Question number 1(a)

We would like to confound the three factor interaction abc with the blocks.

Question number 1(b)

Block 1: 1,ab,ac,bc

Block 2: a,b,c,abc

Question number 2(a)

We would like to confound the two factor interactions ab, ac and bc with the blocks.

Question number 2(b)

Block 1: 1, abc
Block 2: a, bc
Block 3: b ac
Block 4: c ,ab

Question number 3

a<- c(-1, 1, -1, 1,-1, 1, -1, 1)
b<- c(-1, -1, 1,1,-1, -1, 1,1)
c<- c(-1,-1,-1,-1, 1, 1, 1, 1)
ab<-c(1,-1,-1,1,1,-1,-1,1)
ac<-c(1,-1,1,-1,-1,1,-1,1)
bc<-c(1,1,-1,-1,-1,-1,1,1)
abc<-c(-1,1,1,-1,1,-1,-1,1)

tool_life<-c(22,
             32,
             35,
             55,
             44,
             40,
             60,
             39)
dat<-data.frame(a,b,c,ab,ac,bc,abc,tool_life)
dat
##    a  b  c ab ac bc abc tool_life
## 1 -1 -1 -1  1  1  1  -1        22
## 2  1 -1 -1 -1 -1  1   1        32
## 3 -1  1 -1 -1  1 -1   1        35
## 4  1  1 -1  1 -1 -1  -1        55
## 5 -1 -1  1  1 -1 -1   1        44
## 6  1 -1  1 -1  1 -1  -1        40
## 7 -1  1  1 -1 -1  1  -1        60
## 8  1  1  1  1  1  1   1        39
library(DoE.base)
## Loading required package: grid
## Loading required package: conf.design
## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design
## 
## Attaching package: 'DoE.base'
## The following objects are masked from 'package:stats':
## 
##     aov, lm
## The following object is masked from 'package:graphics':
## 
##     plot.design
## The following object is masked from 'package:base':
## 
##     lengths
model<-lm(tool_life~a*b*c, data=dat)
coef(model)
## (Intercept)           a           b           c         a:b         a:c 
##      40.875       0.625       6.375       4.875      -0.875      -6.875 
##         b:c       a:b:c 
##      -2.625      -3.375
halfnormal(model)

## no significant effects

Question number 3(a)

From the halfnormal plot, we can see that there no significant effects.

Question number 3(b)

Since we can see from the halfnormal plot that none of the effects are significant, our blocks also don’t seem to be significant.