# Unlocking the Dynamics of Ball Range: A Comprehensive Design of Experiments Exploration

Embark on a scientific journey delving into the intricacies of ball range through a meticulous Design of Experiments (DOE) exploration. This comprehensive study scrutinizes the influence of rubber bands, cup positions, and ball types, unravelling key factors shaping the dynamics of throws.

## Problem Statement:

Understanding the multifaceted factors influencing ball range remains a critical challenge. In this regression analysis assignment, we address the complexities surrounding rubber band positions, cup placements, and ball types through a rigorous Design of Experiments (DOE) approach. We aim to decipher the intricate interplay of these variables and provide insights that contribute to a deeper comprehension of ball range dynamics.

### Introduction

Our experiment delves into the fascinating world of rubber bands, cups, and balls, exploring the factors that influence the range of throws. By conducting a Design of Experiment (DOE), we aim to unravel the mysteries behind the throw distance and identify key variables affecting it. The experimental factors, outlined in Table 1 (Bontempo, Carandente, & Manna, 2021), include rubber band position, cup position, and two types of balls—golf and squash.

Design Low Level High Level
Rubber Band Position 210 290
Cup position 285 325
Balls Golf Squash
Response Variable Range

Table 1: Experimental factors and response variables

Objectives

• Identify influential factors in ball range.
• Characterize the predictor-response relationship using regression.
• Investigate potential interactions among factors.
• Establish optimal factor combinations for maximizing ball range.

Hypotheses

• Squash balls exhibit a higher range.
• Golf balls achieve the highest range.
• Rubber band and cup position impact the range.

Materials and Methods

We designed a full factorial experiment (2^3) involving rubber band position, cup position, and ball type, resulting in sixteen runs (Table 2). Randomization in Minitab was employed to minimize bias. Uncontrolled variables were addressed through randomization (Chong, et al., 2021), and systematic errors related to Range (R) were mitigated by stabilizing readings.

RunOrder CenterPt Blocks Rubber Band Pos Cup Ball Range ( R )
1 1 2 290 325 Squash 2370
2 1 2 290 285 Golf 3500
3 1 2 210 325 Squash 1940
4 1 2 210 325 Golf 1980
5 1 2 290 285 Squash 5000
6 1 2 210 285 Golf 4250
7 1 2 210 285 Squash 3500
8 1 2 290 325 Golf 2190
9 1 1 290 325 Squash 1945
10 1 1 290 285 Golf 2890
11 1 1 210 325 Squash 1760
12 1 1 210 325 Golf 1840
13 1 1 290 285 Squash 4500
14 1 1 210 285 Golf 3090
15 1 1 210 285 Squash 3750
16 1 1 290 325 Golf 2030

Table 2: DOE Summary

Measurement Method

The experiment measured ball range using various tools, including tape. Each run underwent meticulous cleaning, and measurements were realigned to ensure accuracy. Rubber bands, cups, golf balls, and squash balls constituted the materials.

Factor Levels

Ball range had two levels—low and high—based on literature suggesting their impact on systematic error. Golf balls represented the low level, while squash balls represented the high level due to differences in mass and velocity (El Deeb, Silva, Junior, Hanafi, & Borges, 2021).

Confirmation Runs

Confirmation runs were conducted to validate the model at its optimum, following the same method with rubber bands, cups, and balls.

Results

The regression equation, presented in uncoded units, along with the ANOVA, Pareto Chart, Main Effects Plot, Interaction Plot, and Residual Plots, summarizes our findings.

Table 3: Pareto Chart of Models

Final Model Analysis

The Pareto chart and ANOVA reveal the significance of terms in the final model (Rubber bands, Cups, Balls, and their interactions). The model selection process, supported by p-values and t-statistics, ensures the inclusion of significant variables (Alkiayat, 2021). The adjusted 𝑅-square value of 92.28% attests to the model's explanatory power. Residual plots confirm the model's validity, indicating normal distribution, equal variance, and independence of residuals.

Main Effects Plot

The main effects plot highlights the impact of the cup position, indicating higher extraction rates and increased ball range compared to the rubber band position. Unexpected results in the range factor suggest a complex interplay between rubber bands and cup position, both enhancing ball range (Alkiayat, 2021).

Interaction Plot

The interaction plot reveals varying effects when the rubber band position and cup position are held constant, emphasizing a strong interaction between the rubber band and cup. The distinct lines suggest that cup position influences range differently when rubber band position changes, indicating potential compaction effects on particle size (Rehman, Munir, Raza, & Saeed, 2021).

Cube Plot

The cube plot identifies the optimum conditions for the maximum range (4732.19) with a squash ball and a higher rubber band position. Confirmation runs at these conditions reinforce the model's adequacy and generalizability. Blocks (golf ball and squash ball) prove statistically insignificant, aligning with expectations (Alkiayat, 2021).

Conclusion

Rubber band position, cup position, and ball type emerge as statistically significant factors influencing ball range. The regression model, Range = 14055 + 10.4 rubber band position – 39.5 cup – 16026 ball – 0.0223(Rubber band position cups) + 76.5 (rubber band positionball)+ 48.8 cupball- 0.223 rubber bandscups*balls, confirms balls yield the highest range (4732.19). Hypotheses on ball range, rubber band position, and cup position are validated, except for the golf ball hypothesis.

The experiment's limitations include equipment constraints affecting measurement consistency. Future trials should employ appropriate equipment, consider quantitative measurements, increase replicates, and explore different ball properties for a more comprehensive analysis.

Bibliography

• Alkiayat, M. (2021). A Practical Guide to Creating a Pareto Chart as a Quality Improvement Tool. Global Journal on Quality and Safety in Healthcare, 83-84.
• Bontempo, R., Carandente, R., & Manna, M. (2021). A design of experiment approach as applied to the analysis of diffuser-augmented wind turbines. Energy Conversion and Management, 113924.
• Chong, B. W., Othman, R., Putra Jaya, R., Mohd Hasan, M. R., Sandu, A. V., Nabiałek, M., et al. (2021). Design of experiment on concrete mechanical properties prediction: a critical review. Materials, 1866.
• El Deeb, S., Silva, C. F., Junior, C. S., Hanafi, R. S., & Borges, K. B. (2021). Chiral capillary electrokinetic chromatography: Principle and applications, detection and identification, design of experiment, and exploration of chiral recognition using molecular modelling. Molecules, 2841.
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Appendices

Coded Cofficients

Term Effect Coef SE Coef T - Value P - Value VIF
Constant 2908.4 73.8 39.43 0
Blocks
1 -182.8 73.8 -2.48 0.042 1
Rubber Band Pos 289.4 144.7 73.8 1.96 0.091 1
Cup -1803.1 -901.6 73.8 -12.22 0 1
Ball 374.4 187.2 73.8 2.54 0.039 1
Rubber Band Pos * Cup -35.6 -17.8 73.8 -0.24 0.816 1
Rubber Band Pos * Ball 426.9 213.4 73.8 2.89 0.023 1
Cup " Ball -380.6 -190.3 73.8 -2.58 0.036 1
Rubber Band Pos * Cup Ball -373.1 -186.6 73.8 -2.53 0.039 1

Model Summary

 S R-sq R-sq(adj) R-sq(pred) 295.075 96.40% 92.28% 81.18%

Analysis of Variance

Source DF Adj SS Adj MS F -Value P -Value
Model 8 16305700 2038213 23.41 0.000
Blocks 1 534727 534727 6.14 0.042
Linear 3 13900617 4633539 53.22 0.000
Rubber Band Pos 1 334952 334952 3.85 0.091
Cup 1 13005039 13005039 149.36 0.000
Ball 1 560627 560627 6.44 0.039
2-Way Interactions 3 1313467 437822 5.03 0.036
Rubber Band Pos *Cup 1 5077 5077 0.06 0.816
Rubber Band Pos *Ball 1 728889 728889 8.37 0.023
Cup Ball 1 579502 579502 6.66 0.036
3-Way Interactions 1 556889 556889 6.40 0.039
Rubber Band Pos Cup *Ball 1 556889 556889 6.40 0.039
Error 7 609486 87069
Total 15 16915186