# hw5_solution - STAT 425 FALL 2015 Section T1U T1G Homework...

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STAT 425 - FALL 2015 - Section T1U, T1G Homework V Solution November 11, 2015 Problem 1 (a) The summary of the model is given in Listing 1. Listing 1: Summary R output Call: lm(formula = log( Displacement ) ~ Length + Beam) Residuals: Min 1Q Median 3Q Max -0.40664 -0.05649 0.05264 0.09835 0.23408 Coefficients : Estimate Std. Error t value Pr(>|t|) ( Intercept ) 4.6504403 0.1299365 35.790 < 2e -16 *** Length 0.0039436 0.0009044 4.361 0.000141 *** Beam 0.0352421 0.0066055 5.335 9.04e -06 *** --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 Residual standard error: 0.1487 on 30 degrees of freedom Multiple R-squared: 0.967 , Adjusted R-squared: 0.9648 F- statistic : 439 on 2 and 30 DF , p-value: < 2.2e -16 (b) The summary of the second order polynomial model is given in Listing 2. Listing 2: Summary R output Call: lm(formula = log( Displacement ) ~ Length * Beam + I(Length ^2) + I(Beam ^2)) Residuals: Min 1Q Median 3Q Max -0.144297 -0.032167 -0.007257 0.028312 0.160636 Coefficients : Estimate Std. Error t value Pr(>|t|) ( Intercept ) 2.919e+00 4.639e -01 6.292 9.84e -07 *** Length 6.895e -04 5.723e -03 0.120 0.90500 Beam 1.569e -01 4.459e -02 3.520 0.00155 ** I(Length ^2) 4.801e -05 2.683e -05 1.789 0.08481 . I(Beam ^2) 2.844e -03 1.524e -03 1.866 0.07290 . Length:Beam -8.733e -04 4.175e -04 -2.092 0.04601 * --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 Residual standard error: 0.08167 on 27 degrees of freedom Multiple R-squared: 0.991 , Adjusted R-squared: 0.9894 F- statistic : 596.5 on 5 and 27 DF , p-value: < 2.2e -16 (c) Based on the model in (b), only the beam main effect and the interaction effect are significant. However keeping hierarchy in mind, we cannot remove all of the other terms. We can only remove the quadratic terms ( Length 2 and Beam 2 ). 1
(d) The ANOVA result comparing the models in (a) and (b) are given in Listing 3. Based on the F-test, we should reject the reduced model and use the full polynomial model. Listing 3: Summary ANOVA output Analysis of Variance Table Model 1: log( Displacement ) ~ Length + Beam Model 2: log( Displacement ) ~ Length * Beam + I(Length ^2) + I(Beam ^2) Res.Df RSS Df Sum of Sq F Pr(>F) 1 30 0.66311 2 27 0.18009 3 0.48303 24.14 8.388e -08 *** --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 (e) We remove one term at a time, starting with the ones having the highest p-value and going from quadratic to interaction to main effect hierarchy. The results are given in Listing 4. We see that the final model contains the main effects Length and Beam and their interaction term, Length:Beam .