Untitled
Jestin Job Sibychan
10/10/2021
Q1)Problem 3.23
Ftypes<-c(17.6,18.9,16.3,17.4,20.1,21.6,16.9,15.3,18.6,17.1,19.5,20.3,21.4,23.6,19.4,18.5,20.5,22.3,19.3,21.1,16.9,17.5,18.3,19.8)
z<-c(rep(1,6), rep(2,6), rep(3,6), rep(4,6))
dat<-data.frame(Ftypes,z)
z<-as.factor(dat$z)
boxplot(Ftypes~z)
anv<-aov(Ftypes~z, data = dat)
summary(anv)## Df Sum Sq Mean Sq F value Pr(>F)
## z 1 3.67 3.675 0.874 0.36
## Residuals 22 92.48 4.204plot(anv)
Ftype1<-c(17.6,18.9,16.3,17.4,20.1,21.6)
Ftype2<-c(16.9,15.3,18.6,17.1,19.5,20.3)
Ftype3<-c(21.4,23.6,19.4,18.5,20.5,22.3)
Ftype4<-c(19.3,21.1,16.9,17.5,18.3,19.8)
mean(Ftype1)## [1] 18.65mean(Ftype2)## [1] 17.95mean(Ftype3)## [1] 20.95mean(Ftype4)## [1] 18.81667Here we can see that the value of P is greater than alpha hence we failed to reject the null hypothesis
Here we can see that the mean of fluid 3 is the highest hence it has the longest life
From the boxplot we can see that the basic assumption that the variances are equal is unsatisfied
Q2)Problem 3.51
kruskal.test(Ftypes~z, data = dat)##
## Kruskal-Wallis rank sum test
##
## data: Ftypes by z
## Kruskal-Wallis chi-squared = 6.2177, df = 3, p-value = 0.1015As the value of P is 0.1015 which is greater than alpha hence we fail to reject the null hypothesis
Q2)Problem 3.28
materials<-c(110,157,194,178,1,2,4,18,880,1256,5276,4355,495,7040,5307,10050,7,5,29,2)
y<-c(rep(1,4), rep(2,4), rep(3,4), rep(4,4),rep(5,4))
dat1<-data.frame(materials,y)
y<-as.factor(dat1$y)
avn<-aov(materials~y,data = dat1)
plot(avn)
boxplot(materials~y)
##we can see that the variances are not approximately the same hence we need to equate the variance using boxcox
library(MASS)
boxcox(materials~y)
### here we can see that the lambda is near 0 hence we take lambda as 0.1
lambda<-0.1
material<-materials^(lambda)
boxplot(material~y)
nav<-aov(material~y, data = dat1)
summary(nav)## Df Sum Sq Mean Sq F value Pr(>F)
## y 1 0.028 0.02821 0.103 0.752
## Residuals 18 4.940 0.27446plot(nav)
library(agricolae)
b<-LSD.test(avn, "material", console = TRUE)##
## Study: avn ~ "material"
##
## LSD t Test for materials
##
## Mean Square Error: 8552889
##
## materials, means and individual ( 95 %) CI
##
## materials std r LCL UCL Min Max
## 1 1 NA 1 -6143.2144 6145.214 1 1
## 2 2 0 2 -4342.6156 4346.616 2 2
## 4 4 NA 1 -6140.2144 6148.214 4 4
## 5 5 NA 1 -6139.2144 6149.214 5 5
## 7 7 NA 1 -6137.2144 6151.214 7 7
## 18 18 NA 1 -6126.2144 6162.214 18 18
## 29 29 NA 1 -6115.2144 6173.214 29 29
## 110 110 NA 1 -6034.2144 6254.214 110 110
## 157 157 NA 1 -5987.2144 6301.214 157 157
## 178 178 NA 1 -5966.2144 6322.214 178 178
## 194 194 NA 1 -5950.2144 6338.214 194 194
## 495 495 NA 1 -5649.2144 6639.214 495 495
## 880 880 NA 1 -5264.2144 7024.214 880 880
## 1256 1256 NA 1 -4888.2144 7400.214 1256 1256
## 4355 4355 NA 1 -1789.2144 10499.214 4355 4355
## 5276 5276 NA 1 -868.2144 11420.214 5276 5276
## 5307 5307 NA 1 -837.2144 11451.214 5307 5307
## 7040 7040 NA 1 895.7856 13184.214 7040 7040
## 10050 10050 NA 1 3905.7856 16194.214 10050 10050
##
## Alpha: 0.05 ; DF Error: 18
## Critical Value of t: 2.100922
##
## Groups according to probability of means differences and alpha level( 0.05 )
##
## Treatments with the same letter are not significantly different.
##
## materials groups
## 10050 10050 a
## 7040 7040 ab
## 5307 5307 ab
## 5276 5276 ab
## 4355 4355 ab
## 1256 1256 b
## 880 880 b
## 495 495 b
## 194 194 b
## 178 178 b
## 157 157 b
## 110 110 b
## 29 29 b
## 18 18 b
## 7 7 b
## 5 5 b
## 4 4 b
## 2 2 b
## 1 1 b##a)all five material doesnt have the same mean effect failure time as in boxplot we can see that material 4 differs a lot from the others ##b) from Above table of LSD we can see that the treatments with the same letters are the same
Q3) Problem 3.29
method<-c(31,10,21,4,1,62,40,24,30,35,53,27,120,97,68)
dat2<-c(rep(1,5), rep(2,5), rep(3,5))
cmb<-data.frame(method,dat2)
dat2<-as.factor(cmb$dat2)
library(MASS)
boxplot(method~dat2)
### here we can see that the variances are unequal hence we need to do the boxcox
boxcox(method~dat2)
## we can see that lambda is less than 0.5 hence we take lambda as 0.4
lambda<-0.4
methods<-method^lambda
boxplot(methods~dat2)
methods1<-log(method)
boxplot<-(methods1~dat2)
anva<-aov(methods~dat2, data = cmb)
library(agricolae)
LSD.test(anva, "methods", console = TRUE)##
## Study: anva ~ "methods"
##
## LSD t Test for methods
##
## Mean Square Error: 1.010105
##
## methods, means and individual ( 95 %) CI
##
## methods std r LCL UCL Min Max
## 1 1.000000 NA 1 -1.1712564 3.171256 1.000000 1.000000
## 1.74110112659225 1.741101 NA 1 -0.4301553 3.912358 1.741101 1.741101
## 2.51188643150958 2.511886 NA 1 0.3406301 4.683143 2.511886 2.511886
## 3.37977444523543 3.379774 NA 1 1.2085181 5.551031 3.379774 3.379774
## 3.56520491593201 3.565205 NA 1 1.3939485 5.736461 3.565205 3.565205
## 3.73719281884655 3.737193 NA 1 1.5659364 5.908449 3.737193 3.737193
## 3.89805984091619 3.898060 NA 1 1.7268035 6.069316 3.898060 3.898060
## 3.9495232751503 3.949523 NA 1 1.7782669 6.120780 3.949523 3.949523
## 4.14598014312126 4.145980 NA 1 1.9747238 6.317237 4.145980 4.145980
## 4.37344829577311 4.373448 NA 1 2.2021919 6.544705 4.373448 4.373448
## 4.89452270907168 4.894523 NA 1 2.7232663 7.065779 4.894523 4.894523
## 5.21142720534249 5.211427 NA 1 3.0401708 7.382684 5.211427 5.211427
## 5.40758761965103 5.407588 NA 1 3.2363312 7.578844 5.407588 5.407588
## 6.23316600928143 6.233166 NA 1 4.0619096 8.404422 6.233166 6.233166
## 6.78691638054318 6.786916 NA 1 4.6156600 8.958173 6.786916 6.786916
##
## Alpha: 0.05 ; DF Error: 13
## Critical Value of t: 2.160369
##
## least Significant Difference: 3.07062
##
## Treatments with the same letter are not significantly different.
##
## methods groups
## 6.78691638054318 6.786916 a
## 6.23316600928143 6.233166 ab
## 5.40758761965103 5.407588 abc
## 5.21142720534249 5.211427 abc
## 4.89452270907168 4.894523 abc
## 4.37344829577311 4.373448 abcd
## 4.14598014312126 4.145980 abcd
## 3.9495232751503 3.949523 abcde
## 3.89805984091619 3.898060 abcde
## 3.73719281884655 3.737193 abcde
## 3.56520491593201 3.565205 bcde
## 3.37977444523543 3.379774 bcde
## 2.51188643150958 2.511886 cde
## 1.74110112659225 1.741101 de
## 1 1.000000 eqqnorm(method)
qqline(method)