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Math1302: Design and Analysis of Experiments Submit by the Due Date
Assignment 3 (100 points worth 10%) Due date: As shown on canvas
Student number: Student name:
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INSTRUCTIONS
Please work through all problems using a statistical package where and if required. You will need to create a Word document to copy your codes, outputs, and comments into and prepare your written answers to the questions below, otherwise you might use RMarkdown and submitted as pdf file.
Submit online through Canvas by the due day.
Question 1: An engineer is studying the mileage performance characteristics of 5 types of gasoline additives. In the road test he wishes to use cars as blocks; however, because of a time constraint, he must use an incomplete block design. He runs the balanced design with the five blocks that f o l l o w .
17 14 13 12 14 14 13 10 12 13 12 9 13 11 11 12
11 12 10 8
(a) Verify that this is a balanced incomplete block design.
(b) Test if there is a difference between the five additives?
(c) Obtain the estimates of the treatment means and the treatments effects.
Draw your conclusions using α = 5%.
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(d) Calculate t h e standard error of the differences between two treatment mean estimates (i.e. the standard error of 𝜏𝑖 – 𝜏𝑗 ).
(e) Using the Tukey’s pairwise comparison method, draw your conclusions.
(30 points)
Question 2: 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 full factorial design are run. The results are as follows:
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Combination I
(1) 22 a 32 b 35 ab 55 c 44 ac 40 bc 60 abc 39
31 25 43 29 34 50 47 46 45 38 37 36 50 54 41 47
Treatment Replicate
a) Estimate the factor effects. Which effects appear to be large?
b) Use the analysis of variance to confirm your conclusions for part (a).
c) Write down a regression model for the predicting tool life (in hours) based on the results of
this experiment.
d) Analyze the residuals. Are there any obvious problems?
e) Based on the main effect plots and the interaction plots, what coded factor levels of A, B, and
C would you recommend using?
(30 points)
Question 3: A nickel–titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part because it can lead to non-recoverable failure. A test is run at the parts producer to determine the effect of four factors on cracks. The four factors are pouring temperature (A), titanium content (B), heat treatment method (C), and amount of grain refiner used (D). Two replicates of a 24 design are run, and the length of crack (in mm x 10-2) induced in a sample coupon subjected to a standard test is measured. The data presented in the Table below.
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– + – – + + – – – – + – + – + – – + + – + + + – – – – + + – – + – + – + + + – + – – + + + – + + – + + + + + + +
Combination I
(1) 7.037 a 14.707
b 11.635 ab 17.273 c 10.403 ac 4.368 bc 9.360 abc 13.440 d 8.561 ad 16.867 bd 13.876 abd 19.824 cd 11.846 acd 6.125 bcd 11.190 abcd 15.653
6.376 15.219
12.089 17.815 10.151 4.098 9.253 12.923 8.951 17.052 13.658 19.639 12.337 5.904 10.935 15.053
Treatment Replicate
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a) Estimate the factor effects. Which factor effects appear to be large?
b) Conduct an analysis of variance. Do any of the factors affect cracking? Use α=0.05.
c) Write down a regression model that can be used to predict crack length as a function of the
significant main effects and interactions you have identified in part (b).
d) Analyze the residuals from this experiment.
e) What recommendations would you make regarding process operations? Use interaction
and/or main effect plots to assist in drawing conclusions.
(40 points)
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