EXPERIMENT
INTRODUCTION: CATAPULT Factorial experiments are useful because all possible combinations of the factors (that is, all treatments) are represented for each complete replication of the experiment. With factorial experiments we have the ability to estimate the interaction effects between factors, which is not possible with a one-variable-at-a-time approach. The catapult experiment will give you experience in working with a 23 factorial design. Later in this course you will be required to conduct your own factorial experiment and many of the procedures required in this catapult experiment are similar to those in your experiments.
OBJECTIVE: The objective of this experiment is to model the relationship between the three factors (Release Angle, Stop Angle, and Pivot Height) and the distance the ball is thrown. Follow the lab instructor's directions to set up a 23 factorial with two replicates. Collect data by completing the experiments with the catapult, and measuring the distance the ball is thrown with the tape measure. In subsequent labs you will analyze the data and develp a prediction equation. STUDENT'S RESPONSIBILITY: Together with your team complete the "Checksheet for Catapult Experiments". (Save a copy of the randomized run order and responses to a floppy disk for analyzation in future labs.) PROCEDURE: Define the three levels of the factors as follows: Factors __ i. Release Angle ii. Stop Angle iii. Pivot Height 1. Low Setting_____ Blue: 140 Blue: Approx. 90 Blue: Low Tension High Setting________ Yellow: 170 Yellow: Approx. 130 Yellow: High Tension
In the following table list the experiments (in actual units of the factors) to be run. Decide who will flip the ball, who will measure, etc. Using Minitab, randomize the order of the experiments to prevent biases in the collection of the data. Collect the data by flipping the ball. Record your data on a check sheet and bring it to the next lab.
2. 3.
Checksheet for Catapult Experiments
Date:________________ Release Angle - = 140 (blue) Exper. Condition (Random Sequence) Stop Angle - = 90 (blue) Test Engineer:______________________ Pivot Height - = Low (blue)
Order of Exp. (Run #) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Release Angle
Stop Angle
Pivot Height
Distance Ball Thrown
Comments
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CATAPULT EXPERIMENT (PART 2)
OBJECTIVE: To find an equation that can accurately predict the distance your catapult will throw the ball. STUDENT'S RESPONSIBILITY: Together with your team complete the 5 worksheets in order to find the best prediction equation. WORKSHEET 1 Summary Table for Catapult Experiment T Start Angle - + - + - + - + P H Stop Pivot Angle Height - - + + - - + + - - - - + + + + Degree of Freedom (n-1)
Treatment Combinations (1) A B Ab C Ac Bc abc
Data (Rep1,Rep2) , , , , , , , ,
Average
Variance 2 ( si )
=
Overall Average: y = b0 =
=
=
i =1
2k
Averages
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= _____________
Experimental Variability (called Pooled Variance or MSE): 2k si2 2 s p = k = ________________ i =1 2 Standard Error of an Effect: sE = 2s 2 p n F /2 = _________________ (nF = total # of measurements = 16)
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WORKSHEET 2 Worksheet for computing Catapult Effects Std. Order T P H TP TH PH TPH Average for each Treatment Combination (from worksheet 1)
(1) a b ab c ac bc abc y+ (average at high level) y- (average at low level) y+ - y- (Effect)
- + - + - + - +
- - + + - - + +
- - - - + + + +
+ - - + + - - +
+ - + - - + - +
+ + - - - - + +
- + + - + - - +
WORKSHEET 3
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Cube Plot for Catapult Data Response: (Note: Corners represent average distance for each treatment combination)
1
1
130
1
1
Stop Angle
1
1 High
Pivot Height
90
1 1
Low
140
Start Angle
170
Conclusion:
WORKSHEET 4a
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Two-Way Interaction Plot for Start Angle by Stop Angle (TP) Start Angle - + - + Stop Angle - - + + Pivot Height - + (1)________ (5)________ (2)________ (6)________ (3)________ (7)________ (4)________ (8)________ Average _________ _________ _________ _________ Plotting Symbols
X X
X
X
Stop Angle 90 (-) Stop Angle 110 (-)
Distance
- 140
+ 170
Start Angle
Interpretation:
(Example interpretation for a significant interaction: The effect of Start Angle on the distance depends on the level of Stop Angle.) WORKSHEET 4b Two-Way Interaction Plot for Start Angle by Stop Angle (TH)
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Start Angle - + - +
Pivot Height - - + +
Stop Angle - + (1)________ (5)________ (2)________ (6)________ (3)________ (7)________ (4)________ (8)________
Average _________ _________ _________ _________
Plotting Symbols
X X
X
X
Pivot Height Low (-) Pivot Height High (+)
Distance
- 140
+ 170
Start Angle
Interpretation:
WORKSHEET 4c Two-Way Interaction Plot for Start Angle by Stop Angle (PH)
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Stop Angle - + - +
Pivot Height - - + +
Start Angle - + (1)________ (5)________ (2)________ (6)________ (3)________ (7)________ (4)________ (8)________
Average _________ _________ _________ _________
Plotting Symbols
X X
X
X
Pivot Height Low (-) Pivot Height High (+)
Distance
- 90
+ 130
Stop Angle
Interpretation:
WORKSHEET 5 Summary Table for Determining Significant Effects Catapult Experiment Estimated Coefficients sE t-ratio t* critical Put a check
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Effect from Worksheet 2 T P H TP TH PH TPH
Effect 2
from Worksheet 1
Effect 95% sE
limit confidence value from Table A-5
by the significant Effects
The model used to obtain the predicted distance is:
^ Y = b0 + b1T + b2 P + b3 H + b1 2TP + b1 3TH + b2 3PH + b1 2 3TPH where b0 = overall mean bi = the coefficients for each factor or interaction ^ Y = the predicted distance WRITE THE PREDICTION EQUATION FOR THE DISTANCE THE CATAPULT WILL THROW THE BALL: ^ Y= (Factor levels range from 1 to 1, interpolate if necessary) Given: Start Angle = 160 Stop Angle = 100 Pivot Height = High What is your prediction for the distance that the ball will be thrown? Optional Exercise: Can you repeat all of the calculations done here, using EXCEL?
CATAPULT EXPERIMENT (PART 3)
CATAPULT EXPERIMENT WRITING ASSIGNMENT
Along with the catapult experiment and analysis of data that has been done in lab, there is also a writing assignment involved with this venture. This paper should be a
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culmination of everything that you have learned or done, written in an organized and logical manner. This paper will be directed to a company or person who would be interested in the intricacies of catapult launching. You must convince that person of the validity of your experiment, the accuracy of your results, and the importance of your research in the future of catapult operation. Since the use of catapults in our day is fairly obsolete, your discussion of usage will be fictitious and should not be the major focus of your paper. What will be important is keeping in mind the person reading this paper is neither an engineer nor a statistician (yes there are other fields than these), but is an intelligent person who has a basic understanding of statistical principles such as a mean. Terms such as main effect and pooled variance will need to be explained within the context of the data, as you would explain a difficult topic to a fellow student. You may want to pick someone who best fits this role and visualize that person as you write the paper. You will want to discuss such things as the purpose for the experiment, the results including main effects, interactions, and their interpretation, any problems encountered in the experiment, and suggestions for improvement. It may also be wise to include matters of practical significance such as which settings provide the longest throw. How accurate your predictions will be might also be important to a person seriously considering your experiment. It is also a good idea to explain some of the reasons that you approached the experiment the way you did, i.e. explain the purpose of randomization or why certain factors were held constant. In all cases make sure your discussion covers the subject, but is not too lengthy. The length of the paper is immaterial. Cover the subject but don't beat it to death. In no case should your discussion exceed six pages. Include graphs and figures only if they are important to the clarity of the paper; padding for pages is not necessary. This should be a team effort and not the work of the designated "group scribe". There should be consensus agreement on the content of this report.
The Grading of the Catapult Writing Experience This paper should focus on communicating the correct statistical analysis to a nonstatistically minded audience. In addition to containing the correct information, the paper should be clear in presenting that information. The grader will assume the role of one whom has only a very basic knowledge of statistics. Ideas such as main effect and
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interaction will need to be explained well, while ideas such as mean or random will be assumed to be known. At the minimum, a good paper should include the following: 1. Explanation of the Experiment Explain the purpose for the experiment Explain the design used Explain statistical concepts such as why randomize, or how were measurements taken, etc. 2. Results of the Experiment Reference to a list of your raw data in an appendix Explanation and interpretation of significant effects and interactions Informative tables and plots illustrating the interpretations. Tests of significance. Estimates of variability. Do not pad with non-informative plots or verbiage. 3. Conclusions Remember that unless statistical significance is established, you may be looking at random variability. In stating your conclusions refer back to the purpose for the experiment, was it accomplished? Include the mathematical model and maybe an example of how to use it as well as an interpretation of the accuracy of the model. In writing your paper you might want to ask yourself the following questions: Succinctness: Does the paper adequately express all of the needed ideas without being too verbose? Graphics: Does the paper use only graphs that really help to explain the idea being discussed or are they included for padding purposes only? Explanation: Does the paper explain difficult statistical concepts in a fashion that is easy to understand and correct. Completeness: Does the paper include all relevant discussions including: significant effects, error of estimates, interactions, p-value, etc? Readability: Overall, is the paper constructed well and does it read easily and smoothly? Do the ideas involved flow logically, or is it disorganized and hard to follow.
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