|
The Catapult Has Six Different Factors, Each of Which
Can Be Set At Three Different Levels.
They are:
|
Factor
|
Level I
|
Level II
|
Level III
|
|
Elevation -
|
Low
|
Medium
|
High
|
|
Stationary Arm
|
Low
|
Medium
|
High
|
|
Upright Arm Tension Location
|
Low
|
Medium
|
High
|
|
Pivot Arm Tension Location
|
Low
|
Medium
|
High
|
|
Ball Seat Position
|
Low
|
Medium
|
High
|
|
Turn Table Position
|
Low
|
Medium
|
High
|
|
Ball Type
|
Foam
|
Whiffle
|
Ping
Pong
|
To test each factor at each level would require a full
factorial experiment consisting of 729 different treatment
combinations (36 = 729). If the experiment
is conducted only using the Level I and II the experiment
is reduced to 26 = 64 treatment combinations.
This is still a large experiment, hence the need to develop
some sort of fractional experiment or Taguchi orthogonal
array.
Different types of fractional factorials
can be developed based on the instructors preferences. The following
example is for a one-eighth fractional factorial with two replications
of each treatment. The response variable is the distance the ball
is thrown in inches.
|
Order
|
Elevation
|
Upright
|
Pivot
Arm
|
Ball Seat
|
Turn Table
|
Ball Type
|
Dist
|
|
1
|
Lowest
|
Low
|
High
|
Highest
|
Lowest
|
Foam
|
120
|
|
2
|
Lowest
|
High
|
High
|
Lowest
|
Lowest
|
Ping Pong
|
95
|
|
3
|
Highest
|
High
|
Low
|
Highest
|
Lowest
|
Foam
|
81
|
|
4
|
Highest
|
Low
|
High
|
Lowest
|
Highest
|
Foam
|
44
|
|
5
|
Lowest
|
Low
|
Low
|
Highest
|
Highest
|
Ping Pong
|
39
|
|
6
|
Lowest
|
High
|
Low
|
Lowest
|
Highest
|
Foam
|
41
|
|
7
|
Highest
|
Low
|
Low
|
Lowest
|
Lowest
|
Ping Pong
|
37
|
|
8
|
Highest
|
High
|
Low
|
Highest
|
Lowest
|
Foam
|
79
|
|
9
|
Highest
|
Low
|
Low
|
Lowest
|
Lowest
|
Ping Pong
|
31
|
|
10
|
Lowest
|
High
|
High
|
Lowest
|
Lowest
|
Ping Pong
|
96
|
|
11
|
Lowest
|
Low
|
High
|
Highest
|
Lowest
|
Foam
|
125
|
|
12
|
Lowest
|
Low
|
Low
|
Highest
|
Highest
|
Ping Pong
|
36
|
|
13
|
Highest
|
Low
|
High
|
Lowest
|
Highest
|
Foam
|
52
|
|
14
|
Highest
|
High
|
High
|
Highest
|
Highest
|
Ping Pong
|
88
|
|
15
|
Lowest
|
High
|
Low
|
Lowest
|
Highest
|
Foam
|
42
|
|
16
|
Highest
|
High
|
High
|
Highest
|
Highest
|
Ping Pong
|
95
|
This data has been analyzed by Stat
Ease Design Ease Software. The results are shown on the following
pages. The most significant factors according to the analysis
are Pivot Arm Location, Ball Seat Position, and Turn Table Position.
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
| Source |
Sum of
Squares
|
DF |
Mean
Square |
F
Value |
Prob > F |
|
| Model |
15211.94
|
7 |
2173.13 |
183.97 |
<0.0001 |
significant
|
| |
|
|
|
|
|
|
|
| |
A |
473.06 |
1 |
473.06 |
40.05 |
0.0002 |
|
| |
B |
1105.56 |
1 |
1105.56 |
93.59 |
<0.0001 |
|
| |
C |
6765.06 |
1 |
6765.06 |
572.70 |
<0.0001 |
|
| |
D |
3164.06 |
1 |
3164.06 |
267.86 |
<0.0001 |
|
| |
E |
3220.56 |
1 |
3220.56 |
272.64 |
<0.0001 |
|
| |
F |
280.56 |
1 |
280.56 |
23.75 |
0.0012 |
|
| |
AF |
203.06 |
1 |
203.06 |
17.19 |
0.0032 |
|
| |
Pure
Error |
94.50 |
8 |
11.81 |
|
|
|
| |
Cor Total |
15306.44 |
15 |
|
|
|
|
The Model F-value of 183.97 implies the model is significant.
There is only
a 0.01% chance that a "Model F-Value" this large could
occur due to noise.
Values of "Prob > F" less than 0.0500 indicate model
terms are significant.
In this case A, B, C, D, E, F, AF are significant model terms.
Values greater than 0.1000 indicate the model terms are not significant.
If there are many insignificant model terms (not counting
those required to support hierarchy), model reduction may
improve your model.
|
Std. Dev. m
|
3.44 |
R-Squared |
0.9938 |
|
Mean
|
68.81 |
Adj R-Squared |
0.9884 |
|
C.V.
|
4.99 |
Pred R-Squared |
0.9753 |
|
PRESS
|
378.00 |
Adeq Precision |
36.416 |
The "Pred R-Squared" of 0.9753 is in reasonable agreement
with the "Adj R-Squared" of 0.9884.
In
this case A, B, C, D, E, F, AF are significant model terms.
Values greater than 0.1000 indicate the model terms are not significant.
"Adeq Precision"
measures the signal to noise ratio. A ratio greater than
4 is desirable. Your
ratio of 36.416 indicates an adequate signal.This model can be
used to navigate the design space.
Final Equation in
Terms of Coded Factors:
| Distance |
= |
| +68.81 |
|
| -5.44 |
*A |
| +8.31 |
*B |
| +20.56 |
*C |
| +14.06 |
*D |
| -14.19 |
* E |
| -4.19 |
* F |
| +3.56 |
* A * F |
|