Statistical Science 9544A Case Study II

Steel Testing thirty-six samples were taken from one end of a coil of steel that is approximately 700 meters long.

Description of the Experiment Twelve strips of five meters in length and thirty centimeters wide were taken from the coil. Then samples were taken from across the width of each strip (two edges E1 and E2 and the center CTR). Each of the sample types E1, E2 and CTR were mixed amongst themselves and divided into three lots of four samples. All the samples were subjected to the same heat treatment which could add sample to sample variation. A lot of each of E1, E2, and CTR was sent to each of three different labs for destructive testing. The measurement taken are the bend angles at which the samples were destroyed.

Data

Research Questions Primary Do the labs give the same results? Secondary Are there differences in where the sample is taken (E1, E2 and CTR)?

First Step: Plot the Data > boxplot(Angle~Part+Lab)

ANOVA > boxplot(Angle~Part+Lab) > steellm<-lm(Angle~Part*Lab) > anova(steellm) Analysis of Variance Table Response: Angle Df Sum Sq Mean Sq F value Pr(>F) Part 2 46.30 23.150 2.6223 0.09104 . Lab 2 317.95 158.975 18.0074 1.074e-05 *** Part:Lab 4 19.06 4.766 0.5398 0.70776 Residuals 27 238.37 8.828 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual Analysis > resids<-residuals(steellm) > qqnorm(resids) > qqline(resids)

Where are the Two Residuals From? > resids 1 2 3 4 5 6 7 8 9 10 11 -0.425 1.175 -1.125 0.375 2.475 -6.925 -1.625 6.075 -1.425 -0.825 0.975 12 13 14 15 16 17 18 19 20 21 22 1.275 -3.575 2.925 -1.875 2.525 0.825 -2.475 -0.775 2.425 -1.000 -0.600 23 24 25 26 27 28 29 30 31 32 33 1.700 -0.100 1.700 0.700 -6.200 3.800 -2.200 -2.500 2.100 2.600 0.475 34 35 36 -0.725 -1.925 2.175 6 62.3 E1 B 27 70.5 E2 C

Post Hoc Tests – First Step First extract the mean square error > anova(steellm) Analysis of Variance Table Response: Angle Df Sum Sq Mean Sq F value Pr(>F) Part 2 46.30 23.150 2.6223 0.09104 . Lab 2 317.95 158.975 18.0074 1.074e-05 *** Part:Lab 4 19.06 4.766 0.5398 0.70776 Residuals 27 238.37 8.828 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > mse<-anova(steellm)[["Mean Sq"]][4] > mse [1] 8.828333

Post Hoc Test – Do Pairwise Comparisons > labdata<-split(Angle,Lab) > labmeans<-sapply(labdata,mean) > labmeans A B C 68.88333 68.92500 75.20833 > (labmeans[1]-labmeans[2])/sqrt(2*mse/12) -0.03434986 > (labmeans[1]-labmeans[3])/sqrt(2*mse/12) -5.214309 > (labmeans[2]-labmeans[3])/sqrt(2*mse/12) -5.179959

Why is Lab C different from the other two? Back to the Client Why is Lab C different from the other two?