Stat 502
Homework 4
Assigned 10/25/07
Due 11/1/07
1. Clay surface roughness: A 3x3 factorial design was used to explore the effects
of three sources of clay (I, II, III) and three ceramic molds (A, B, C) on surface
roughness. Five replicates of each clay-mold combination were prepared and all
45 randomly placed within a furnace before firing. The response is a measure of
surface roughness. The data are in the file claymold.rough.dat.
(a) Examine these data graphically. What do you think about the data and
effects of the two factors (clay source and ceramic molds) based just on this
graphical examination of the data. Can you see any features of the data that
you must pay attention to in formal statistical analysis?
(b) Might a transformation be useful for these data? Why?
(c) Write out the treatment effects model for the experiment and compute the
least squares estimates of the parameters of this model. (This model may be
on a transformed scale, depending on your answer to part (b).)
(d) Compute the ANOVA table, including a column for the “Expected mean
squares” (which you won’t get from R output). Is an additive model justified?
Summarize your conclusions about the effects of clay source and ceramic
molds.
(e) Compute a 95% confidence interval for the contrast in cell means that you
would use to contrast the difference between molds A and B using clay from
source II with the difference between molds A and B using clay from source
III. If you use the “Least Significant Difference” framework for hypothesis
testing, do the results in part (d) authorize you to proceed with a test of the
significance of this contrast?
(f) Is there an optimal combination of clay source and mold if we are interested
in minimizing the measure of surface roughness? That is, is there one
combination of clay source and ceramic mold that has a mean score
significantly less than all the other treatment combinations? Explain what
statistical methodology you use to answer this question.
2. Plastic laminate cutting: The data below are production data for four different
machines (1, 2, 3, 4) cutting four different types of laminated plastics (A, B, D, D).
(a) Plot these data so that you can visualize the machine and plastic effects and
comment. Do you observe any “bad values”? Explain.
(b) There are no replications for the combinations of the “treatments” of Machine
and Plastic, so consider an additive model. Then, for a range of values of the
exponent λ (e.g., {-1, -.5, 0, 1/3, .5} evaluate the F statistic for the test of a
machine effect on production using the power transformation Yλ where Y
denotes production. Similarly, evaluate the F statistic for the effect of the
type of plastic over these same values of λ. Does this help you determine an
appropriate transformation? Choose your preferred transformation and plot
the usual diagnostics using the residuals from the fit of the additive model on
the transformed scale. What conclusions do you draw?
Plastic 1
2
3
4
A
1271
1440
612
605
Machines
B
C
4003
1022
1651
372
1664
829
2001
258
D
2643
5108
1944
2607