Homework
Peihang Li
2024-10-25
5.2
a.
MS_A <- 0.0002
SS_B <- 180.378
SS_Interaction <- 8.479
SS_Error <- 158.797
SS_Total <- 347.653
DF_A <- 1
DF_Interaction <- 3
DF_Error <- 8
DF_Total <- 15
DF_B <- 3
SS_A <- MS_A * DF_A
MS_B <- SS_B / DF_B
MS_Interaction <- SS_Interaction / DF_Interaction
MS_Error <- SS_Error / DF_Error
MS_Total <- SS_Total / DF_Total
F_A <- MS_A / MS_Error
F_B <- MS_B / MS_Error
F_Interaction <- MS_Interaction / MS_Error
P_A <- pf(F_A, DF_A, DF_Error, lower.tail = FALSE)
P_B <- pf(F_B, DF_B, DF_Error, lower.tail = FALSE)b.
Levels in B = DF_B + 1 = 4
c.
DF_Error= I*J*(K-1)=8
I=2, J=4
K=2
d.
None of the factors have significant effects on the observations at 95% level.
5.4
a.
feedrate <- c(rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3))
depth <- c(rep(0.15,9),rep(0.18,9),rep(0.20,9),rep(0.25,9))
obs <- c(74,64,60,92,86,88,99,98,102,79,68,73,98,104,88,104,99,95,82,88,92,99,108,95,108,110,99,99,104,96,104,110,99,114,111,107)
depth <- as.factor(depth)
feedrate <- as.factor(feedrate)
dat <- data.frame(depth,feedrate,obs)
model <- aov(obs~depth+feedrate+depth*feedrate, data = dat)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## depth 3 2125.1 708.4 24.663 1.65e-07 ***
## feedrate 2 3160.5 1580.2 55.018 1.09e-09 ***
## depth:feedrate 6 557.1 92.8 3.232 0.018 *
## Residuals 24 689.3 28.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Both factors and their interaction have significant effects at 0.05 level.
b.
plot(model)
The data seem to be normally distributed and the variances appear to be constant.
c.
means <- aggregate(obs ~ feedrate, data = dat, mean)
sd <- aggregate(obs ~ feedrate, data = dat, sd)
print(means)## feedrate obs
## 1 0.2 81.58333
## 2 0.25 97.58333
## 3 0.3 103.83333print(sd)## feedrate obs
## 1 0.2 14.336592
## 2 0.25 8.005207
## 3 0.3 6.072791d.
The p-values for feed rate, depth and their interaction are respectively 1.65e-07,1.09e-09 and, 0.018.
5.9
Feed_rate<-c(rep(0.015,4),rep(0.030,4),rep(0.045,4),rep(0.060,4))
Drill_speed<-c(125,125,200,200,125,125,200,200,125,125,200,200,125,125,200,200)
obs<-c(2.7,2.78,2.83,2.86,2.45,2.49,2.85,2.8,2.6,2.72,2.86,2.87,2.75,2.86,2.94,2.88)
data<-data.frame(Feed_rate,Drill_speed,obs)
data$Feed_rate <- as.factor(data$Feed_rate)
data$Drill_speed <- as.factor(data$Drill_speed)
model <- aov(obs~Feed_rate+Drill_speed+Feed_rate*Drill_speed,data = data)
summary(model)## Df Sum Sq Mean Sq F value Pr(>F)
## Feed_rate 3 0.09250 0.03083 11.859 0.00258 **
## Drill_speed 1 0.14823 0.14823 57.010 6.61e-05 ***
## Feed_rate:Drill_speed 3 0.04187 0.01396 5.369 0.02557 *
## Residuals 8 0.02080 0.00260
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1plot(model)
Both factors and their interaction significantly affect the observations at 95% level.
5.34
feedrate <- c(rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3),rep(0.2,3),rep(0.25,3),rep(0.3,3))
depth <- c(rep(0.15,9),rep(0.18,9),rep(0.20,9),rep(0.25,9))
obs <- c(74,64,60,92,86,88,99,98,102,79,68,73,98,104,88,104,99,95,82,88,92,99,108,95,108,110,99,99,104,96,104,110,99,114,111,107)
depth <- as.factor(depth)
feedrate <- as.factor(feedrate)
block <- c(seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3),seq(1,3))
block <- as.factor(block)
dat1 <- data.frame(block,depth,feedrate,obs)
model1 <- aov(obs~block+depth+feedrate+depth*feedrate,data = dat1)
summary(model1)## Df Sum Sq Mean Sq F value Pr(>F)
## block 2 180.7 90.3 3.907 0.03532 *
## depth 3 2125.1 708.4 30.637 4.89e-08 ***
## feedrate 2 3160.5 1580.2 68.346 3.64e-10 ***
## depth: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 ' ' 1Introducing the block reduced their p-values, therefore it appears to be useful.
13.5
library(GAD)
pos <- c(rep(1,3),rep(2,3),rep(1,3),rep(2,3),rep(1,3),rep(2,3))
temp <- c(rep(800,6),rep(825,6),rep(850,6))
obs <- c(570,565,583,528,547,521,1063,1080,1043,988,1026,1004,565,510,590,526,538,532)
pos <- as.random(pos)
temp <- as.fixed(temp)
dat2 <- data.frame(pos,temp,obs)
model2 <- aov(obs~pos*temp)
gad(model2)## $anova
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## pos 1 7160 7160 15.998 0.0017624 **
## temp 2 945342 472671 1155.518 0.0008647 ***
## pos:temp 2 818 409 0.914 0.4271101
## Residuals 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Position and temperature both significantly affect the observations, but their interaction does not.
13.6
library(GAD)
part <- 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 <- 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)
part <- as.random(part)
operator <- as.fixed(operator)
dat3 <- data.frame(part,operator,obs)
model3 <- aov(obs~operator*part)
gad(model3)## $anova
## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## operator 1 0.417 0.4167 0.6923 0.4269
## part 9 99.017 11.0019 7.3346 3.216e-06 ***
## operator:part 9 5.417 0.6019 0.4012 0.9270
## Residuals 40 60.000 1.5000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Only the part significantly affect the observations.