design.
(b) [1] What are the resolutions of these two designs?
(c) [1] Design with higher resolutions are preferred. Accordingly, which
design is a better design?
(d) [1] What effects are confounded with the main effect of factor 4 in
design A?
[6 marks] 4. A door panel stamping experiment was conducted to study the effects
of 6 factors on the formality of a panel. One measure of formality is
the thinning percentage of the stamped panel at a critical position.
The 6 factors, each at 2 levels, are lubricant concentration (A), panel
thickness (B), force on outer portion of the panel (C), force on inner
portion of the panel (D), punch speed (E), and lubricant thickness (F).
The experiment was done in two days. “Day” was considered as a
blocking factor (G) to reduce the influence of day-to-day variation,
with “−” representing day 1 and “+” day 2.
The experiment had a 27−2 resolution IV design with
I = ABCDEF = −ABFG = −CDEG
(We have k = 6, p = 1 and b = 1 in our notation, yet it is called 27−2
because G is regarded as a factor). The design matrix and response y
data are in the file stamping.dat on Canvas.
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(a) [2] Find all order-2 and order-3 interactions that are aliased with
factor A, with factor B, and with 2-factor interaction AB respectively.
Which interactions are left if the ones containing blocking factor G are
removed?
(b) [2] Because the logit transformation logit(y) = log(y/(1− y)) maps
the interval (0, 1) to the real line, it is often used for percentage data.
Analyze logit(y) values in terms of the factorial effects and the block
effect.
(c) [1] Use half-normal plot to identify significant effects (based on your
discretion).
(d) [1] What would be your recommended factor settings to reduce/minimize
percentage thinning and why?
Remember: statistics is like an art sometimes when working on (c) and
(d).
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