HW6
STAT 3515Q

Due by April 2, 2021, 11:59pm

This homework covers material from chapter 6-8.
All data sets for the following questions are supplied in an Excel file.

Question 1 (20 points)

An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle
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 as follows:

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 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 analysis of main effects and interaction plots, what levels of A, B, and C
would you recommend using?

Question 2 (30 points)

A nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a
potentially serious problem in the final part, as it can lead to non-recoverable failure. A test is run
at the parts producer to determine the effects of four factors on cracks. The four factors are pouring
temperature (A), titanium content (B), heat treatment method (C), and the amount of grain refiner
used (D). Two replicated of a 24 design are run, and the length of crack (in μm) induced in a sample
coupon subjected to a standard test is measured. The data are shown below:

a. Estimate the factor effects. Which factors appear to be large?
b. Conduct an analysis of variance. Do any of the factors affect cracking?
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. Is there an indication that any of the factors affect the variability in cracking?
f. What recommendations would you make regarding process operations? Use interaction
and/or main effect plots to assist in drawing conclusions.

Question 3 (10 points)

An experiment was run in a semiconductor fabrication plant in an effort to increase yield. Five
factors, each at two levels, were studied. The factors (and levels) were A = aperture setting (small,
large), B = exposure time (20% below nominal, 20% above nominal), C = development time (30
s, 45 s), D = mask dimension (small, large), and E = etch time (14.5 min, 15.5 min). The
unreplicated 25 design shown below was run.

a. Construct and analyze a design in two blocks with ABCDE confounded with blocks.
b. Assume that four blocks are necessary. Suggest a reasonable confounding scheme.
c. Suppose that it was necessary to run this design in four blocks with ACDE and BCD (and
consequently ABE) confounded. Analyze the data from this design.

Question 4 (10 points)

Construct a 23 design with ABC confounded in the first two replicates and BC confounded in the
third. Outline the analysis of variance and comment on the information obtained.

Question 5 (10 points)

Researchers wanted to perform an experiment to improve the yield of a chemical process. Four
factors were selected, and two replicates of a completely randomized experiment were run.
However, suppose it was only possible to run a one-half fraction of the 24 design. Construct the
design and perform the statistical analysis, using the data from replicate 1.

Question 6 (20 points)

An article by J.J. Pignatiello, Jr. And J.S. Ramberg in the Journal of Quality Technology, (Vol. 17,
1985, pp. 198-206) describes the use of a replicated fractional factorial to investigate the effects
of five factors on the free height of leaf springs used in an automotive application. The factors are
A = furnace temperature, B = heating time, C = transfer time, D = hold down time, and E = quench
oil temperature.
a. Write out the alias structure for this design. What is the resolution of this design?
b. Analyze the data. What factors influence the mean free height?
c. Analyze the residuals from this experiment, and comment on your findings.
d. Is this the best possible design for five factors in 16 runs? Specifically, can you find a
fractional design for five factors in 16 runs with a higher resolution than this one?  