A 16-run experiment was performed in a semiconductor manufacturing plant
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A 16-run experiment was performed in a semiconductor manufacturing plant

8.22 A 16-run experiment was performed in a semiconductor manufacturing plant to study the effects of six factors on the curvature or camber of the substrate devices produced. The six variables and their levels are shown in Table P8.2.

a. What type of design did the experimenters use?

b. What are the alias relationships in this design?

c. Do any of the process variables affect average camber?

d. Do any of the process variables affect the variability in camber measurements?

e. If it is important to reduce camber as much as possible, what recommendations would you make?

8.41 A 16-run fractional factorial experiment in nine factors was conducted by Chrysler Motors Engineering and described in the article "Sheet Molded Compound Process Improvement," by P. I. Hsieh and D. E. Goodwin (Fourth Symposium on Taguchi Methods, American Supplier Institute, Dearborn, MI, 1986, pp. 13-21). The purpose was to reduce the number of defects in the finish of sheet-molded grill opening panels. The design, and the resulting number of defects, c, observed on each run, is shown in Table P8.12. This is a resolution III fraction with generators E= BD, F= BCD. G = AC, H = ACD, and J = AB.

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 number of defects. 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 count, you should have expected this.) The previous table also shows a transformation on c, the square root, that is a widely used variance stabilizing transformation for count 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 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. F&T's modification to the square root transformation is

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.)

9.1 The effects of developer strength (A) and development time (B) on the density of photographic plate film are being studied. Three strengths and three times are used, and four replicates of a 32 factorial experiment are run. The data from this experiment follow. Analyze the data using the standard methods for factorial experiments.

9.4 An experiment is run in a chemical process using a 32 factorial design. The design factors are temperature and pressure, and the response variable is yield. The data that result from this experiment are as follows.

a. Analyze the data from this experiment by conducting an analysis of variance. What conclusions can you draw?

b. Graphically analyze the residuals. Are there any concerns about underlying assumptions or model adequacy?

c. Verify that if we let the low, medium, and high levels of both factors in this design take on the levels -1,0, and +1, then a least squares fit to a second-order model for yield is

d. Confirm that the model in part (c) can be written in terms of the natural variables temperature (T) and pressure (P) as

e. Construct a contour plot for yield as a function of pressure and temperature. Based on examination of this plot, where would you recommend running this process?

StatisticsThe normal probability plot refers to a graphical method used to assess if a data set is nearly distributed normally or not. Data are usually plotted against theoretical normal distribution in a way that the points are able to form a straight line roughly....

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