Question 5.4
library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.
feedrate<-c(rep(1,12),rep(2,12),rep(3,12))
depthcut<-rep(seq(1,4),9)
Observations<-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,depthcut,Observations)
feedrate<-as.fixed(feedrate)
depthcut<-as.fixed(depthcut)
model<-aov(Observations~depthcut+feedrate+depthcut*feedrate)
GAD::gad(model)
## Analysis of Variance Table
##
## Response: Observations
## Df Sum Sq Mean Sq F value Pr(>F)
## depthcut 3 2125.11 708.37 24.6628 1.652e-07 ***
## feedrate 2 3160.50 1580.25 55.0184 1.086e-09 ***
## depthcut: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 ' ' 1
B)From the Normality plot and residual vs factor plot we can say that our assumption of normality and constant variance is somewhat correct. Hence our data is adequate
mean_0.20<-mean(dat$Observations[1:12])
mean_0.25<-mean(dat$Observations[13:24])
mean_0.30<-mean(dat$Observations[25:36])
mean_0.20
## [1] 81.58333
mean_0.25
## [1] 97.58333
mean_0.30
## [1] 103.8333
C)The point estimate of the mean surface finish at each feed rate is as follows
Mean at 0.20=81.5833
Mean at 0.25=97.5833
Meant at 0.30=103.833
D)Following are the P values
P value of depth of cut = 1.652e-07(Significant)
P value of feed rate = 1.086e-09(significant)
P value of interaction between depth of cut and feed rate = 0.01797(significant)
Question 5.34
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))
blocking<-as.fixed(block)
dat2<-data.frame(depthcut,feedrate,blocking,Observations)
model1<-aov(Observations~depthcut+feedrate+blocking+depthcut*feedrate)
GAD::gad(model1)
## Analysis of Variance Table
##
## Response: Observations
## Df Sum Sq Mean Sq F value Pr(>F)
## depthcut 3 2125.11 708.37 30.6373 4.893e-08 ***
## feedrate 2 3160.50 1580.25 68.3463 3.635e-10 ***
## blocking 2 180.67 90.33 3.9069 0.035322 *
## depthcut:feedrate 6 557.06 92.84 4.0155 0.007258 **
## Residual 22 508.67 23.12
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
From the above data we can say that blocking has a significant effect of 3.64e-10 which is less than our alpha i.e 0.05
Variance components i=4,j=3,k=3,
variance component for block is (Mean square of blocking-Mean square error)/i*j
=(90.3-23.1)/4*3
=5.6
Question 13.5
Furnace_position<-c(rep(1,9), rep(2,9))
Furnace_Temperature<-rep(seq(1,3),6)
Observation1<-c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
dat3<-data.frame(Furnace_position,Furnace_Temperature,Observation1)
Furnace_Temperature<-as.fixed(Furnace_Temperature)
Furnace_position<-as.random(Furnace_position)
model2<-aov(Observation1~Furnace_position+Furnace_Temperature+Furnace_position*Furnace_Temperature)
GAD::gad(model2)
## Analysis of Variance Table
##
## Response: Observation1
## Df Sum Sq Mean Sq F value Pr(>F)
## Furnace_position 1 7160 7160 15.998 0.0017624 **
## Furnace_Temperature 2 945342 472671 1155.518 0.0008647 ***
## Furnace_position:Furnace_Temperature 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
From the above table we can say that the p value of the interaction between temperature and position is larger than alpha i.e 0.05 hence they are not significant while the p value of temperature and position as main effect is less than alpha so they are significant
Q 13.6
Part_No<-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))
Operator_Measurements<-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))
Observation2<-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)
dat4<-data.frame(Part_No,Operator_Measurements,Observation2)
Part_No<-as.random(Part_No)
Operator_Measurements<-as.fixed(Operator_Measurements)
model3<-aov(Observation2~Part_No+Operator_Measurements+Part_No*Operator_Measurements)
GAD::gad(model3)
## Analysis of Variance Table
##
## Response: Observation2
## Df Sum Sq Mean Sq F value Pr(>F)
## Part_No 9 99.017 11.0019 7.3346 3.216e-06 ***
## Operator_Measurements 1 0.417 0.4167 0.6923 0.4269
## Part_No:Operator_Measurements 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 ' ' 1
From the above table we can conclude that the p value of the interction between operator and part is greater than alpha hence they are not significant
Also the p value of main effect of operator is greater than alpha hence it is non significant too
But the p value of part no is less than 0.05 hence it is significant
