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日期：2018-04-23 08:45

Design Analysis of Experiments

“The Longest Putt”

4 Components:

1 Design – what will the experiment consist of, what are you testing

2 Modeling – Regression Analysis of a 24

Factorial design

3 Analysis – Minitab output

4 Scientific Inference – Conclusions

Design/Background:

The design of this experiment is simple; determine which characteristics of

a golf ball will make it go the furthest. A number of problems arise with this

initially simple proposal: which characteristics to test (diameter, core structure,

exterior makeup), under what conditions do you test them under (outside, inside,

hot, cold, driving, putting), etc…

In doing some research into this top of golf ball and distance there is a great deal

of liturgy on the trajectory and spin of a gold ball in flight, mathematically

analyzing the rotation of the golf ball in the air (asu.edu, 2007).

In tournaments officials use a device called a stemp meter to analyze the

speed of the greens, they do this by rolling a golf ball down a ramp at a certain

angle on the green and calculate how far the ball rolls. They do this 2 times in

opposite directions and take the average distance. In trying to design a

repeatable method such as this for striking the golf ball I determined there would

be a great deal of variation in my swing or anyone else’s for that matter to get an

accurate representation of distance the ball traveled with a relatively small

number of trials.

The United States Golf Association specifies that a golf ball must weigh no more

than 1.620 ounces, be spherical in shape and be no less than 1.68-inches in

diameter. This is called "The American Ball" (leaderboard, 2007). The "British

Ball" is a bit smaller, at 1.62-inches. That may seem like too tiny a difference to

mean anything tangible. But that 3.7% difference in diameter translates into 7.5%

less wind that the ball must cut through during flight. Conversely, there is a

thought about how the weight of the golf ball affects the distance traveled, the

heavier the ball the further is goes. Under this assumption Top Flite designed a

golf ball called the Magna. Increasing the weight of a golf ball tends to increase

the distance it will travel and lower the trajectory. A ball having greater

momentum is better able to overcome drag (Patent, 1998).

For the design of my experiment I chose the following factors to test:

Factor Variable High (+) Low (-)

Ball shell A Hard Soft

Ball Diameter B Large (172mm) Standard (162mm)

Ball Compression C High (100) Low (80)

Ball core D solid wound

Data Collection

16 golf balls were obtained with the various characteristics described in

the above diagram. A steel Ping Anser 2 putter was used for hitting the golf

balls. A wooded stand protruding 20 inches above the ground was used as a

back stop when I drew the putter back before striking the ball. There was a tape

measure on the ground beside the path the ball would travel on the green. The

putter was drawn back by hand to the back rest 20 inches above the ground and

24 inches behind the original position of the ball. The putter was released at a

freefall motion within my hands to avoid any influence of force. This aspect of

the experiment was unable to be controlled for and may have contributed to a

variation in the distance the ball traveled from the 3 replicates of each run. This,

however, was a much more controlled force technique than the driving distance

concern described in the introduction where the United States Golf Association

has a mechanical swing arm calibrated for testing different golf balls introduced

to the market. The ball cannot exceed 280 yards when hit with this mechanical

device. For my experiment it was infeasible to design such a device so I will just

point out the possible limitations of the experiment design with having my hands

control the fall of the putter from the predetermined stopping point. Each run was

repeated 3 times and the average of the replicates was computed and used in

the design of experiment analysis in Minitab.

The random order for each of the subjects was created using Excel. A

column containing sixteen cells was given a random assignment of numbers

between 0 and 1 using the excel command “rand()”. Then each of these cells

was numbered one through sixteen. Excel was then asked to put the sixteen

numbers in chronological order, rearranging the numbers one through sixteen.

The resulting order of the numbers 1-16 were then recorded and printed out.

This was repeated three times to get three random assignments of the numbers

1-16 for each of the 3 trials.

Table 1

Factors Replicates (in feet)

Shell (A) Diameter (B) Compression (C) Core (D) 1 2 3

Average

(y bar)

- - - - 18.25 19.05 18.4 18.57333

+ - - - 18.2 17.5 18.5 18.06667

- + - - 17.8 18.1 17.6 17.83333

+ + - - 17.4 17.9 17 17.43333

- - + - 17.2 17.5 16.8 17.16667

+ - + - 19.2 18.5 18.6 18.76667

- + + - 18.1 17.4 16.9 17.46667

+ + + - 18.5 18.2 17.9 18.2

- - - + 17.6 18.1 17.2 17.63333

+ - - + 18.7 19.25 19.15 19.03333

- + - + 17.25 16.85 16.5 16.86667

+ + - + 17.55 17.85 16.9 17.43333

- - + + 18.65 19.25 19.15 19.01667

+ - + + 20.05 19.82 19.65 19.84

- + + + 18.05 17.4 17.25 17.56667

+ + + + 18.3 19.2 19.1 18.86667

Planning Matrix:

Planning Matrix

Run

Order Shell Diameter Compression Core

1 soft 162 80 solid

2 hard 162 80 solid

3 soft 172 80 solid

4 hard 172 80 solid

5 soft 162 100 solid

6 hard 162 100 solid

7 soft 172 100 solid

8 hard 172 100 solid

9 soft 162 80 wound

10 hard 162 80 wound

11 soft 172 80 wound

12 hard 172 80 wound

13 soft 162 100 wound

14 hard 162 100 wound

15 soft 172 100 wound

16 hard 172 100 wound

Modeling:

The generic model to be fit to the above data is:

y=μ + (A/2)x1 + (B/2)x2 + (C/2) x3 + (D/2)x4 + (AB/2)x1x2 + (AC/2)x1x3 + (AD/2)x1x4

+ (BC/2)x2x3 + (BD/2)x2x4 + (CD/2)x3x4 + (ABC/2)x1x2x3 + (ABD/2)x1x2x4 +

(ACD/2)x1x3x4 + (BCD/2)x2x3x4 + (ABCD/2)x1x2x3x4 + ε

In this model A, B, C, and D correspond to the four factors (see Table 1). The x1,

x2, x3 and x4 correspond to the factor variables A, B, C, and D, respectively. The

xi (i=1,2,3,4) variables = -1 if the factor is at its low lever and =1 if the factor is at

its high level. The ε independently and identically distributed with mean 0 and

constant variance, IIDN (0,σ2

). ε= y – yhat, where yhat is the fitted value.

Best fitting model for the data: (from the Estimated Effects and Coefficients for

Data, with a p-value < .01) *Note this is a slightly different model than the

Normal Probability Plot of the Standardized Effects would lead us to believe. To

simplify the model as much as possible I used the p-value justification. The

Hierarchical Ordering Principle appears to be verified by my analysis. All four

main effects are significant and there are only two 2 factor interactions and one 3

factor interactions that are also significant in the model. The Effect Sparsity

Principle is also being verified from the results, 8 of the 16 effects are significant.

y=μ + (A/2)x1 + (B/2)x2 + (C/2) x3 + (D/2)x4 + (AC/2)x1x3 + (BD/2)x2x4 + (CD/2)x3x4

+ (ABD/2)x1x2x4 + ε

Therefore the model for the 24

Factorial Design is:

y= 18.0969 + (0.3573/2)x1 + (-0.4135/2)x2 + (0.2635/2)x3 + (0.1844/2)x4 +

(-0.1844/2)x2x4 + (0.2760/2)x3x4 + (-0.1802/2)x1x2x4

From this model it can be seen that in order to make y the largest we should

have the following results :

x1 : + Hard Shell

x2 : - Stand Diameter

x3 : + High Compression

x4 : + Solid Core

With this result and entering in the appropriate values we can obtain a value of :

y = 19.026

Analysis:

Welcome to Minitab, press F1 for help.

Full Factorial Design

Factors: 4 Base Design: 4, 16

Runs: 48 Replicates: 3

Blocks: 1 Center pts (total): 0

All terms are free from

aliasing.

Factorial Fit: Data versus A, B, C, D

Estimated Effects and Coefficients for Data (coded units)

Term Effect Coef SE Coef T P

Constant 18.0969 0.06171 293.24 0

A 0.7146 0.3573 0.06171 5.79 0 Significant

B -0.8271 -0.4135 0.06171 -6.7 0 Significant

C 0.5271 0.2635 0.06171 4.27 0 Significant

D 0.3688 0.1844 0.06171 2.99 0.005 Significant

A*B -0.1146 -0.0573 0.06171 -0.93 0.36 Not Significant

A*C 0.3979 0.199 0.06171 3.22 0.003 Significant

A*D 0.3063 0.1531 0.06171 2.48 0.019 Not Significant

B*C 0.1563 0.0781 0.06171 1.27 0.215 Not Significant

B*D -0.3687 -0.1844 0.06171 -2.99 0.005 Significant

C*D 0.5521 0.276 0.06171 4.47 0 Significant

A*B*C 0.0187 0.0094 0.06171 0.15 0.88 Not Significant

A*B*D 0.0271 0.0135 0.06171 0.22 0.828 Not Significant

A*C*D -0.3604 -0.1802 0.06171 -2.92 0.006 Significant

B*C*D -0.1687 -0.0844 0.06171 -1.37 0.181 Not Significant

A*B*C*D 0.3104 0.1552 0.06171 2.51 0.017 Not Significant

S = 0.427566 R-Sq = 84.18% R-Sq(adj) = 76.77%

Analysis of Variance

Source DF Seq SS Adj SS Adj MS F P

Main Effects 4 19.302 19.302 4.8255 26.4 0

2-Way Interactions 6 8.765 8.765 1.4609 7.99 0

3-Way Interactions 4 1.914 1.914 0.4784 2.62 0.053

4-Way Interactions 1 1.156 1.156 1.1563 6.33 0.017

Residual Error 32 5.85 5.85 0.1828

Pure Error 32 5.85 5.85 0.1828

Total 47 36.987

Alias Structure

I

A

B

C

D

A*B

A*C

A*D

B*C

B*D

C*D

A*B*C

A*B*D

A*C*D

B*C*D

A*B*C*D

Scientific Inference:

The motivation for the project is to determine the optimum type of golf ball

to use to achieve the maximum distance in driving. If optimal drive distance

criteria can not be determined it will at least be worth while to know if there is a

difference in distance based on the factors listed above. The cost of golf balls

varies greatly based on these factors. I chose to conduct the experiment with the

putter for a more controlled and easy to measure experiment, however I believe

the distances recorded can be extrapolated to reflect distances achieved in

driving the ball as well. As I later learned through research, distance depends a

great deal on air resistance and force of impact, which the putter provides little

relation to a person driving the ball. So in the interest of doing a worthwhile

experiment it is limited in scope to the impact of the ball at a low force and as

predicted, the hardness of the shell had the most significant effect on the

distance the ball rolled. This was illustrated by the Main Effects Plot (Figure 3).

A major difference between driving the ball and putting is the type of

friction on the ball in the two scenarios, putting being mainly friction with the

ground and driving being mainly with the air. I am not trying to determine a

distance the ball will travel if struck with a certain force but merely which ball will

travel the furthest under the 4 factors chosen for this experiment based on

availability of gold balls.

The Residual Plots for Data in Figure 1 verify the distribution is normal and

the residuals vs. fitted values are evenly distributed around 0. The Histogram is

slightly away from the normal curve but appears acceptable for the purposes of

this project. The Normal Probability Plot of the Standardized Effects for an alpha

value of 0.05 appears linear but the linear fit line is slightly skewed away from the

data points and I cannot explain this other than to say it was auto generated by

Minitab.

From the Interaction Plots illustrated in figure 4 there are two that are

interesting. The first one being the synergistic plot between A and B that shows

there is little relationship between the two factors. Factor A is the hardness of

the shell and factor B is the ball diameter, this makes sense from a practical

standpoint that each factor had a significant effect on its own for the distance.

This is illustrated in the Main Effects Plots seen in Figure 3. The second

interesting relationship is the antagonistic plot of C and D. This relationship is

more difficult to explain from a practical standpoint of the experiment as factor C

(ball compression) and factor D (core composition) would seem to have a small

affect on distance with the light forces exerted by the putter. None the less,

further experimentation would be needed to determine this relationship further.