Week 10 - Homework 8
Mary Adepoju
11/5/2021
5.4: Feed Rate and Depth of Cut Block Experiment
Hypothesis:
\[ \\ H_o:\alpha_i = 0 \;\; H_a: \alpha_i \neq 0 \\ H_o:\beta_j = 0 \;\; H_a: \beta_j \neq 0 \\ H_o:\alpha\beta_{ij} = 0 \;\; H_a: \alpha\beta_{ij} \neq 0 \]
## Analysis of Variance Table
##
## Response: responses
## Df Sum Sq Mean Sq F value Pr(>F)
## feedRate 2 3160.50 1580.25 55.0184 1.086e-09 ***
## depthOfCut 3 2125.11 708.37 24.6628 1.652e-07 ***
## feedRate:depthOfCut 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
The normal distribution plot appears to be relatively straight. On the other hand there is some minute curvenees to the graph. Furthermore, the rediusals plot fors somewhat of a straight line. This tells us that this data set is grod engouht to reun a form of the ANOVA test. The interaction plot does not show parallelism which shows the interaction are significant.
As we can see from the graph, the p-value of the main effects as well as the interaction are significant. Therefore we reject the null hypothesis. The p-value of the feedRate is 1.086e-09 and the p-value of the depth of Cut is 1.652e-07. The p-value of the interaction is 0.01797. The mean of the feed rate 0.2 is 74. The mean of feed rate 0.25 is 92. The mean of feed rate 0.3 is 99.
5.34: Feed Rate and Depth of Cut Block Factorial Experiment
## Analysis of Variance Table
##
## Response: yields
## Df Sum Sq Mean Sq F value Pr(>F)
## blocks 2 180.67 90.33 3.9069 0.035322 *
## feedRate 2 3160.50 1580.25 68.3463 3.635e-10 ***
## depthOfCut 3 2125.11 708.37 30.6373 4.893e-08 ***
## feedRate:depthOfCut 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 ' ' 1The \(MS_{Error}\) decreased slightly when blocks where implemneted. This did not significantly alter the results.
13.5: Mixed Experiment: Feed Rate and Depth of Cut Factorial
\[ \\ H_o:\alpha_i = 0 \;\; H_a: \alpha_i \neq 0 \\ H_o:\sigma_{\beta}^2 = 0 \;\; H_a: \sigma_{\beta}^2 \neq 0 \\ H_o:\sigma_{\alpha\beta}^2 = 0 \;\; H_a: \sigma_{\alpha\beta}^2 \neq 0 \]
## Analysis of Variance Table
##
## Response: data
## Df Sum Sq Mean Sq F value Pr(>F)
## furnacePosition 1 7160 7160 15.998 0.0017624 **
## tempreture 2 945342 472671 1155.518 0.0008647 ***
## furnacePosition:tempreture 2 818 409 0.914 0.4271101
## Residual 12 5371 448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The p-values from the ANOVA table shows us that the main effects are significant and the interaction effect is insignificant.
13.6: Operator and Part Number Mixed Experiments
\[ \\ H_o:\alpha_i = 0 \;\; H_a: \alpha_i \neq 0 \\ H_o:\sigma_{\beta}^2 = 0 \;\; H_a: \sigma_{\beta}^2 \neq 0 \\ H_o:\sigma_{\alpha\beta}^2 = 0 \;\; H_a: \sigma_{\alpha\beta}^2 \neq 0 \]
## Analysis of Variance Table
##
## Response: data2
## Df Sum Sq Mean Sq F value Pr(>F)
## partNum 9 99.017 11.0019 7.3346 3.216e-06 ***
## operator 1 0.417 0.4167 0.6923 0.4269
## partNum:operator 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 ' ' 1The p-values from the ANOVA table shows us that the only main effects that is significant is the part number effect. The interaction effect and the effect of the operator factor are insignificant.
All Code
library(GAD)
#5.4
feedRate<-c(rep(0.20,12), rep(0.25,12), rep(0.30,12))
depthOfCut<-rep(c(0.15,0.18,0.20,0.25), 9)
responses<-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)
feedRate <- as.fixed(feedRate)
depthOfCut <- as.fixed(depthOfCut)
model<-aov(responses~feedRate+depthOfCut+feedRate*depthOfCut)
gad(model)
interaction.plot(feedRate, depthOfCut, responses)
plot(model)
feedRate20<-mean(74, 79, 82, 99, 64, 68, 88, 104, 60, 73, 92, 96)
feedRate25<-mean(92, 98, 99, 104, 86, 104, 108, 110, 88, 88, 95, 99)
feedRate30<-mean(99, 104, 108, 114,98, 99, 110, 111, 102, 95, 99, 107)
#5.34
k <- c(rep (1,4), rep(2,4), rep(3,4))
blocks <- rep(k,3)
yields<-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)
blocks<-as.fixed(blocks)
feedRate<-c(rep(0.20,12), rep(0.25,12), rep(0.30,12))
depthOfCut<-rep(c(0.15,0.18,0.20,0.25), 9)
feedRate <- as.fixed(feedRate)
depthOfCut <- as.fixed(depthOfCut)
dat3 <- cbind(blocks,feedRate,depthOfCut,yields)
dat3<-as.data.frame(dat3)
blockFactorial<-lm(yields~ blocks+feedRate+depthOfCut+ feedRate*depthOfCut)
gad(blockFactorial)
#13.5
tempreture<-c(rep(c(800,825,850),6))
furnacePosition <-c(rep(1,9), rep(2,9))
data<-c(570,1063,565,
565,1080,510,
583,1043,590,
528,988,526,
547,1026,538,
521,1004,532)
tempreture<-as.fixed(tempreture)
furnacePosition<-as.random(furnacePosition)
mixedModel<-aov(data~furnacePosition+tempreture+tempreture*furnacePosition)
gad(mixedModel)
#13.6
data2 <-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)
partNum <-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))
kReplicate <- c(rep(seq(1,3),20))
operator <- rep(c(1,1,1,2,2,2),10)
operator <- as.fixed(operator)
partNum <- as.random(partNum)
#mixedModel2 <- cbind(data2,partNum,operator,kReplicate)
#mixedModel2<-as.data.frame(mixedModel2)
anovaModel <- aov(data2 ~partNum+operator+partNum*operator)
gad(anovaModel)