Null and Alternative Hypotheses

Ho: \(\alpha_{i} = 0\) - Null Hypothesis

Ha: \(\alpha_{i} \ne 0\) - Alternative Hypothesis

Ho: \(\beta_{i} = 0\) - Null Hypothesis

Ha: \(\beta_{i} \ne 0\) - Alternative Hypothesis

Ho: \(\alpha \beta_{ij} = 0\) - Null Hypothesis

Ha: \(\alpha \beta_{ij} \ne 0\) - Alternative Hypothesis

Level of Significance

\(\alpha\) = 0.05

Model Equation

\(y_{ijk} = \mu + \alpha_{i} + \beta_j + \alpha \beta_{ij} + \epsilon_{ijk}\)

Proposed Layout with a Randomized Run Order

##    plots r A B
## 1    101 1 1 2
## 2    102 2 1 2
## 3    103 1 2 3
## 4    104 1 2 1
## 5    105 1 1 1
## 6    106 2 2 1
## 7    107 1 1 3
## 8    108 1 2 2
## 9    109 3 2 1
## 10   110 2 1 3
## 11   111 3 1 2
## 12   112 2 2 3
## 13   113 2 2 2
## 14   114 3 2 2
## 15   115 3 2 3
## 16   116 2 1 1
## 17   117 3 1 1
## 18   118 3 1 3

In the layout, factor A(Pin.Location) represents Pin Elevation and it has levels 1 and 2 for settings 1 and 3 respectively. factor B(Angle) represents the Release Angle with levels 1,2 and 3 for corresponding angles 110, 140 and 170 degrees. Number of replications is 3 which gives a total of 18 observations in the experiment

Collected Data on Proposed Layout

##    Replication Pin.Location Angle Distance...Inches.
## 1            1            1   140                 25
## 2            2            1   140                 35
## 3            1            3   170                 55
## 4            1            3   110                 32
## 5            1            1   110                 24
## 6            2            3   110                 23
## 7            1            1   170                 48
## 8            1            3   140                 36
## 9            3            3   110                 24
## 10           2            1   170                 56
## 11           3            1   140                 37
## 12           2            3   170                 61
## 13           2            3   140                 52
## 14           3            3   140                 48
## 15           3            3   170                 72
## 16           2            1   110                 30
## 17           3            1   110                 26
## 18           3            1   170                 33

Testing the Hypotheses

## Analysis of Variance Table
## 
## Response: BungeeEx$Distance...Inches.
##                                      Df  Sum Sq Mean Sq F value    Pr(>F)    
## BungeeEx$Pin.Location                 1  440.06  440.06  3.5616 0.1997555    
## BungeeEx$Angle                        2 2305.33 1152.67 19.4817 0.0001704 ***
## BungeeEx$Pin.Location:BungeeEx$Angle  2  247.11  123.56  2.0883 0.1666387    
## Residual                             12  710.00   59.17                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Firstly, we tested the interaction hypothesis that the pin location and the angle had an effect on the shooting distance. If we failed to reject the interaction null hypothesis, we tested the main effects the pin location and angle effects on the distance.

From the interaction result, interaction effects has fo value is 2.0883 with a corresponding p-value of 0.1666387 >0.05. Since 0.1666387 >0.05, we failed to reject the interaction null hypothesis that the interaction between pin location and the angle have an effect on the shooting distance.

The next section we removed the interaction effect and tested the main effects.

Model Equation

\(y_{ijk} = \mu + \alpha_{i} + \beta_j + \epsilon_{ijk}\)

model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location+BungeeEx$Angle)
gad(model)
## Analysis of Variance Table
## 
## Response: BungeeEx$Distance...Inches.
##                       Df  Sum Sq Mean Sq F value   Pr(>F)    
## BungeeEx$Pin.Location  1  440.06  440.06  6.4368 0.023703 *  
## BungeeEx$Angle         2 2305.33 1152.67 16.8605 0.000187 ***
## Residual              14  957.11   68.37                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From the pin location result fo value is 6.4368 which corresponds to a p-value of 0.023703. The angle result fo value is 16.8605 which corresponds to a p-value of 0.000187

We concluded that the pin location and angle have an effect on the shooting distance of the ball

Pin.Location: 0.023703 <0.05

Angle: 0.000187 <0.05

ANOVA Test Plot and Interaction Plot

There seems to be nothing unusaual about the plots. the data seems to follow a straight line on the normal probability plot with 2 extreme outliers on the tail ends of the data distribution. Other than that, everything is fairly normal.

trts<-c(2,3)
design<-design.ab(trt=trts, r=3, design="crd",seed=878900)
design$book

BungeeEx<-read.csv("https://raw.githubusercontent.com/Rusty1299/Projects/main/Factorial%20Design%20Project.csv")
library(GAD)
BungeeEx$Pin.Location<-as.fixed(BungeeEx$Pin.Location)
BungeeEx$Angle<-as.random(BungeeEx$Angle)

model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location*BungeeEx$Angle)
gad(model)

model<-aov(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location+BungeeEx$Angle)
gad(model)

interaction.plot(BungeeEx$Angle,BungeeEx$Pin.Location,BungeeEx$Distance...Inches., type = "l", col = 5:7 ,main ="Interraction Plot", ylab = "Distance", xlab = "Release Angles", trace.label = "Pin Elevation", lwd = 3, lty = 1)

plot(model)
boxplot(BungeeEx$Distance...Inches.~BungeeEx$Angle, col = 6:9:3, main = "Boxplot for Relaease Angle", xlab = "Release Angle", ylab = "Distance")
boxplot(BungeeEx$Distance...Inches.~BungeeEx$Pin.Location, col = 2:4, main = "Boxplot for Pin Elevation", xlab = "Pin Elevation", ylab = "Distance")