Applied Statistics and Probability for Engineers

14-7 2kFACTORIAL DESIGNS 535

t-statistic is reported for each effect instead of the F-statistic used in Table 14-17. Now the square

of a trandom variable with ddegrees of freedom is an Frandom variable with 1 numerator and

ddenominator degrees of freedom. Thus the square of the t-statistic reported by Minitab will be

equal (apart from rounding errors) to the F-statistic in Table 14-17. To illustrate, for the main ef-

fect of feed Minitab reports t4.32 (with eight degrees of freedom), and t^2 (4.32)^2 18.66,

which is approximately equal to the F-ratio for feed reported in Table 14-17 (F18.69). This

F-ratio has one numerator and eight denominator degrees of freedom.

The lower panel of the Minitab output in Table 14-18 is an analysis of variance summary

focusing on the types of terms in the model. A regression model approach is used in the pres-

entation. You might find it helpful to review Section 12-2.2, particularly the material on the

partial F-test. The row entitled “main effects’’under source refers to the three main effects

Table 14-17 Analysis of Variance for the Surface Finish Experiment

Source of Degrees of

Variation Sum of Squares Freedom Mean Square f 0 P-Value

A 45.5625 1 45.5625 18.69 0.0025

B 10.5625 1 10.5625 4.33 0.0709

C 3.0625 1 3.0625 1.26 0.2948

AB 7.5625 1 7.5625 3.10 0.1162

AC 0.0625 1 0.0625 0.03 0.8784

BC 1.5625 1 1.5625 0.64 0.4548

ABC 5.0625 1 5.0625 2.08 0.1875

Error 19.5000 8 2.4375

Total 92.9375 15

Table 14-18 Minitab Analysis for Example 14-4

Estimated Effects and Coefficients for Roughness

Term Effect Coef StDev Coef T P

Constant 11.0625 0.3903 28.34 0.000

Feed 3.3750 1.6875 0.3903 4.32 0.003

Depth 1.6250 0.8125 0.3903 2.08 0.071

Angle 0.8750 0.4375 0.3903 1.12 0.295

Feed*Depth 1.3750 0.6875 0.3903 1.76 0.116

Feed*Angle 0.1250 0.0625 0.3903 0.16 0.877

Depth*Angle 0.6250 0.3125 0.3903 0.80 0.446

Feed*Depth*Angle 1.1250 0.5625 0.3903 1.44 0.188

Analysis of Variance for Roughness

Source DF Seq SS Adj SS Adj MS F P

Main Effects 3 59.188 59.188 19.729 8.09 0.008

2-Way Interactions 3 9.187 9.187 3.062 1.26 0.352

3-Way Interactions 1 5.062 5.062 5.062 2.08 0.188

Residual Error 8 19.500 19.500 2.437

Pure Error 8 19.500 19.500 2.437

Total 15 92.938

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