Question: Solder Thickness. Bisgaard, in a 1994 Journal of Quality Technology paper, discussed an experiment designed to improve solder layer mean thickness and thickness uniformity on printed circuit boards. (A uniform solder layer of a desired thickness provides good electrical contacts.) To stay competitive, it was important for a manufacturer to solve the problem of uneven solder layers. A team of engineers focused on the operation of a hot air solder leveler (HASL) machine. A 16-run screening experiment involving six two-level factors was run. A measure of solder layer uniformity was obtained from each of the 16 runs (y is a sample variance thickness based on 24 thickness measurements). The design generators were E↔ABC and F↔BCD. The 16 combinations of levels of factors A through D and corresponding values of the response variable are given in Table.

(a) Rearrange the rows in the table to produce Yates standard order for factors A, B, C, and D. Then add two columns to the table giving the levels of factors E and F that were used.

(b) Discuss why, even though there were 24 thickness measurements (probably from a single PC board) involved in the computation of "y" for a given treatment combination, this is really an experiment with no replication (m = 1).

(c) Find the defining relation for this 26-2 fractional factorial experiment.

(d) The effects are aliased in 16 groups of four effects each. For each of the six main effects (α2, β2, γ2, δ2, 2, and φ2), find the sums involving those which can be estimated using data from this experiment. (Use notation like α2 +βγ222 +αβγδφ22222 +δφ222.)

(e) Transform the responses y by taking natural logarithms, y' = ln(y). Then use the Yates algorithm to find the estimated sums of effects on y'.

(f) Make a normal probability plot for the last 15 estimated sums of effects in (e). Then make a half-normal plot for the absolute values of these estimates.

(g) Based on your plots in (f), do you detect any important effects on solder layer uniformity? Why or why not? What is the simplest possible interpretation of the importance of these? Based on your results from the full normal plot and using the simplest interpretation (assume all interaction effects are 0), what combination or combinations of levels of the six factors do you predict will have the best uniformity of solder layer? Why?

(h) Would you be willing to recommend adoption of your projected best treatment combination(s) from (g) with no further experimentation? Explain.