66. The article "Applying Regression Analysis to Improve Dyeing Process Quality: A Case Study" (Intl. J. of Advanced Manuf. Tech., 2010: 357-368) examined the practice of adjust pH of dye liquor at a large manufac- turer of automotive carpets. The investigation was based on a data set consisting of 114 observations included in the article). The dependent variable is y = pH before addition of dyes, and the predictors are x, = carpet density (oz/yd?), x = carpet weight (lb), x, = dye weight (g), x4 = dye weight as a percentage of carpet weight (%), and x, = pH after addition of dyes. a. Here is output from Minitab's Best Subsets Regression option. Which model(s) would you rec- ommend, and why? Mallows 1 2 3 Vars R-Sq R-Sq(adj) Cp 4 63.7 63.4 16.6 0.34971 1 4.4 3.5 223.6 0.56773 2 68.7 68.1 1.2 0.32630 2 68.6 68.0 1.6 0.32684 X X 3 69.0 68.2 2.2 0.32616 Mallowe Vara R-8q R-8q(adj) 1 2 68.9 68.0 2.5 0.32668 4. 69.0 67.9 4.1 0.32754 4 69.0 67.9 4.1 0.32759 X x 69.0 67.6 6.0 0.32894 X x x x x b. The cited article recommended the model with just x and x, as predictors. The following Minitab output resulted from fitting that model. Predictor Coef SE Coef P Constant 0.9402 0.2814 3.34 0.001 X3 -0.00004639 0.00001104 -4.20 0.000 x5 0.73710 0.04813 15.31 0.000 S = 0.326304 R-Sq = 68.78 R-Sq (adj) = 68.1 Analyeia of Variance Source DF MS Regression 2 25.925 12.962 121.74 0.000 Residual Error 111 11.819 0.106 Total 113 37.744 Does this model appear to specify a useful relation- ship between the response variable and the predic- tors? (Note: The pattern in a normal probability plot of the standardized residuals is very linear. The plots of standardized residuals against both x, and x, show no discernible pattern. There is one observation whose x, value is more than twice as large as for any other observation, but with n = 114, this observation has very little influence on the fit.] c. Should either one of the two predictors be eliminated from the model provided that the other predictor is retained? Explain your reasoning. d. Calculate and interpret 95% Cls for the B cocfi- cients of the two model predictors. e. The estimated standard deviation of when x,- 1000 and x, = 6 is .0336. Obtain and interpret a 95% CI for true average pH before addition of dyes under these circumstances.