A 16-run fractional factorial experiment in 10 factors on sand-casting of engine manifolds was conducted by engineers at the Essex Aluminum Plant of the Ford Motor Company and described in the article “Evaporative Cast Process 3.0 Liter Intake Manifold Poor Sandfill Study,” by D. Becknell

(Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, 1986, pp. 120–130). The purpose was to determine which of 10 factors has an effect on the proportion of defective castings. The design and the resulting proportion of nondefective castings ̂p observed on each run are shown in Table P8.13. This is a resolution III fraction with generators E = CD, F = BD, G = BC, H = AC, J = AB, and K = ABC.

Assume that the number of castings made at each run in the design is 1000.

(a) Find the defining relation and the alias relationships in this design.

(b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors.

(c) Fit an appropriate model using the factors identified in part (b).

(d) Plot the residuals from this model versus the predicted proportion of nondefective castings. Also prepare a normal probability plot of the residuals. Comment on the adequacy of these plots.

(e) In part

(d) you should have noticed an indication that the variance of the response is not constant. (Considering that the response is a proportion, you should have expected this.) The previous table also shows a transformation on ̂p, the arcsin square root, that is a widely used variance stabilizing transformation for proportion data (refer to the discussion of variance stabilizing transformations in Chapter 3). Repeat parts

(a) through

(d) using the transformed response and comment on your results.
Specifically, are the residual plots improved?

(f) There is a modification to the arcsin square root transformation, proposed by Freeman and Tukey (“Transformations Related to the Angular and the Square Root,” Annals of Mathematical Statistics, Vol. 21, 1950, pp. 607–611), that improves its performance in the tails.
F&T’s modification is [arcsin √
n̂p∕(n + 1)
+arcsin √
(n̂p + 1)∕(n + 1)]∕2 Rework parts

(a) through

(d) using this transformation and comment on the results. (For an interesting discussion and analysis of this experiment, refer to

“Analysis of Factorial Experiments with Defects or Defectives as the Response,” by S. Bisgaard and H. T. Fuller, Quality Engineering, Vol. 7, 1994–95, pp. 429–443.)