Ensait - Design of Experiment (Level 2)

2009-05-25 (10h00)
duration 2h00

All documents are allowed

Factorial "catapult" experiment

(from http://www.itl.nist.gov/div898/handbook/pri/section4/pri472.htm).
This experiment was conducted on a catapult launching golf balls - a table-top device used to teach design of experiments and statistical process control. The experiment has five factors that might affect the distance the golf ball travels. The purpose is to determine the significant factors that affect the distance the ball is thrown by the catapult, and to determine the settings required to reach 3 different distances (30, 60 and 90 inches). The response variable is the distance Y in inches from the front of the catapult to the spot where the ball lands. The variables are:

• Factor 1 = band height (height of the pivot point for the rubber bands - levels were 2.25 and 4.75 inches with a centerpoint level of 3.5)
• Factor 2 = start angle (location of the arm when the operator releases- starts the forward motion of the arm - levels were 0 and 20 degrees with a centerpoint level of 10 degrees)
• Factor 3 = rubber bands (number of rubber bands used on the catapult- levels were 1 and 2 bands)
• Factor 4 = arm length (distance the arm is extended - levels were 0 and 4 inches with a centerpoint level of 2 inches)
• Factor 5 = stop angle (location of the arm where the forward motion of the arm is stopped and the ball starts flying - levels were 45 and 80 degrees with a centerpoint level of 62 degrees)

Table 1: Experimental data

#	band height	start angle	rbands	arm length	stop angle	Y: distance
1	3.25	0	1	0	80	28
2	4	10	2	2	62	99
3	4.75	20	2	4	80	126.5
4	4.75	0	2	4	45	126.5
5	3.25	20	2	4	45	45
6	4.75	0	1	0	45	35
7	4	10	1	2	62	45
8	4.75	20	1	0	80	28.25
9	4.75	0	1	4	80	85
10	3.25	20	1	0	45	8
11	4.75	20	1	4	45	36.5
12	3.25	0	1	4	45	33
13	4	10	2	2	62	84.5
14	4.75	20	2	0	45	28.5
15	3.25	0	2	0	45	33.5
16	3.25	20	2	0	80	36
17	4.75	0	2	0	80	84
18	3.25	20	1	4	80	45
19	4	10	1	2	62	37.5
20	3.25	0	2	4	80	106

1 Describing

1. How many trials ($jx$) are occurring? How many factors ($ix$)? What is, for each factor, the number of levels ?
2. Draw the Cartesian map relative to (start angle ; stop angle). Any comments ?
3. Draw the Cartesian map relative to (band height and rbands ; arm length). Describe how you are proceeding, and comment the result.

2 Discrete model

1. What is a discrete model ? What is the size of the code adapted to the experiments ? What is the size of the global space $\Omega$ ?
2. Describe how to code the experiments, and give the result for the first one.
3. What is the number of remaining experiments, that can be use to increase our confidence in the model ?

3 Continuous model

1. Since all factors are continuous apart from a two levels factor, a continuous model can be used. What is the code length of a first degree model ? Knowing that computations are leading to $rawvar=1276$ and $redvar=411$, what is the value of est(VRF), i.e. the measure of forecasting ability ?
2. What other kinds of model can be used ? Knowing that computations are leading now to $redvar=38$, what is the resulting value of est(VRF) ?
3. Now, we replace column 1 and 3 by another column, namely the product element by element of these two columns. This process results in a description with four entries. What could be the rationales for acting that way ? The model of second kind gives now $redvar=46$. What are your conclusions ?

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