12 Airplane Sidewall Panels

EXAMPLE 12-12 Airplane Sidewall Panels

  Sidewall panels for the interior of an airplane are formed in a

  y

  1500-ton press. The unit manufacturing cost varies with the production lot size. The data shown below give the average

  x

  cost per unit (in hundreds of dollars) for this product ( y) and

  y

  the production lot size (x). The scatter diagram, shown in Fig. 12-11, indicates that a second-order polynomial may be

  x

  appropriate.

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  12-6 ASPECTS OF MULTIPLE REGRESSION MODELING

  y 1.60 1.50 1.40

  1.30 Average cost per unit,

  Figure 12-11 Data

  for Example 12-11.

  Lot size, x

  We will fit the model

  Solving the normal equations X¿X ␤ ˆ X¿y gives the fitted

  2 Y model

  0 1 x

  11 x

  The y vector, the model matrix X and the

  y ˆ 2.19826629 0.02252236x 0.00012507 x ˛ ␤ vector are as follows: 2

  Conclusions: The test for significance of regression is shown

  in Table 12-13. Since f 0 1762.3 is significant at 1, we

  conclude that at least one of the parameters 1 and 11 is not

  zero. Furthermore, the standard tests for model adequacy do

  not reveal any unusual behavior, and we would conclude that

  0 this is a reasonable model for the sidewall panel cost data.

  y

  X 1.30 1 60 3600 ␤ £ 1 §

  In fitting polynomials, we generally like to use the lowest-degree model consistent with the data. In this example, it would seem logical to investigate the possibility of dropping the quadratic term from the model. That is, we would like to test

  H 0 : 11 0

  H 1 : 11 0

  Table 12-13 Test for Significance of Regression for the Second-Order Model in Example 12-12

  Source of

  Sum of

  Degrees of

  f 0 P-value

  Regression

  1762.28 2.12E-12

  Error

  Total

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  CHAPTER 12 MULTIPLE LINEAR REGRESSION

  Table 12-14 Analysis of Variance for Example 12-12, Showing the Test for H 0 : 11 0

  Source of

  Degrees of

  Mean

  Variation

  Sum of Squares

  Freedom

  Square

  f 0 P-value

  Regression

  SS R 1 , 11 0 0 2 0.52516

  SS R 1 0 0 2 0.49416

  SS R 1 11 0 0 , 1 2 0.03100

  The general regression significance test can be used to test this hypothesis. We need to deter-

  mine the “extra sum of squares” due to 11 , or

  SS R 1 11 0 1 , 0 2 SS R 1 , 11 0 2 SS R 1 0 2 The sum of squares SS R 1 , 11 0 2 0.52516 from Table 12-13. To find SS R 1 0 0 2 , we fit a

  simple linear regression model to the original data, yielding y ˆ 1.90036313 0.00910056x

  It can be easily verified that the regression sum of squares for this model is

  SS R 1 0 0 2 0.4942

  Therefore, the extra sum of the squares due to 11 , given that 1 and 0 are in the model, is SS R 1 11 0 1 , 0 2 SS R 1 , 11 0 2 SS R 1 0 2

  The analysis of variance, with the test of H 0 : 11 0 incorporated into the procedure, is

  displayed in Table 12-14. Note that the quadratic term contributes significantly to the model.

  12-6.2 Categorical Regressors and Indicator Variables

  The regression models presented in previous sections have been based on quantitative vari- ables, that is, variables that are measured on a numerical scale. For example, variables such as temperature, pressure, distance, and voltage are quantitative variables. Occasionally, we need to incorporate categorical, or qualitative, variables in a regression model. For example, suppose that one of the variables in a regression model is the operator who is associated with each observation y i . Assume that only two operators are involved. We may wish to assign different levels to the two operators to account for the possibility that each operator may have a different effect on the response.

  The usual method of accounting for the different levels of a qualitative variable is to use indicator variables. For example, to introduce the effect of two different operators into a regression model, we could define an indicator variable as follows:

  x e

  0 if the observation is from operator 1

  1 if the observation is from operator 2

  12-6 ASPECTS OF MULTIPLE REGRESSION MODELING

  493

  In general, a qualitative variable with r-levels can be modeled by r

  1 indicator variables,

  which are assigned the value of either zero or one. Thus, if there are three operators, the different levels will be accounted for by the two indicator variables defined as follows:

  x 1 x 2

  if the observation is from operator 1 if the observation is from operator 2 if the observation is from operator 3

  Indicator variables are also referred to as dummy variables. The following example [from Montgomery, Peck, and Vining (2006)] illustrates some of the uses of indicator variables; for other applications, see Montgomery, Peck, and Vining (2006).

  0 1