Homework10
Ayodeji, Md Ariful Haque
2022-11-05
Question 5.4
Feedrate<- c(rep(1,12), rep(2,12), rep(3,12))
cut<- rep(seq(1,4),9)
obs <- c(74, 79, 82, 99, 64, 68, 88, 104, 60, 73, 92, 96, 92, 98, 99,
104, 86, 104, 108, 110, 88, 88, 95, 99, 99, 104, 108, 114,
98, 99, 110, 111, 102, 95, 99, 107)
dat<- data.frame(Feedrate,cut,obs)Question 5.4 a
Testing at level of significance=0.05
Using the model equation of
\(Yijk\)=\(\mu\)+\(\alpha i\)+\(\beta j\)+\(\alpha\beta ij\)+\(\varepsilon ijk\)
where \(\varepsilon ijk\) \(\sim \left( 0,\sigma^{2} \right)\)
Stating our null and alternative hypothesis we have
Null hypothesis for interaction effects are
\(\alpha\beta ij=0\) for all ij
Alternative hypothesis for interaction effects are
\(\alpha\beta ij\neq 0\) for some ij
Main effect hypothesis
For Null hypothesis
\(\alpha i=0\) for all i
for alternative hypothesis
\(\alpha i\neq 0\) for some i
Null hypothesis
\(\beta j=0\) for all j
\(\beta j\neq 0\) for some j
library(GAD)
Feedrate<- as.fixed(Feedrate)
cut <- as.fixed(cut)
Model <- aov(obs~cut+Feedrate+cut*Feedrate)
GAD::gad(Model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## cut 3 2125.11 708.37 24.6628 1.652e-07 ***
## Feedrate 2 3160.50 1580.25 55.0184 1.086e-09 ***
## cut:Feedrate 6 557.06 92.84 3.2324 0.01797 *
## Residual 24 689.33 28.72
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the values obtained from our model we can see that
the p-value for the interaction model (0.01797) is less than our reference significant level of alpha(0.05) , hence we are stating that we are rejecting the null hypothesis and saying that there is interaction between the feedrate and depth of cut in our analysis
The p-value for our feed-rate was highly significant with p=1.086e-09, which was less than 0.05
The p-value for our depth of cut was also highly significant with p=1.652e-07, which was less than 0.05
interaction.plot(Feedrate,cut,obs)
The above interaction plot concludes that we have an interaction effects between the two factors considered (depth of cut and feed-rate)
Question 5.4 b
plot(Model)
From the residual plot, we can see that the Normal probability plot of the residuals falls fairly on a straight line and hence we can validate normality
Since the residual vs fitted plot falls fairly on a straight line ,we can roughly assume constant variance in our model
Question 5.4 c
mean(dat$obs[1:12])## [1] 81.58333var(dat$obs[1:12])## [1] 205.5379mean(dat$obs[13:24])## [1] 97.58333var(dat$obs[13:24])## [1] 64.08333mean(dat$obs[25:36])## [1] 103.8333var(dat$obs[25:36])## [1] 36.87879point estimate for feed-rate 0.2
mean= 81.58 and variance=205.53
point estimate for feed-rate 0.25
mean= 97.58 and variance=64.08
point estimate for feed-rate 0.3
mean=103.83 and variance=36.87
Part D
Find the P-values for model in part (a).
P-value of Depth of Cut & Feedrate Interaction: 0.01797(significant)
P-value of Depth of Cut : 1.652e-07 (significant)
P-value of Feedrate : 1.086e-09(significant)
QUESTION 5.34
Using the model equation of
\(Yijkl\)=\(\mu\)+\(\alpha i\)+\(\beta j\)+\(\gamma k\)+\(\alpha\beta ij\)+\(\varepsilon ijk\)
\(\gamma k\)=block effect
where \(\varepsilon ijk\) \(\sim \left( 0,\sigma^{2} \right)\)
Stating our null and alternative hypothesis we have
Null hypothesis for interaction effects are
\(\alpha\beta ij=0\) for all ij
Alternative hypothesis for interaction effects are
\(\alpha\beta ij\neq 0\) for some ij
Main effect hypothesis
For Null hypothesis
\(\alpha i=0\) for all i
for alternative hypothesis
\(\alpha i\neq 0\) for some i
Null hypothesis
\(\beta j=0\) for all j
\(\beta j\neq 0\) for some j
Feedrate<- c(rep(1,12), rep(2,12), rep(3,12))
cut<- rep(seq(1,4),9)
block<- c(rep(1,4),rep(2,4),rep(3,4),rep(1,4), rep(2,4),rep(3,4),rep(1,4),rep(2,4),rep(3,4))
obs<- c(74, 79, 82, 99, 64, 68, 88, 104, 60, 73, 92,96, 92, 98, 99,104, 86, 104, 108, 110, 88, 88, 95,99, 99, 104, 108, 114, 98, 99, 110, 111, 102, 95, 99, 107)
dat<- data.frame(cut,Feedrate,block,obs)library(GAD)
Feedrate<- as.fixed(Feedrate)
cut <- as.fixed(cut)
block <- as.fixed(block)
Model2<- aov(obs~cut+Feedrate+block+cut*Feedrate)
summary(Model2)## Df Sum Sq Mean Sq F value Pr(>F)
## cut 3 2125.1 708.4 30.637 4.89e-08 ***
## Feedrate 2 3160.5 1580.2 68.346 3.64e-10 ***
## block 2 180.7 90.3 3.907 0.03532 *
## cut:Feedrate 6 557.1 92.8 4.015 0.00726 **
## Residuals 22 508.7 23.1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Checking out the p-value(0.00726<0.05) of the interaction effects, we can see there exists an interaction effect between the two factors.
We can also see that the p-values of the main effects are also significant
p-value of depth of cut - 4.89e-08 (highly significant at alpha=0.05)
p-value of feed-rate - 3.64e-10 ((highly significant at alpha=0.05)
The variance components of blocks is
Msb=90.3
Mse=23.1
i=3 , j=4
i*j=12
var<- (90.3-23.1)/12
var## [1] 5.6The variance of the block= 5.6
from our model the p-value of blocks is 0.03532 which is less than our reference significant level of alpha(0.05), we can confidently state that blocking had a significant effect in this case and therefore we conclude that it was important to block.
Interaction plot
interaction.plot(Feedrate,cut,obs)
The above interaction plot concludes that we have an interaction effects between the two factors considered (depth of cut and feed-rate).
Question 13.5
Using the model equation of
\(Yijk\)=\(\mu\)+\(\alpha i\)+\(\beta j\)+\(\alpha\beta ij\)+\(\varepsilon ijk\)
where \(\varepsilon ijk\) \(\sim \left( 0,\sigma^{2} \right)\)
Stating our null and alternative hypothesis we have
Null hypothesis for interaction effects are
\(\sigma^{2}_{\alpha\beta}=0\)
Alternative hypothesis for interaction effects are
\(\sigma^{2}_{\alpha\beta}\neq 0\)
Main effects (hypothesis)
Null hypothesis
\(\sigma^{2}_{\alpha}=0\)
Alternative hypothesis
\(\sigma^{2}_{\alpha}\neq 0\)
Null hypothesis
\(\sigma^{2}_{\beta}=0\)
Alternative hypothesis
\(\sigma^{2}_{\beta}\neq 0\)
p<- c(rep(1,9), rep(2,9))
t<- rep(seq(1,3),6)
obs<- c(570, 1063, 565, 565, 1080, 510, 583, 1043, 590, 528, 988, 526, 547, 1026,
538, 521, 1004, 532)
dat<- data.frame(p,t,obs)library(GAD)
p<- as.random(p)
t<- as.fixed(t)
model3<- aov(obs~p+t+p*t)
GAD::gad(model3)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## p 1 7160 7160 15.998 0.0017624 **
## t 2 945342 472671 1155.518 0.0008647 ***
## p:t 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Checking out the p-value(0.4271101 >0.05) of the interaction effects(position and temperature), we can see there exists no interaction effect between the two factors.(we are failing to reject the null hypothesis)
But
We can also that the p-values of the main effects are all significant
p-value of position - 0.0017624 (highly significant at alpha=0.05)
p-value of temperature - 0.0008647 (highly significant at alpha=0.05)
Question 13.6
Using the model equation of
\(Yijk\)=\(\mu\)+\(\alpha i\)+\(\beta j\)+\(\alpha\beta ij\)+\(\varepsilon ijk\)
where \(\varepsilon ijk\) \(\sim \left( 0,\sigma^{2} \right)\)
Stating our null and alternative hypothesis we have
Null hypothesis for interaction effects are
\(\sigma^{2}_{\alpha\beta}=0\)
Alternative hypothesis for interaction effects are
\(\sigma^{2}_{\alpha\beta}\neq 0\)
Main effects (hypothesis)
Null hypothesis
\(\sigma^{2}_{\alpha}=0\)
Alternative hypothesis
\(\sigma^{2}_{\alpha}\neq 0\)
Null hypothesis
\(\sigma^{2}_{\beta}=0\)
Alternative hypothesis
\(\sigma^{2}_{\beta}\neq 0\)
parts<- c(rep(1,6), rep(2,6), rep(3,6), rep(4,6), rep(5,6), rep(6,6), rep(7,6), rep(8,6), rep(9,6), rep(10,6))
operators<- c(rep(1,3), rep(2,3), rep(1,3), rep(2,3),rep(1,3), rep(2,3), rep(1,3),rep(2,3), rep(1,3), rep(2,3), rep(1,3),rep(2,3), rep(1,3), rep(2,3),
rep(1,3), rep(2,3), rep(1,3), rep(2,3), rep(1,3), rep(2,3))
obs<- c(50, 49, 50, 50, 48, 51, 52, 52, 51, 51, 51, 51, 53, 50, 50, 54, 52, 51, 49, 51, 50, 48, 50, 51, 48, 49, 48, 48, 49, 48, 52, 50, 50, 52, 50, 50,51, 51, 51, 51, 50, 50, 52, 50, 49, 53, 48, 50, 50, 51, 50, 51, 48, 49,47, 46, 49, 46, 47, 48)
dat3<- data.frame(parts,operators,obs)library(GAD)
operators<- as.fixed(operators)
parts<-as.random(parts)
model4<- aov(obs~parts+operators+parts*operators)
GAD::gad(model4)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## parts 9 99.017 11.0019 7.3346 3.216e-06 ***
## operators 1 0.417 0.4167 0.6923 0.4269
## parts:operators 9 5.417 0.6019 0.4012 0.9270
## Residual 40 60.000 1.5000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Checking out the p-value (0.9270>0.05) of the interaction effects(operator and parts), we can see there exists no interaction effect between the two factors.(we are failing to reject the null hypothesis)
But
p-value of parts - 3.216e-06 (highly significant at alpha=0.05)
p-value of operator - 0.4269 (insignificant at alpha=0.05)