Part 1 Recalculation:

Part 1 of the project is involved, among other things, sample size recalculation of the sample size. The calculation is shown here, but the data and analysis will be submitted as part of the final report. This was recalculated using the the Intermediate formula for the F-Value.

library(pwr)
alphaLevel <- .05
confidence <- .75
d <- .5
numPop <- 3

fVal <- (d/2)*sqrt((numPop+1)/(3*(numPop-1)))

pwr.anova.test(k = numPop,n = NULL,f = fVal,sig.level = alphaLevel,power = confidence)
## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 3
##               n = 69.73415
##               f = 0.2041241
##       sig.level = 0.05
##           power = 0.75
## 
## NOTE: n is number in each group

70 samples are need from each group.

Project Part 2

Perform a designed experiment to determine the effect of Pin Elevation and Release Angle on distance in which a ball is thrown when Fire Angle is 100 degrees, Bungee Position is 150mm, and Cup Elevation is 250mm. Settings of the Pin Elevation at 100, 150, and 200mm should be investigated as a fixed effect, as well as settings of the Release Angle corresponding to 110, 140, and 170 degrees as a random effect. The design should be replicated three times.

Part A

State model equation with the null and alternative hypotheses to be tested. In addition, state the level of significance that will be used in your analysis.

The linear effects equation for this analysis is \[y_{ijk}=\mu+\alpha_i+\beta_j+\alpha\beta_{ij} + \epsilon_{ijk}\] The Hypotheseis are as follow:

\[H_0: \alpha\beta_{ij} = 0\ for\ all\ ij\] \[H_1: \alpha\beta_{ij} \ne 0\ for\ at\ least\ one\ ij\] \[H_0: \alpha_i = 0\ for\ all\ ij\] \[H_1: \alpha_i \ne 0\ for\ at\ least\ one\ i\] \[H_0: \beta_j = 0\ for\ all\ ij\] \[H_1: \beta_j \ne 0\ for\ at\ least\ one\ j\] In this analysis, a standard \(\alpha = 0.05\) will be used.

Part B

Propose a layout with a randomized run order

An experiment with 2 Factors with 3 levels per factor and 3 replications will be conducted.

library(agricolae)

trts <- c(3,3)
seedNum <- 1234567
experiment <- design.ab(trt = trts, r=3,design="crd",seed = seedNum)

The random data collection order is shown here below.

experiment$book
##    plots r A B
## 1    101 1 1 2
## 2    102 1 1 3
## 3    103 1 3 3
## 4    104 1 3 2
## 5    105 1 2 3
## 6    106 1 2 2
## 7    107 1 2 1
## 8    108 2 1 2
## 9    109 1 1 1
## 10   110 2 2 3
## 11   111 2 3 3
## 12   112 2 3 2
## 13   113 3 1 2
## 14   114 2 1 1
## 15   115 2 1 3
## 16   116 1 3 1
## 17   117 3 3 2
## 18   118 2 2 2
## 19   119 3 2 3
## 20   120 2 3 1
## 21   121 2 2 1
## 22   122 3 1 3
## 23   123 3 2 1
## 24   124 3 2 2
## 25   125 3 1 1
## 26   126 3 3 3
## 27   127 3 3 1

Part C

Collect data and record observations on the layout proposed in part B

The data was collected using the online catapult per the data collection order shown above.

#read in the data
expData <- read.csv("C:/Users/fay/Documents/Scott Texas Tech/DoE/Project/Part2_Data.csv")
observation <- expData$Output
library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.
pinEl <- as.fixed(expData$Pin.Elevation)
relAng <- as.random(expData$Release.Angle)

The following table shows collected data.

Data Collected from Online Catapult
Replicate Pin.Elevation Release.Angle Output
1 100 140 124
1 100 170 178
1 200 170 273
1 200 140 186
1 150 170 227
1 150 140 156
1 150 110 53
2 100 140 123
1 100 110 39
2 150 170 229
2 200 170 277
2 200 140 188
3 100 140 124
2 100 110 38
2 100 170 183
1 200 110 60
3 200 140 186
2 150 140 156
3 150 170 228
2 200 110 63
2 150 110 52
3 100 170 180
3 150 110 53
3 150 140 155
3 100 110 38
3 200 170 280
3 200 110 63

Part D

Test the hypotheses and state conclusions, determining those effects that are significant. Show any plots that might be useful/necessary to show your findings. You may also show residual plots and make appropriate comments, but do not transform the data (i.e. use the raw data regardless of normality and variance constancy)

The Analysis of Variance follows:

aov1 <- aov(observation~pinEl+relAng+pinEl*relAng)
gad(aov1)
## Analysis of Variance Table
## 
## Response: observation
##              Df Sum Sq Mean Sq    F value  Pr(>F)    
## pinEl         2  16749    8374     8.4192 0.03685 *  
## relAng        2 142985   71492 25398.5921 < 2e-16 ***
## pinEl:relAng  4   3979     995   353.3684 < 2e-16 ***
## Residual     18     51       3                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
qqnorm(aov1$residuals)
qqline(aov1$residuals)

boxplot(aov1$residuals~aov1$fitted.values)

interaction.plot(pinEl,relAng,observation)

interaction.plot(relAng,pinEl,observation)

The data looks to be normally distributed except for are few observations at the tails.

Would have some concerns about constant variance. However from the box plot, the residuals seem to be within whiskers listed.

Based on results of anova, the interaction of Pin Elevation and release angle is highly significant.

All Code Used

Below you will find a block containing all of the code used to generate this document

#Part 1 Rework
library(pwr)
alphaLevel <- .05
confidence <- .75
d <- .5
numPop <- 3
fVal <- (d/2)*sqrt((numPop+1)/(3*(numPop-1)))
pwr.anova.test(k = numPop,n = NULL,f = fVal,sig.level = alphaLevel,power = confidence)

#Part 2
library(agricolae)
trts <- c(3,3)
seedNum <- 1234567
experiment <- design.ab(trt = trts, r=3,design="crd",seed = seedNum)
experiment$book

#Part 2 Read in Data
expData <- read.csv("C:/Users/fay/Documents/Scott Texas Tech/DoE/Project/Part2_Data.csv")
observation <- expData$Output
library(GAD)
pinEl <- as.fixed(expData$Pin.Elevation)
relAng <- as.random(expData$Release.Angle)

#Part 2 Analysis 
aov1 <- aov(observation~pinEl+relAng+pinEl*relAng)
gad(aov1)
qqnorm(aov1$residuals)
qqline(aov1$residuals)
boxplot(aov1$residuals~aov1$fitted.values)
interaction.plot(pinEl,relAng,observation)
interaction.plot(relAng,pinEl,observation)

```