Question number 1 (Blocked case)

Linear Effects Model:

\(Y_{ij}=\mu+\alpha_i+\beta_j+\varepsilon_{ij}\)

where,

\(\mu\) = Grand mean

\(\alpha_i\) = fixed effects for treatment i

\(\beta_j\) = Block effect for j

\(\varepsilon_{ij}\) = Random error

Hypothesis:

\(H_{0}: \mu_1=\mu_2=\mu_3=\mu_4\)

\(H_{a}\): at least one \(\mu_i\) differs

library(GAD)
obs<- c(73, 68, 74, 71, 73, 67, 75, 72, 75, 68, 78, 73, 73, 71, 75, 75)
chemical<-c(rep(1,4), rep(2,4), rep(3,4), rep(4,4))
bolt<-c(rep(seq(1,4),4))
chemical<-as.fixed(chemical)
bolt<-as.random(bolt)

model<-lm(obs~chemical + bolt)
gad(model)
## $anova
## Analysis of Variance Table
## 
## Response: obs
##           Df  Sum Sq Mean Sq F value    Pr(>F)    
## chemical   3  14.187   4.729  2.7349 0.1057217    
## bolt       3 104.188  34.729 20.0843 0.0002515 ***
## Residuals  9  15.563   1.729                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Here, the p-value (0.1057217) is less than alpha (0.15). So we reject null hypothesis.

Question number 2 (Unblocked case)

Linear effect equation:

\(Y_{ij}=\mu +\alpha_{i}+\varepsilon _{ij}\)

where, \(\mu\) = Grand mean

\(\alpha_i\) = fixed effects for treatment i

\(\varepsilon_{ij}\) = Random error

Hypothesis:

\(H_{0}: \mu_1=\mu_2=\mu_3=\mu_4\)

\(H_{a}\): at least one \(\mu_i\) differs

model2<- lm(obs~chemical)
gad(model2)
## $anova
## Analysis of Variance Table
## 
## Response: obs
##           Df  Sum Sq Mean Sq F value Pr(>F)
## chemical   3  14.187  4.7292  0.4739 0.7062
## Residuals 12 119.750  9.9792

Here, the p-value(0.7062) is greater than alpha (0.15). Hence, we fail to reject the null hypothesis.

Question number 3

While comparing the p-value with the alpha , we rejected the null hypothesis in question number 1 whereas we failed to reject the null hypothesis in question number 2. We can see that the effect of blocking included nuisance variability and led to rejection of null hypothesis. So, we can conclude that the bolt of cloth represents a significant amount of nuisance variability.

Complete R-code

library(GAD)
obs<- c(73, 68, 74, 71, 73, 67, 75, 72, 75, 68, 78, 73, 73, 71, 75, 75)
chemical<-c(rep(1,4), rep(2,4), rep(3,4), rep(4,4))
bolt<-c(rep(seq(1,4),4))
chemical<-as.fixed(chemical)
bolt<-as.random(bolt)

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

model2<- lm(obs~chemical)
gad(model2)