7. Weld strength experiment The data shown in Table 6.22 are a subset of the data given by Anderson and McLean (1974) and show the strength of a weld in a steel bar. Two factors of interest were gage bar setting (the distance the weld die travels during the automatic weld cycle) and time of welding (total time of the automatic weld cycle). Assume that the levels of both factors were selected to be equally spaced.

(a) Using the cell-means model (6.2.1) for these data, test the hypothesis that there is no difference in the effects of the treatment combinations on weld strength against the alternative hypothesis that at least two treatment combinations have different effects.

(b) Suppose the experimenters had planned to calculate confidence intervals for all pairwise comparisons between the treatment combinations, and also to look at the confidence interval for the difference between gage bar setting 3 and the average of the other two. Write down the contrasts in terms of the parameters τij of the cell-means model, and suggest a strategy for calculating all intervals at overall level “at least 98%.”

(c) Consider the intervals in part (b). Give the formulae and calculate the actual interval for τ13 −τ15 (the difference in the true mean strengths at the 3rd and 5th times of welding for the first gage bar setting), and explain what this interval tells you. Also calculate the actual interval for the difference between gage bar setting 3 and the average of the other two, and explain what this interval tells you.

(d) Calculate an upper 90% confidence limit for σ2.

(e) If the experimenters were to repeat this experiment and needed the pairwise comparison intervals in

(b) to be of width at most 8, how many observations should they take on each treatment combination? How many observations is this in total?