Question

Question 3 A nickel-titanium alloy is used to make aerospace components. Cracking is a potentially serious problem in the final parts because it can lead to failre. An experiment was conducted to determine the effect of three factors on cracks. Three factors at two levels each were selected and two replicates of a completely randomized experiment were run. The factors are Temperature (A), titanium content (B) and heat treatment (C). Treatment Replicate 54 50 42 45 46 43 42 46 49 45 48 bc 47 48 54 56 abc Determine the followings: (a) Calculate the factor effects. Which factor is the largest? (b) Calculate Sum of Square using contrast for all factor factors. (c) Construct the ANOVA table for the above experiment. (Approved by UAC) (d) Propose an appropriate refined mathematical model for predicting length of crack based on the experiment above?

Answer 1

Yates Method for 23 Ereriment: Treatment combination Total from all replications Mean ef fect-Col(3)/8 SS Col(2/16 Col(1) Col(3) Col(2) 54 50 104 87+104 191 179 191-370 393 370763 -13/81.625 (-13)/1610.5625 5/8 0.625 25/8 3.125 23/8 2.875 39/8 4.875 29/8 3.625 9/8 1.125 87 42 + 45 85+94= 179 205+188=393 13-26=-13 52/ 16 1 .5625 (25)2/16 39.0625 (23)2/16 33.0625 (39)2/ 16 95.0625 (29)2/16-52.5625 92/16 5.0625 48 + 46 94 93 +95 -188 9-17=-26 17-12=5 ab 43 42 = 85 110 + 95-205 15 - 2-13 17+8-25 87-104-1 7 46 49 95 179-191=-12 393-370=23 45 48 93 85-94 =-9 205-188 = 17 13+26 = 39 ac -9178 47 +48-95 54 56- 110 bc 17 + 12 -29 93 -95--2 10-95-15 abc 15+2-17 17-89

(a) Following table show the factor ef fects. From the above table AC is the largest since Mean ef fect of AC, 4.875 Factorial effect Mean effect Col (3)/8 -13/8 =-1.625 5/8 0.625 25/8 3.125 23/8 2.875 39/8 = 4.875 29/8 3.625 9/8-1.125 AB AC BC ABC (b) Following table shows the SS due dif ferent factorial ef fects: Factorial effect SS=Col(3)12 /16 (-13)2/16 10.5625 52/16= 1.5625 (25)2/16 39.0625 (23)2/16 33.0625 (39)2/16 95.0625 (29)2/16 52.5625 92/16 5.0625 AB AC BC ABC (c) SS due total SST = 54° 50° 42° 45°-+ 482 +462 432 422 + 462 +492 +452 +482 +472 +482 +542 +562 - 7632/16 - 263.4375 SS due to error263.4375-(10.5625+1.5625+39.0625+33.0625+95.0625+ 52.5625 + 5.0625) 26.5

ANOVA table F- ratio Factorial effect SS df MS = SS/df Decision crit F0.05, 1.8=5.3177 Insignificant Fo.05,1,8 5.3177 Insigni ficanit 39.0625 39.0625/3.3125 11.7925 Fo.05,1,8 5.3177 Significant 33.0625 33.0625/3.3125-9.9811 Fo05,185.3177 Significant 95.062595.0625/3.3125 28.6981 Fo.05,1,8 5.3177 Significant 52.5625 52.5625/3.3125-15.8679 Fo05,1,85.3177 Significant 5.0625/3.3125- 1.5283 Fo.o5,1,8 5.3177 Insigni ficant 10.5625 10.5625/3.3125-3.1887 10.5625 1.5625 1.5625 1.5625/3.3125-0.4717 AB 39.0625 33.0625 95.0625 AC BC 52.5625 5.0625 5.0625 ABC 26.5/8 - 3.3125 26.5 ETror Total 263.4375 16-1=15 (d) Since AB, C, AC, BC are significant then refined mathematical model is у 95.375 + 3. 125xAB + 2.875TC + 4.8752·AC + 3.625xBC