1 Assignment Description

A rocket propellant manufacturer is studying the burning rate of propellant from three production processes. Four batches of propellant are randomly selected from the output of each process, and three determinations of burning rate are made on each batch. The results follow. State the model equation and hypotheses to be tested. Perform the analysis and draw conclusions, using alpha=0.05 where applicable.

1.1 Solution

Model Equation:

\[ Y_{ijk}=\mu +\tau_{i}+\alpha_{j(i)}+\epsilon _{k(ij)} \]

Where:

Ï„ represents the process effect

α represents the batch (random) effect

library(GAD)
results<-c(25,19,15,15,
           19,23,18,35,
           14,35,38,25,
           30,28,17,16,
           17,24,21,27,
           15,21,54,29,
           26,20,14,13,
           14,21,17,25,
           20,24,50,33)
batch<-as.random(rep(seq(1,4),9))
obs<-as.fixed(c(rep(1,12),rep(2,12),rep(3,12)))
process<-as.fixed(rep(c(rep(1,4),rep(2,4),rep(3,4)),3))
datafr<-data.frame(process,batch,obs,results)
model<-lm(results~process+batch%in%process)
gad(model)
## Analysis of Variance Table
## 
## Response: results
##               Df  Sum Sq Mean Sq F value    Pr(>F)    
## process        2  676.06  338.03  1.4643    0.2815    
## process:batch  9 2077.58  230.84 12.2031 5.477e-07 ***
## Residual      24  454.00   18.92                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion:

Considering a 95% significance, it seems that the process used is not significant due to the P value being 0.2815 (>.05), but the batches within the process does seem to be significant on the mean burning rate since the P value is very small.

plot(model)

Based on the graphs shown above, we can see the residuals follows a trend line and the normal Q-Q plot seems to be straight. Therefore, the assumptions of normality and constant variance is adequate for this model.

2 Complete R Code

library(GAD)
library(GAD)
results<-c(25,19,15,15,
           19,23,18,35,
           14,35,38,25,
           30,28,17,16,
           17,24,21,27,
           15,21,54,29,
           26,20,14,13,
           14,21,17,25,
           20,24,50,33)
batch<-as.random(rep(seq(1,4),9))
obs<-as.fixed(c(rep(1,12),rep(2,12),rep(3,12)))
process<-as.fixed(rep(c(rep(1,4),rep(2,4),rep(3,4)),3))
datafr<-data.frame(process,batch,obs,results)
model<-lm(results~process+batch%in%process)
gad(model)
#####
plot(model)