Question 1

## Loading required package: carData
#Input Data
obs <- c(73,68,74,71,67,73,67,75,72,70,75,68,78,73,68,73,71,75,75,69)
bolt <- c(seq(1,5),seq(1,5),seq(1,5),seq(1,5))
chem <- c(1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4)
chem <- as.fixed(chem)
bolt <- as.fixed(bolt)

Ho: mu1=mu2=mu3=mu4=mu5 Ha: at least one differs Yij = mu + Ti + Bj + Ei

model <- lm(obs~chem+bolt)
gad(model)
## $anova
## Analysis of Variance Table
## 
## Response: obs
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## chem       3  12.95   4.317  2.3761    0.1211    
## bolt       4 157.00  39.250 21.6055 2.059e-05 ***
## Residuals 12  21.80   1.817                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Reject the Null based on a p-value based on an alpha of 0.15

Question 2

Ho: mu1=mu2=mu3=mu4=mu5 Ha: at least one differs Yij = mu + Ti + Ei

##             Df Sum Sq Mean Sq F value Pr(>F)
## chem         3  12.95   4.317   0.386  0.764
## Residuals   16 178.80  11.175

Fail to reject Ho at an alpha of 0.15.

Question 3: The two tests have significantly different p-values as you cannot reject unless you block the design by the bolts. Therefore, the bolt does have a significant amount of nuisance variability. (This is supported by the p-value of the bolts outputted in question 1)

Complete Code

library(GAD)


obs <- c(73,68,74,71,67,73,67,75,72,70,75,68,78,73,68,73,71,75,75,69)
bolt <- c(seq(1,5),seq(1,5),seq(1,5),seq(1,5))
chem <- c(1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4)
chem <- as.fixed(chem)
bolt <- as.fixed(bolt)


model <- lm(obs~chem+bolt)
gad(model)


model <- lm(obs ~ chem)
summary(aov(model))