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problem 1

Confound the ABC interaction with the blocks because the three factor interactions are usually assumed to be negligible.

Two blocks, each containing four corner points.

Block Treatment combinations

1 (1), ab, ac, bc

2 a, b, c, abc

problem 2

Confound AB, AC, and BC with blocks. Taking AB and AC as block generators and their product

BC with blocks.

(AB)*(AC)= BC

Four blocks, each containing two corner points.

block Treatment combinations

1 (1), abc

2 a, bc

3 b, ac

4 c, ab

Problem 3

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

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)

toollife<-c(22,32,35,55,44,40,60,39)

d12<-data.frame(A,B,C,toollife)
d12

##    A  B  C toollife
## 1 -1 -1 -1       22
## 2  1 -1 -1       32
## 3 -1  1 -1       35
## 4  1  1 -1       55
## 5 -1 -1  1       44
## 6  1 -1  1       40
## 7 -1  1  1       60
## 8  1  1  1       39

mod<-lm(toollife~ A*B*C, data=d12)

halfnormal(mod)

## no significant effects

No effects appear to be significant based on the half normal plot.

The block doesn’t appear to be significant because ABC, which is confounded with blocks, is not significant.