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Math 558- Design of Experiments Final Exam : W 2023
Exam Release Date: April 17, 2023 – Time: 9:00 am Due Date: April 20, 2023 – Time: 9:00 am
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Question 1.
[20 points]
Steel is normalized by heating above the critical temperature, soaking, and then air cooling. This process increases the strength of the steel, refines the grain, and homog- enizes the structure. An experiment is performed to determine the effect of tempera- ture and heat treatment time on the strength of normalized steel. Two temperatures and three times are selected. The experiment is performed by heating the oven to a randomly selected temperature and inserting three specimens. After 10 minutes one specimen is removed, after 20 minutes the second specimen is removed, and after 30 minutes the final specimen is removed. Then the temperature is changed to the other level and the process is repeated. Four shifts are required to collect the data, which are shown below. Analyze the data and draw conclusions, assuming both factors are fixed. Attach R-output and R-codes.
Temperature(0F )
Shift Time(min) 1500 1600
1 20 54 91 30 61 62 10 50 80
2 20 52 72 30 59 69 10 48 73
3 20 74 81 30 71 69 10 54 88
4 20 48 92 30 59 64
Solution: Q1.
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Question 2.
[20 Points] An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are given in the table below.
Treatment Replicates A B C Combinations I II
– – – (1) 22 31
+– a 32 43 -+- b 35 34 ++- ab 55 47 –+ c 44 45 +-+ ac 40 37 -++ bc 60 50 +++ abc 39 41
a. Write down a regression model for predicting tool parameters
III 25 29 50 46 38 36 54 47
life and estimate the model
b. Estimate the factor effects. Which effects appear to be large?
c. Based on the results from parts (a) and (b) write down the reduced model and estimate its parameters.
Note: Attach R-output and R-codes.
Solution Q2.
Question 3.
a. [12 Points] Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in five-gallon milk containers. The analysis is done in a laboratory, and only three trials can be run on any day. Because days could represent a potential source of variability, the experimenter decides to use a randomized block design. Observations are taken for four days, and the data are shown here. Analyze the data from this experiment (use
α = 0.05) and draw conclusions. Attach R-output and R-codes.
Days Solution 1 2 3 4
1 13 22 18 39
2 16 24 17 44
3 5 4 1 22
b. [8 Points] Show that for a (t, b, r, k, λ) BIBD,
b ≥ t + (r − k)
Where, t is the number of treatments, b is the number of blocks, r is the replication, k is the block size and λ is the concurrence of the pair (i,j) for all i and j.
Solution Q3.
Question 4.
a. [14 Points] Seven different hardwood concentrations are being studied to determine their effect on the strength of the paper produced. However, the pilot plant can only produce three runs each day. As days may differ, the analyst uses the balanced incomplete block design that follows. Analyze this experiment (use α = 0.05) and draw conclusions. Attach R-output and R-codes.
Hardwood Days Concentration% 1 2 3 4 5 6 7
2 4 6 8 10 12 14
136 130 127
b. [3 Points] Give the incidence matrix for the design given in part (a).
c. [3 Points] Give the concurrence matrix for the design given in part (a).
Solution Q4.
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Question 5.
a. [8 Points] Show that for a (t, b, r, k, λ) BIBD, r≥k
λ(t−1)>k2 −k
Where, t is the number of treatments, b is the number of blocks, r is the replication, k is the block size and λ is the concurrence of the pair (i,j) for all i and j.
b. [6 Points] Describe the sort of design that would be most appropriate for each of the following situations:
(i) investigating a new drug which purports(claims) to reduce high blood pressure.
(ii) comparing four diets for piglets to see which gives the greatest weight gain in the first six weeks of life.
(iii) seeing whether either of the two proposed new drugs to help people give up smoking has any effect.
c. [3 Points] You want to compare four treatments in blocks of two runs. Find a Balanced Incomplete Block Design for this experiment with six blocks.(You just have to give all the parameters of this BIBD)
d. [3 Points] Verify that a BIBD with the parameters a = 8, r = 8, k = 4, and b = 16 does not exist.
Solution Q5.
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