Problem 14.3
Machine <- c(rep(1,8),rep(2,8),rep(3,8))
Spindle <- rep(c(rep(1,4),rep(2,4)),3)
Obs <- c(12,9,11,12,8,9,10,8,14,15,13,14,12,10,11,13,14,10,12,11,16,15,15,14)
dat <- data.frame(Machine,Spindle,Obs)
dat
## Machine Spindle Obs
## 1 1 1 12
## 2 1 1 9
## 3 1 1 11
## 4 1 1 12
## 5 1 2 8
## 6 1 2 9
## 7 1 2 10
## 8 1 2 8
## 9 2 1 14
## 10 2 1 15
## 11 2 1 13
## 12 2 1 14
## 13 2 2 12
## 14 2 2 10
## 15 2 2 11
## 16 2 2 13
## 17 3 1 14
## 18 3 1 10
## 19 3 1 12
## 20 3 1 11
## 21 3 2 16
## 22 3 2 15
## 23 3 2 15
## 24 3 2 14
Machine <- as.factor(Machine)
Spindle <- as.factor(Spindle)
model <- lm(Obs~Machine+Spindle%in%Machine)
aovmodel <- aov(model)
summary(aovmodel)
## Df Sum Sq Mean Sq F value Pr(>F)
## Machine 2 55.75 27.875 18.934 3.74e-05 ***
## Machine:Spindle 3 43.75 14.583 9.906 0.000443 ***
## Residuals 18 26.50 1.472
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(aovmodel)
From the p-values, it is implied that the machine factor and the spindle (nested) factor are both significant. The aov() function was used instead of the gad() so that both factors could be analyzed as fixed. Additionally, from the residuals plot and normal probability plot, the model has constant variance and normality.
R Code
Machine <- c(rep(1,8),rep(2,8),rep(3,8))
Spindle <- rep(c(rep(1,4),rep(2,4)),3)
Obs <- c(12,9,11,12,8,9,10,8,14,15,13,14,12,10,11,13,14,10,12,11,16,15,15,14)
dat <- data.frame(Machine,Spindle,Obs)
dat
Machine <- as.factor(Machine)
Spindle <- as.factor(Spindle)
model <- lm(Obs~Machine+Spindle%in%Machine)
aovmodel <- aov(model)
summary(aovmodel)
plot(aovmodel)