Example 12.1 Process Manufacturing (Regression Model Specification) The production manager for Flexible Circuits, Inc., has asked for your assistance in studying a manufacturing process. Flexible circuits are produced from a continuous roll of flexible resin material with a thin film of copper-conducting material bonded to its surface. Copper is bonded to the resin by passing the resin through a copper-based solution. The thickness of the copper is critical for high-quality circuits. Copper thick- ness depends, in part, on the temperature of the copper solution, speed of the produc- tion line, density of the solution, and thickness of the flexible resin material. To control the thickness of the bonded copper, the production manager needs to know the effect of each of these variables. You have been asked for assistance in developing a multiple regression model. Solution Model development begins with a careful analysis of the problem context. The first step for this example would be an extended discussion with product design and manufacturing engineers so that you understand the process being modeled in detail. In some cases, you would study existing literature related to the process. The process must be understood and agreed to by the engineers and analysts before a useful model can be developed using multiple regression analysis. In this example the dependent variable, Y, is the copper thickness. Independent variables include temperature of the copper solution, Xi; speed of the production line, Xy; density of the solution, Xy; and thickness of the flexible resin material, X4. These variables were identified as potential predictors of copper thickness, Y, by engineers and scientists that understand the technology of the plating process. Based on the study of the process, the resulting model specification is as follows: Y = Be + B,X, + B2X2 + B3X3 + B4X4 In this linear model the B;s are constant linear coefficients of the independent variables X, that indicate the conditional effect of each independent variable on the determination of the dependent variable, Y, in the population. Thus, the coef- ficients B, are parameters in the linear regression model. A series of production runs would then be made to obtain measurements of various combinations of in- dependent and dependent variables. (See the discussion of experimental design in Section 13.2.) x19 Example 12.2 Store Location (Model Specification) The director of planning for a large retailer was dissatisfied with the company's new- store development experience. In the past 4 years 25% of new stores failed to obtain their projected sales within the 2-year trial period and were closed, with substantial economic losses. The director wanted to develop better criteria for choosing store loca- tions and decided that the historical experience of successful and unsuccessful stores should be studied