UNIVERSITY OF SOUTHAMPTON MATH6027W1
SEMESTER 1 EXAMINATION 2020/21
MATH6027 Design of Experiments
Duration: 120 minutes (plus 60 minutes to upload PDF solutions to Blackboard)
This paper contains FOUR questions.
Answer ALL questions.
An outline marking scheme is shown in brackets for each question.
All students: This is an open book assessment. You may consult books, notes or internet
sources. You are permitted to use calculators or mathematical software, but to obtain full
marks, you must show and explain your working as well as the final answer.
The assessment must be carried out in accordance with the University Academic integrity
regulations. It is not permitted to communicate with anyone else (be it private or online)
about the content of this exam during the whole time it is open.
• Start a new question on a fresh sheet of paper.
• Make sure your page is in portrait orientation.
• Write in Black or Blue pen.
• On each page, write your page number in the top left and your module and student ID
in the top right.
• Show all working
• In addition to the given duration of this paper (2 hours), you have an hour to scan and
upload your work.
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1. [25 marks] In planning an experiment to compare five active treatments B,. . .,F, it
has been suggested that the treatments should each be compared with a control
treatment, A, and hence with each other.
The incomplete block design shown below for 6 treatments in 5 blocks of 2 units has
been suggested. Block 1 contains treatments A and B, Block 2 has treatments A and
C, etc. The response obtaind from each unit is shown in the table.
Treatments
A B C D E F
1 12.79 10.70
2 5.26 2.64
Blocks 3 8.89 6.14
4 6.55 3.53
5 7.93 5.67
(a) [4 marks] Write down an appropriate and estimable linear model for this
experiment and give the X and Z matrices explicitly. What interpretation do your
model parameters have? State the normal equations for estimation of these
parameters.
(b) [10 marks] Complete the following ANOVA table.
Source Degrees of Sum of squares Mean Square
freedom
Blocks 71.90
Extra due to
treatments
Residual
Total
(c) [11 marks] Write down a matrix L such that Lβˆ compares treatments A and B,
and treatments B and C , where βˆ are the treatment parameters from your
model. Find the form of the variance-covariance matrix of Lβˆ in terms of
matrices X and Z .
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2. [25 marks]
(a) Two choices of a 26−2 factorial design are being considered for use in an
experiment with factors labelled 1, . . . , 6. One design, d1, aliases interactions
13456 and 1236 with the mean; the other design, d2, aliases interactions 2456
and 1234 with the mean.
(i) [3 marks] Write down the full defining relation for each design.
(ii) [2 marks] State the resolution of each design.
(iii) [2 marks] Which design would you prefer? Give your reasons.
(b) The minimum aberration resolution 25−2 fractional factorial design has defining
words 134 and 125.
(i) [2 marks] Write down the full defining relation for this design.
(ii) [3 marks] If we had originally only wanted to investigate four factors, is there
an 8 run design with higher resolution than the design above that we could
have used? If so, state its defining relation.
(iii) [10 marks] Returning to the original 25−2 design, find 8 extra treatments to
add to the design that will result in the 16 treatments forming a 25−1 fractional
factorial design of resolution IV.
(iv) [3 marks] If we had originally wanted to use 16 runs, is there a better design
we could have used? If so, state its defining relation.
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TURN OVER
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3. [25 marks] An experiment was carried out in order to fit a second order polynomial
response surface model to describe the effects of temperature (◦C), pressure (bar)
and residence time (minutes) on the yield of a chemical reaction. The following
central composite design was used:
Temperature Pressure Time Temperature Pressure Time
1 170.00 0.00 0.00 12 185.00 15.00 20.00
2 200.00 0.00 0.00 13 159.77 15.00 20.00
3 170.00 30.00 0.00 14 210.23 15.00 20.00
4 200.00 30.00 0.00 15 185.00 -10.23 20.00
5 170.00 0.00 40.00 16 185.00 40.23 20.00
6 200.00 0.00 40.00 17 185.00 15.00 -13.64
7 170.00 30.00 40.00 18 185.00 15.00 53.64
8 200.00 30.00 40.00 19 185.00 15.00 20.00
9 185.00 15.00 20.00 20 185.00 15.00 20.00
10 185.00 15.00 20.00 21 185.00 15.00 20.00
11 185.00 15.00 20.00 22 185.00 15.00 20.00
(a) [3 marks] Define coded variables in such a way that the factorial points take
values −1 and 1 for each variable, i.e. give a formula for each coded variable,
x1, x2 and x3, relating their values to the temperature, pressure and time
respectively.
(b) [4 marks] State the value of α used for the axial points and nf , na and nc, the
numbers of factorial, axial and centre points respectively.
(c) [3 marks] Show whether or not the design is rotatable.
(d) [5 marks] Show whether or not the design is orthogonal.
(e) [10 marks] The least squares estimates of the parameters are
βˆ0 = 77 ,
βˆ1 = 12 , βˆ2 = 36 , βˆ3 = −16 ,
βˆ11 = −48 , βˆ22 = −90 , βˆ33 = −39 ,
βˆ12 = 0 , βˆ13 = 0 , βˆ23 = 26 .
Find the stationary point of the estimated response surface. Is this point a
maximum? Find the estimated response at this point.
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4. [25 marks] Consider the continuous design
ξ =
{ −1 0 1
1
3
1
3
1
3
}
for an experiment with one variable, x, and a second-order model
Y (x) = β0 + β1x+ β2x
2 + ε, where the errors ε are independent and identically
distributed. The variable x varies across the standardised range, −1 ≤ x ≤ 1.
(a) [10 marks] Without explicitly calculating the information matrix M(ξ) or its
inverse, show that the standardised variance is equal to p, the number of
parameters in the model, at each of the support points of ξ.
(b) [11 marks] Find and plot the standardised variance for ξ as a function of x.
(c) [4 marks] Show that this design is both D- and G-optimal.
END OF PAPER
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