Experimental design (KKR031, KBT120) Tuesday 23/8 2011

Experimental design (KKR031, KBT120)
Tuesday 23/8 2011 - 8:30-13:30 V
Jan Rodmar will be available at ext 3024 and will visit the examination
room at ca 10:30.
The examination results will be available for review at the earliest
Friday 9/9 2011.
Time for examination = 5 h
Examination aids:
Textbook (Douglas C. Montgomery: Design and Analysis of Experiments) with
notes. No calculation examples (in book or on paper) are allowed as aid. All type
of calculators are allowed. Standard Math. Tables, TEFYMA table, Beta
Mathematics Handbook or Handbook of Chemistry and Physics are accepted.
Problem 1 (5 credits)
An article in Solid State Technology describes an experiment to determine the effect of C2F6 flow
rate on the uniformity of the etch on a silicon wafer used in integrated circuit manufacturing. Data for
two flow rates are as follows:
C2F6
Flow rate
125
200
1
2.7
4.6
2
4.6
3.4
Uniformity observation
3
4
2.6
3.0
2.9
3.5
5
3.2
4.1
6
3.8
5.1
A: Does the C2F6 flow rate affect average etch uniformity?
B: Does the C2F6 flow rate affect the wafer-to-wafer variability in etch uniformity?
Use α = 0.05.
Problem 2 (7 credits)
A consumer product-testing organization has compared the annual power consumption for five
different brands of dehumidifier. Because power consumption depends on the prevailing humidity
level, it was decided to monitor each brand at four different humidity levels. Within each level, brands
were randomly assigned to the five selected locations. The resulting amount of power consumption
(kWh/year) appears in the following table.
Brands
1
2
3
4
5
1
685
722
733
811
828
Humidity level
2
3
792
838
806
893
802
880
888
952
920
978
4
875
953
941
1005
1023
A: Evaluate if the brands have a significant (95%) effect on the power
consumption.
B: Calculate even if there is a significant difference between brands:
B1: 2 and 3
B2: 4 and 5
C: Comment the results in A and B.
Problem 3 (8 credits)
It is of interest to investigate the effects of five factors on the colour of a chemical product. The
factors are A = solvent, B = catalyst, C = temperature, D = reactant purity and E = pH. Experiments
with 16 runs have been performed and the results are as follows:
A: What kind of a fractional factorial design is used and what is the alias structure?
Four-factor interactions can be omitted.
B: Calculate the main effects and the two-factor interaction effects AB, AC, AD and CD
and examine their significance ( α = 0.05). All other factor effects can be neglected.
C: Calculate the residuals and plot them versus the fitted values. Comment on the plot.
SS(corr) = ∑ (y i − y) 2 = 114.7
Problem 4 (6 credits)
One wishes to develop a new substrate for production of a single cell protein consisting of N-source
(N), Carbon-source (C) and a Potassium content (K). One has earlier shown that
N > 10 %
C > 60 %
K >2%
In order to find a good optimum one wants to fit a polynomial with both linear, interaction and
A: Suggest a suitable experimental design and calculate the composition of the substrate in
the suggested experiments.
There must be at least four degrees of freedom for determination of the experimental error.
B: Calculate even the first row in the calculation matrix for the suggested design.
Problem 5 (8 credits)
3
3
3
i =1
i =1
j>i
yˆ = b0 + ∑ bi x i + ∑∑ bij x i x j has been fitted to the following experimental
The model
data
Y
12.48
44.14
26.32
23.71
19.93
24.67
24.16
17.50
25.49
24.45
31.15
12.75
17.66
16.15
41.80
22.20
X1
-1.00
-1.00
-1.00
1.00
1.00
0
1.00
1.00
1.00
1.00
-1.00
1.00
1.00
1.00
-1.00
0
X2
-1.00
1.00
-1.00
1.00
0
0
-1.00
1.00
0
1.00
1.00
-1.00
1.00
1.00
1.00
0
X3
-1.00
1.00
1.00
-1.00
1.00
0
-1.00
1.00
0
-1.00
0
1.00
1.00
1.00
1.00
0
with the result:
bT = [23.29 -2.51 3.46 2.24 -3.03 -6.51 1.22]
The diagonal elements in (XTX)-1 have been calculated to :
[0.09 0.10 0.12 0.12 0.11 0.15 0.13]
Useful sums of squares:
SST = ∑ (y i − y) 2 = 1205.07
SSE1 = ∑ (yi − yˆ i )2 = 54.97
A: Decide if the model is significant.
B: Calculate 95 % confidence interval for parameter b1 (-2.51).
C: Examine if the model has systematic errors.
When even quadratic terms were added to the model the residual sum of squares decreased to
SSE 2 = ∑ (y i − yˆ i ) 2 = 10.01
D: Calculate if the added parameters significantly improved the model.
Use α = 0.05
Problem 6 (6 credits)
A: Describe how one can find the optimum in an experimental region with
1. Augmented experimental design and canonical analysis
2. Steepest ascent (Gradient search)
3. EVOP
B: Also state under which conditions the different methods are suitable.