Final Project Submission

Katarina Kentera, Mary Adepoju, Ryan Davis

IE 5342 - Design of Experiments

Dr. Timothy Matis

December 1, 2021

In this project, the team completed three different experiments with different design. The supplies of this experiment were a statapult and varioius bouncy balls. Throughtout different stages of the experiment, the team tested the significance of the pin elevation, bungee position and the release angle on the balls traveled distance.

Part 1

Sample Size Determination

The team first completed a power test to find the necessary number of samples to conduct the experiment. We used pwr.anova.test, where $k$, is the number of populations, and $f$ is the effect. We used pwr.anova.test because we do not have any pre-samples.

$k = 3$ (the different ball)

$f = d\sqrt{(k^2-1)/(2k)}$ in the case of maximum variability.

After running the power analysis we derived 53 samples would be needed for each population.

## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 3
##               n = 52.55574
##               f = 0.2357023
##       sig.level = 0.05
##           power = 0.75
## 
## NOTE: n is number in each group

A completely randomized design was constructed using R and the resulting order of balls can be seen below.

After we retrieved the order we then started to collect the data. You can find the observed data in the table below.

The hypotheses that will be tested are:

$H_0: \mu_1=\mu_2=\mu_3$

$H_a:$ at least one $\mu_i$ differs

$ _1= $ ball 1 = red ball

$ _2= $ ball 2 = yellow ball

$ _3= $ ball 3 = brown ball

To check the data for normality, we will plot the normal probability plots in R. These plots are shown below.

+ From the plots we can see that the data follows a straight line, so we will move forward with the assumption that the data is normal.

From the boxplot, we can tell that the variances are not constant. Therefore, we will need to perform a transformation on the data to see if we can make the variances appear constant.

Below we will transform the data using a log transform and then consider whether or not to use this.

+ From the new boxplot of the logged data, we can see that the variances are more constant than they were before. The range of numbers on the y axis reduced significantly, therefore we can say that this data can now be assumed to have a constant variance.

Below, we will construct new normal probability plots of the logged data to be sure that it is still normal.

From these normal probability plots, we can see that the data still exhibits normal behavior.

Below, we will run a parametric ANOVA test.

##              Df Sum Sq Mean Sq F value Pr(>F)
## ind           2  0.044 0.02222    0.25  0.779
## Residuals   156 13.861 0.08885

+ The P value is found to be 0.499 which is greater than $\alpha$ of 0.05. Therefore we will fail to reject the null hypothesis and conclude that there is not a significant difference between the means of distance that each ball traveled.

The residuals are plotted below for further investigation.

+ From the residual plot, we can see that the variance plots are very similar in height because of the small scaling on the y-axis. Because they are so close, we can conclude that the variances really are constant (or just not significantly different) from one another.

Part 2

In this section of the experiment, the team performed a design experiment to determine the effect of the pin elevation and release angle on distance. In this experiment, the bungee position was fixed at position two and the ball used was the red ball. Furthermore, the factor, Pin elevation was investigated as fixed at position one and three. The Release Angle was investigated at a random effect at 110, 140 and 170 degrees.

The model equation for a full factorial model is:
$y_{ijk}=\mu+\alpha_i+\beta_j+\alpha\beta_{ij}+\epsilon_{ijk}$

Hypotheses:
$H_O: \alpha_i=0$
$H_a: \alpha_i \neq 0$
$H_O: \sigma_{\beta}^2=0$
$H_a: \\sigma_{\beta}^2\neq 0$
$H_O: \sigma_{\alpha\beta}^2=0$
$H_a: \\sigma_{\alpha\beta}^2\neq 0$

Part 3

In the last section of the experiment, the team performed a design that tested the significane of all four factors

Factor	Low Level(-1)	High Level(+1)
Pin Elevation	Position 1	Position 3
Bungee Position	Position 2	Position 3
Release Angle	140 degrees	170 degrees
Ball Type	Yellow	Red

GroupProject

Mary Adepoju

12/1/2021

Final Project Submission

Katarina Kentera, Mary Adepoju, Ryan Davis

IE 5342 - Design of Experiments

Dr. Timothy Matis

December 1, 2021

Part 1

Sample Size Determination

The team first completed a power test to find the necessary number of samples to conduct the experiment. We used pwr.anova.test, where \(k\), is the number of populations, and \(f\) is the effect. We used pwr.anova.test because we do not have any pre-samples.

\(k = 3\) (the different ball)

\(f = d\sqrt{(k^2-1)/(2k)}\) in the case of maximum variability.

After running the power analysis we derived 53 samples would be needed for each population.

A completely randomized design was constructed using R and the resulting order of balls can be seen below.

After we retrieved the order we then started to collect the data. You can find the observed data in the table below.

The hypotheses that will be tested are:

\(H_0: \mu_1=\mu_2=\mu_3\)

\(H_a:\) at least one \(\mu_i\) differs

$ _1= $ ball 1 = red ball

$ _2= $ ball 2 = yellow ball

$ _3= $ ball 3 = brown ball

To check the data for normality, we will plot the normal probability plots in R. These plots are shown below.

+ From the plots we can see that the data follows a straight line, so we will move forward with the assumption that the data is normal.

Below we will transform the data using a log transform and then consider whether or not to use this.

+ From the new boxplot of the logged data, we can see that the variances are more constant than they were before. The range of numbers on the y axis reduced significantly, therefore we can say that this data can now be assumed to have a constant variance.

Below, we will construct new normal probability plots of the logged data to be sure that it is still normal.

Below, we will run a parametric ANOVA test.

+ The P value is found to be 0.499 which is greater than \(\alpha\) of 0.05. Therefore we will fail to reject the null hypothesis and conclude that there is not a significant difference between the means of distance that each ball traveled.

The residuals are plotted below for further investigation.

+ From the residual plot, we can see that the variance plots are very similar in height because of the small scaling on the y-axis. Because they are so close, we can conclude that the variances really are constant (or just not significantly different) from one another.

Part 2

Part 3

In the last section of the experiment, the team performed a design that tested the significane of all four factors

GroupProject

Mary Adepoju

12/1/2021

Final Project Submission

Katarina Kentera, Mary Adepoju, Ryan Davis

IE 5342 - Design of Experiments

Dr. Timothy Matis

December 1, 2021

Part 1

Sample Size Determination

The team first completed a power test to find the necessary number of samples to conduct the experiment. We used pwr.anova.test, where \(k\), is the number of populations, and \(f\) is the effect. We used pwr.anova.test because we do not have any pre-samples.

\(k = 3\) (the different ball)

\(f = d*\sqrt{(k^2-1)/(2*k)}\) in the case of maximum variability.

After running the power analysis we derived 53 samples would be needed for each population.

A completely randomized design was constructed using R and the resulting order of balls can be seen below.

After we retrieved the order we then started to collect the data. You can find the observed data in the table below.

The hypotheses that will be tested are:

\(H_0: \mu_1=\mu_2=\mu_3\)

\(H_a:\) at least one \(\mu_i\) differs

$ _1= $ ball 1 = red ball

$ _2= $ ball 2 = yellow ball

$ _3= $ ball 3 = brown ball

To check the data for normality, we will plot the normal probability plots in R. These plots are shown below.

+ From the plots we can see that the data follows a straight line, so we will move forward with the assumption that the data is normal.

Below we will transform the data using a log transform and then consider whether or not to use this.

+ From the new boxplot of the logged data, we can see that the variances are more constant than they were before. The range of numbers on the y axis reduced significantly, therefore we can say that this data can now be assumed to have a constant variance.

Below, we will construct new normal probability plots of the logged data to be sure that it is still normal.

Below, we will run a parametric ANOVA test.

+ The P value is found to be 0.499 which is greater than \(\alpha\) of 0.05. Therefore we will fail to reject the null hypothesis and conclude that there is not a significant difference between the means of distance that each ball traveled.

The residuals are plotted below for further investigation.

+ From the residual plot, we can see that the variance plots are very similar in height because of the small scaling on the y-axis. Because they are so close, we can conclude that the variances really are constant (or just not significantly different) from one another.

+ We will recommend that no further test is necessary because the assumptions made (normal distribution of data and constant variance) held true after transforming the data logarithmically.

Part 2

Part 3

In the last section of the experiment, the team performed a design that tested the significane of all four factors

\(f = d\sqrt{(k^2-1)/(2k)}\) in the case of maximum variability.