Derringer and Suich ("Simultaneous Optimization of Several Response Variables," Journal of Quality Technology, 1980) studied the relationship of an abrasion index for a tire tread compound in terms of three factors: $x_{1}$, hydrated silica level; $x_{2}$, silane coupling agent level; and $x_{3}$, sulfur level. The following table gives the actual results.

$x_{1}$	$x_{2}$	$x_{3}$	$y$
-1	-1	1	102
1	-1	-1	120
-1	1	-1	117
1	1	1	198
-1	-1	-1	103
1	-1	1	132
-1	1	1	132
1	1	-1	139
0	0	0	133
0	0	0	133
0	0	0	140
0	0	0	142
0	0	0	145
0	0	0	142

a. Perform a thorough analysis of the results including residual plots.

b. Perform the appropriate test for lack of fit.