DOE Homework week 14

A manufacturing engineer is studying the dimension- al variability of a particular component that is produced on three machines. Each machine has two spindles, and four components are randomly selected from each spindle. The results follow. Analyze the data, assuming that machines and spindles are fixed factors.

Model Equation:

\[Y_{ijk}=\mu + \alpha_i+\beta_{j(i)} + \epsilon_{ijk}\] where,

\(\alpha_i\)= Factor A(Machine) with 3 levels “i”

\(\beta_{j(i)}\)= Factor B(Spindle) with 2 eves “j”

\(\epsilon_{ijk}\)= Standard Error Term

library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.
Mchn <- c(rep(1,8),rep(2,8),rep(3,8))
Spndl <- rep(c(rep(1,4),rep(2,4)),3)
obs <- c(12,9,11,12,8,9,10,8,14,15,13,14,12,10,11,13,14,10,12,11,16,15,15,14)
dat <- data.frame(Mchn,Spndl,obs)
dat
##    Mchn Spndl obs
## 1     1     1  12
## 2     1     1   9
## 3     1     1  11
## 4     1     1  12
## 5     1     2   8
## 6     1     2   9
## 7     1     2  10
## 8     1     2   8
## 9     2     1  14
## 10    2     1  15
## 11    2     1  13
## 12    2     1  14
## 13    2     2  12
## 14    2     2  10
## 15    2     2  11
## 16    2     2  13
## 17    3     1  14
## 18    3     1  10
## 19    3     1  12
## 20    3     1  11
## 21    3     2  16
## 22    3     2  15
## 23    3     2  15
## 24    3     2  14

Hypothesis:

Nested Factor-Spindle:

Null Hypothesis: \[H_o:{\sigma_\beta}^2=0\]

Alternate Hypothesis: \[H_a:{\sigma_\beta}^2\not=0\]

Principle Factor-Machine:

Null Hypothesis: \[\alpha_i=0\]

Alternate Hypothesis: \[\alpha_i\not=0\]

Analysis:

Mchn <- as.fixed(Mchn)
Spndl <- as.random(Spndl)
Test<- lm(obs~Mchn+Spndl%in%Mchn)
gad(Test)
## Analysis of Variance Table
## 
## Response: obs
##            Df Sum Sq Mean Sq F value    Pr(>F)    
## Mchn        2  55.75 27.8750  1.9114 0.2915630    
## Mchn:Spndl  3  43.75 14.5833  9.9057 0.0004428 ***
## Residual   18  26.50  1.4722                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(Test)

The p-value is 0.2915630, which is greater than the alpha value of 0.05. As a result, we are unable to reject the null hypothesis. The Spindle has a considerable influence, whereas the Machine has no meaningful effect.