library(GAD)

chemical<-rep(1:4,each=5)
bolt<-rep(1:5,4)

strength<-c(73,68,74,71,67,
            73,67,75,72,70,
            75,68,78,73,68,
            73,71,75,75,69)

chemical<-as.fixed(chemical)
bolt<-as.random(bolt)

Question 1.

a. Linear Effects Model and Hypotheses

Linear effects model for the RCBD:

\[y_{ij}=\mu+\alpha_i+\beta_j+\epsilon_{ij}\]

where:

  • \(\mu\) is the overall mean tensile strength
  • \(\alpha_i\) is the fixed effect of chemical \(i\)
  • \(\beta_j\) is the random effect of bolt \(j\)
  • \(\epsilon_{ij}\) is the random error

Hypotheses:

Chemical treatments: \[H_0:\mu_1=\mu_2=\mu_3=\mu_4\]

\[H_a:\text{At least one } \mu_i\ {differs.}\]

Random block effect: \[H_0:\sigma_\beta^2=0\]

\[H_a:\sigma_\beta^2>0\]

b. ANOVA

rcbd.model<-aov(strength~chemical+bolt)
summary(rcbd.model)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## chemical     3  12.95    4.32   2.376    0.121    
## bolt         4 157.00   39.25  21.606 2.06e-05 ***
## Residuals   12  21.80    1.82                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

c. Conclusions

Since the p-value for Chemical = 0.1211 is less than \(\alpha=0.15\), we reject \(H_0\). There is sufficient evidence to conclude that at least one chemical has a different mean tensile strength.

Since the p-value = 0.000021 is than \(\alpha=0.15\), we reject \(H_0\) for the block effect. There is sufficient evidence that the bolts contribute significant variability.

Question 2. CRD

a. Linear Effects Model and Hypotheses

Linear effects model for CRD:

\[y_{ij}=\mu+\alpha_i+\epsilon_{ij}\]

where:

  • \(\mu\) is the overall mean tensile strength
  • \(\alpha_i\) is the fixed effect of Chemical \(i\)
  • \(\epsilon_{ij}\) is the random error

Hypotheses:

\[H_0:\mu_1=\mu_2=\mu_3=\mu_4\]

\[H_a:\text{At least one } \mu_i\ {differs.}\]

b. ANOVA

crd.model<-aov(strength~chemical)
summary(crd.model)
##             Df Sum Sq Mean Sq F value Pr(>F)
## chemical     3  12.95   4.317   0.386  0.764
## Residuals   16 178.80  11.175

c. Conclusion

Since p-value = 0.7644 is greater than \(\alpha=0.15\), we fail to reject \(H_0\). There is insufficient evidence to conclude that the mean tensile strengths differ among the four chemicals.

Question 3.

The RCBD found a significant difference between the chemical treatments at \(\alpha=0.15\), while the CRD did not. The error mean square for the RCBD was approx. 1.8167 compared to 11.1750 for the CRD, meaning that blocking on Bolt substantially reduced the unexplained variability. Since the block effect was significant at \(\alpha=0.15\), Bolt represents a signficant amount of nuisance variability. Thus, the RCBD is more effective than the CRD for this experiment.

R Script

library(GAD)

chemical<-rep(1:4,each=5)
bolt<-rep(1:5,4)

strength<-c(73,68,74,71,67,
            73,67,75,72,70,
            75,68,78,73,68,
            73,71,75,75,69)

chemical<-as.fixed(chemical)
bolt<-as.random(bolt)
rcbd.model<-aov(strength~chemical+bolt)
summary(rcbd.model)
crd.model<-aov(strength~chemical)
summary(crd.model)