\(H_0:\tau_i=0\)
\(H_1:\tau_i\neq0\)
chemical<-c(rep(1,5),rep(2,5),rep(3,5),rep(4,5))
chemical <- as.factor(chemical)
bolt<-c(seq(1,5),seq(1,5),seq(1,5),seq(1,5))
bolt <- as.factor(bolt)
ob <- c(73,68,74,71,67,73,67,75,72,70,75,68,78,73,68,73,71,75,75,69)
dat <- data.frame(ob,chemical,bolt)
model <- aov(ob~chemical+bolt,data = dat)
summary(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 ' ' 1As evident from these results we can’t reject the null hypothesis with \(\alpha=0.05\) therefore we can’t say that there is a significant difference between these chemicals.
Linear equation model:
\(X_{ij}=\mu+\tau_i+\beta{j}+\epsilon_{ij}\)
In the question 4.3, \(\tau_i\) is the effect of the treatments (chemical).
\(\mu=\dfrac{35}{20},\tau_1=\dfrac{-23}{20},\tau_2=\dfrac{-7}{20},\tau_3=\dfrac{13}{20}\) \(\tau_4=\dfrac{17}{20},\beta_1=\dfrac{35}{20},\beta_2=\dfrac{-65}{20},\beta_3=\dfrac{75}{20},\) \(\beta_4=\dfrac{20}{20},\beta_5=\dfrac{-65}{20}\)
\(H_0:\tau_i=0\)
\(H_1:\tau_i\neq0\)
Batch <- c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))
day <- c(seq(1,5),seq(1,5),seq(1,5),seq(1,5),seq(1,5))
ingredient <- c('A','B','D','C','E','C','E','A','D','B','B','A','C','E','D','D','C','E','B','A','E','D','B','A','C')
obs <- c(8,7,1,7,3,11,2,7,3,8,4,9,10,1,5,6,8,6,6,10,4,2,3,8,8)
Batch <- as.factor(Batch)
day <- as.factor(day)
ingredient <- as.factor(ingredient)
data <- data.frame(obs,Batch,day,ingredient)
model <- aov(obs~Batch+day+ingredient,data = data)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Batch 4 15.44 3.86 1.235 0.347618
## day 4 12.24 3.06 0.979 0.455014
## ingredient 4 141.44 35.36 11.309 0.000488 ***
## Residuals 12 37.52 3.13
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As the resulted p-value for our factor of interest (ingredient) is 0.000488 and lower than our specified \(\alpha\) (0.05), we reject the null hypothesis and conclude that the choice of ingredient has a significant effect on the reaction time.