In continuation to the experimental studies of the Catapult, we are performing a designed experiment to determine the effect of Pin Elevation and Release Angle on distance in which the red ball is thrown when the Bungee Position is fixed at the second position.

The Settings one and three of Pin Elevation will 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 is replicated three times

We identify Two factors: Factor A: Release Angle with 3 levels ( 110, 140, 170) therefore the degrees of freedom for factor A , will be i-1=3-1=2.

Factor B: Pin elevation with 2 levels ( 1 and 3) therefore the degrees of freedom for factor B, will be j-1=2-1=1.

Part A: Model equation with the null and alternative hypotheses to be tested We choose alpha to be 0.05 in this analysis. a) Model Equation \(Y_{ijkl}\) = \(\mu\) + \(\alpha_i\) + \(\beta_j\) + \(\alpha\beta_{ij}\) + \(\epsilon_{ijkl}\)

b)Hypothesis :

NUll Hypothesis : \(\alpha\beta_{ij} = 0\) For all ij Alternative Hypothesis : \(\alpha\beta_{ij} \neq 0\) for some ij

NUll Hypothesis :\(\alpha_i = 0\) Alternative Hypothesis: \(\alpha_i \neq 0\)

NUll Hypothesis: \(\beta_j=0\) Alternative Hypothesis: \(\beta_j\neq0\)

Part B: Proposed layout with a randomized run order

trt0<- c(2,3)
design<- design.ab(trt=trt0,r=3,design ="crd",seed=1000)
design$book
##    plots r A B
## 1    101 1 1 1
## 2    102 1 1 3
## 3    103 1 2 3
## 4    104 2 1 1
## 5    105 2 1 3
## 6    106 2 2 3
## 7    107 1 1 2
## 8    108 3 2 3
## 9    109 3 1 3
## 10   110 1 2 2
## 11   111 1 2 1
## 12   112 2 2 2
## 13   113 2 2 1
## 14   114 3 2 2
## 15   115 2 1 2
## 16   116 3 1 1
## 17   117 3 1 2
## 18   118 3 2 1

Part C: data Collection and observations recording on the layout proposed

observation <- c(27,59,88,27,55,83,42,86,59,
                 52,27,47,30,48,42,28,40,31)

Part D: We will Test the hypotheses and state conclusions by determining those effects that are significant. We will Show any plots that might be useful/necessary to show our findings. We will also show residual plots and make appropriate comments.

Release_Angle<-c(rep(110,3),rep(140,3),rep(170,3))
Pin_Elevation<-c(rep(1,9),rep(3,9))
obs<-c(27,27,28,42,42,40,59,55,59,
       21,30,31,52,47,48,88,83,86)
data.frame(Release_Angle,Pin_Elevation,obs)
##    Release_Angle Pin_Elevation obs
## 1            110             1  27
## 2            110             1  27
## 3            110             1  28
## 4            140             1  42
## 5            140             1  42
## 6            140             1  40
## 7            170             1  59
## 8            170             1  55
## 9            170             1  59
## 10           110             3  21
## 11           110             3  30
## 12           110             3  31
## 13           140             3  52
## 14           140             3  47
## 15           140             3  48
## 16           170             3  88
## 17           170             3  83
## 18           170             3  86
Release_Angle<-as.random(Release_Angle)
Pin_Elevation<-as.fixed(Pin_Elevation)

datpr<-data.frame(Release_Angle,Pin_Elevation,obs)
modelpr<-aov(obs~Release_Angle+Pin_Elevation+Release_Angle*Pin_Elevation, data=datpr)
GAD::gad(modelpr)
## Analysis of Variance Table
## 
## Response: obs
##                             Df Sum Sq Mean Sq  F value    Pr(>F)    
## Release_Angle                2 5971.4 2985.72 353.5724 2.159e-11 ***
## Pin_Elevation                1  636.1  636.06   2.0253    0.2907    
## Release_Angle:Pin_Elevation  2  628.1  314.06  37.1908 7.187e-06 ***
## Residual                    12  101.3    8.44                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(modelpr) 
##                             Df Sum Sq Mean Sq F value   Pr(>F)    
## Release_Angle                2   5971  2985.7  353.57 2.16e-11 ***
## Pin_Elevation                1    636   636.1   75.32 1.62e-06 ***
## Release_Angle:Pin_Elevation  2    628   314.1   37.19 7.19e-06 ***
## Residuals                   12    101     8.4                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(modelpr)

Conclusion

The p-value is less than alpha=0.05 , we reject Ho. therefore factor Release_Angle, Pin_Elevation, and Release_Angle-Pin_Elevation are all significant.

From the plot we can see that the spread of the residuals tends to be equal to fitted values. The residual and normality plots show that the variance is approximately constant for our experiment and the model is adequate in terms of normality (see below).

We started by exploring the highest order interaction(\(\alpha\beta_{ij}\)). We observed that the two factor interaction effect is significant under Ho is true. This means that we should stop exploration of the associated factors and we generated the interaction plot of the factors A,B, and AB as shown bellow.

interaction.plot(datpr$Release_Angle,datpr$Pin_Elevation,datpr$obs)