1 Question 5.2:

1.1 Solution:

PART A:

---> Finding degrees of freedom for source B:

Dof_total<-15
Dof_error<-8
Dof_interaction<-3
Dof_A<-1

Dof_B<-Dof_total-Dof_error-Dof_interaction-Dof_A
print(Dof_B)
## [1] 3

---> Finding SSA:

MSA<-0.0002
SSA<-MSA*Dof_A
print(SSA)
## [1] 2e-04

---> Finding MSB, MSAB and MSE:

SSB<-180.378
SSAB<-8.479
SSE<-158.797

MSB<-SSB/Dof_B
MSAB<-SSAB/Dof_interaction
MSE<-SSE/Dof_error

print(MSB)
## [1] 60.126
print(MSAB)
## [1] 2.826333
print(MSE)
## [1] 19.84962

----> Finding F-Statistic Values:

Fa<-MSA/MSE
Fb<-MSB/MSE
Fab<-MSAB/MSE

print(Fa)
## [1] 1.007576e-05
print(Fb)
## [1] 3.029075
print(Fab)
## [1] 0.1423872

----> Finding P-Values:

pf(0.00001007576,1,8, lower.tail = FALSE)
## [1] 0.9975451
pf(3.0290,3,8, lower.tail = FALSE)
## [1] 0.09335136

PART B: (Levels of Source B)

LevelsofB<-Dof_B+1
b<-LevelsofB
print(b)
## [1] 4

---> Levels of B = 4

PART C: (Replicates of the Experiment)

LevelsofA<-Dof_A+1
a<-LevelsofA

#Dof_Total: 15 = a*b*n - 1
#a*b*n = 16
#n=16/a*b

n<-16/(a*b)
print(n)
## [1] 2

---> Replicates are runs with the same levels/factor combination, i.e. k=2 replicates per level set

PART D: (Conclusions)

Comparing F-Statistic Values with Critical Values of F:

Testing Interaction Effect:

Fcritical_interaction=qf(0.9317,3,8)
print(Fcritical_interaction)
## [1] 3.52881
—> Since F-Statistic (0.1424) < F-Critical (3.528), we fail to reject Null Hypothesis and say that interaction is insignificant.

Testing Main Effects:

Fcritical_A=qf(0.998,1,8)
print(Fcritical_A)
## [1] 20.25712
—> Since F-Statistic (0.00000101) < F-Critical (20.25), we fail to reject Null Hypothesis and say that Factor A is insignificant at any reasonable level of alpha. 
Fcritical_B=qf(0.093,3,8)
print(Fcritical_B)
## [1] 0.180005
—> Since F-Statistic (3.029) > F-Critical (0.18), we reject Null Hypothesis and say that Factor B is significant.

2 Question 5.9:

2.1 Solution:

Effects Model Equation:

\[ y_{i,j,k}=\mu+\alpha_{i}+\beta_{j}+\alpha\beta_{i,j}+\epsilon_{i,j,k} \]

where,

\(\alpha_{i}\) is the main effect of the ith treatment of Feed Rate used


\(\beta_{j}\) is the main effect of the jth treatment of Drill Speed used


\(\alpha\beta_{i,j}\) is the interaction effect of the ijth treatment of Feed Rate * Drill Speed, and


\(\epsilon_{i,j,k}\) is the random error term

Hypothesis:

Feed Rate Main Effect:

\[ H_o: \alpha_{i}=0 \space\forall\space"i" \]

\[H_a: \alpha_{i}\neq0\space"for\space ateast\space one\space"i"\]
Drill Speed Main Effect:

\[ H_o:\beta_{j}=0\space\forall "i" \]

\[ H_a: \beta_{i}\neq0\space"for\space ateast\space one\space"j"\]

Feed Rate * Drill Speed Interaction Effect:

\[ H_o:\alpha\beta_{i,j}=0\space \forall\space"i" \]

\[\alpha\beta_{i,j}\neq0\space"for\space ateast\space one\space"i,j"\]

FeedRate <- rep(seq(1,4),4)
DrillSpeed <- c(rep(1,8),rep(2,8))
ThrustForce <- c(2.70,2.45,2.60,2.75,
                 2.78,2.49,2.72,2.86,
                 2.83,2.85,2.86,2.94,
                 2.86,2.80,2.87,2.88)
dataframe1 <- data.frame(FeedRate,DrillSpeed,ThrustForce)
library(GAD)
FeedRate <- as.fixed(FeedRate)
DrillSpeed <- as.fixed(DrillSpeed)
model <- aov(ThrustForce~FeedRate+DrillSpeed+FeedRate*DrillSpeed)
GAD::gad(model)
## Analysis of Variance Table
## 
## Response: ThrustForce
##                     Df   Sum Sq  Mean Sq F value    Pr(>F)    
## FeedRate             3 0.092500 0.030833 11.8590  0.002582 ** 
## DrillSpeed           1 0.148225 0.148225 57.0096 6.605e-05 ***
## FeedRate:DrillSpeed  3 0.041875 0.013958  5.3686  0.025567 *  
## Residual             8 0.020800 0.002600                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion:

—> With P-values < 0.05 for both factors and the interaction factor, we reject H0 and conclude that the drill speed, feed rate, and the interaction are significant.