R代码代写、R编程与程序辅导
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.
http://www.daixie0.com/contents/18/1276.html
本团队核心人员组成主要包括硅谷工程师、BAT一线工程师，国内Top5硕士、博士生，精通德英语！我们主要业务范围是代做编程大作业、课程设计等等。
我们的方向领域：window编程 数值算法 AI人工智能 金融统计 计量分析 大数据 网络编程 WEB编程 通讯编程 游戏编程多媒体linux 外挂编程 程序API图像处理 嵌入式/单片机 数据库编程 控制台 进程与线程 网络安全 汇编语言 硬件编程 软件设计 工程标准规等。其中代写代做编程语言或工具包括但不限于以下范围:
C/C++/C#代写
Java代写
IT代写
Python代写
辅导编程作业
Matlab代写
Haskell代写
Processing代写
Linux环境搭建
Rust代写
Data Structure Assginment 数据结构代写
MIPS代写
Machine Learning 作业 代写
Oracle/SQL/PostgreSQL/Pig 数据库代写/代做/辅导
Web开发、网站开发、网站作业
ASP.NET网站开发
Finance Insurace Statistics统计、回归、迭代
Prolog代写
Computer Computational method代做
因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：[email protected]
微信：codinghelp