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Transcript of Helicopter Paper
Helicopter
BY:
RENDY 004201200003
RIFI PRASETYO 004201200033
GANDI SUHARTINAH 004201200036
INDUSTRIAL ENGINEERING DEPARTMENT
ENGINEERING FACULTY – PRESIDENT UNIVERSITY
2015
Table of Contents
CHAPTER I.........................................................................................................................2
INTRODUCTION...............................................................................................................2
1.1. Background...............................................................................................................2
1.2. Objective...................................................................................................................3
1.3. Tools and Equipment................................................................................................3
1.4. Steps..........................................................................................................................3
CHAPTER II........................................................................................................................3
LITERATURE STUDY.......................................................................................................8
Experiment with Three-level, mixed-level and fractional factorial designs..........Error! Bookmark not defined.
Experiment with Generating a Mixed Three-Level and Two-Level Design.........Error! Bookmark not defined.
CHAPTER III......................................................................................................................8
DATA COLLECTION......................................................................................................18
3.1. Experiment Procedure.............................................................................................18
3.2. Response Measurement...........................................................................................19
3.3. Experiment Hypothesis...........................................................................................21
CHAPTER IV....................................................................................................................21
DATA ANALYSIS............................................................................................................26
4.1. Pre Test...................................................................................................................26
4.2. Effect Plot................................................................Error! Bookmark not defined.
4.3. Interaction Plot........................................................................................................28
4.4. ANOVA Test..........................................................................................................29
4.5. Residual Plot and Model Adequacy........................................................................32
4.6. Hypothesis Testing..................................................................................................33
4.7. Regression Model...................................................................................................35
CHAPTER V......................................................................................................................36
CONCLUSION..................................................................................................................36
REFERENCE.....................................................................................................................37
APPENDIX 1: Documentation of Experiment..................Error! Bookmark not defined.
APPENDIX 2: Minitab Output..........................................Error! Bookmark not defined.
CHAPTER I
INTRODUCTION
Helicopter Project | Design of Experiment Page 1Industrial Engineering 2012 | President University
1.1. Background
Design of Experiment is a method to determine the relationship between
factors that affecting the process and the output of that process. In other words, it
is used to find cause-and-effect relationships between factors to the output or
mostly known as response. Design of Experiment can be done in many aspects
including the daily life operation.
One application of experimental design is by doing an experiment to
measure downward speed of the paper helicopter. In this occasion, the
experiments used paper helicopter as the material of the experiments and then
give different treatment to measure the differences. The paper helicopter is a
simple construction that shares this property of autorotation when falling to the
ground and the objective of the project is to build a paper helicopter that takes the
longest time to fall to the ground from a given height. Helicopters rely on a
phenomenon called autorotation to slow their descent to the ground when they
lose power. The air-flow past the rotors generated by the downward speed causes
the rotor to spin and generate drag that slows down the fall. However, there are
several factors that might be effecting the downward speed time to fall faster, such
as: body lenghts size, tail width size, tail lenght size and paperclip size. Thus, this
experiment aims to analyze which factors that have significant effect to the
downward speed time.
This experiment is using 4 factors that believe as the factors that have
significant effect to the growth rate (response); those factors are Size of Body
Lenght, Size of Tail Widht, Size of Tail Lenght and Size of Paper Clip. The size
of body length are differing into two (small size and large). The tail width is
divided into two; one small size and large large. The tail lenght is divided into two
levels (small size and large size). By using three replications, so there are 48
experiments.
This experiment is using the method of full factorial design. The
experiments used that method as experimental units of homogeneous materials or
being considered homogeneous and different treatment, which was the diversity
of the response brought about only through by the treatment. The observation is
Helicopter Project | Design of Experiment Page 2Industrial Engineering 2012 | President University
noted every day and being compared whether there is any influence that was due
to the provision of the treatment against heavy and high of the mung bean.
This report contains of six main chapters, which are: Introduction,
Literature Study, Data Collection, Data Analysis, Conclusion, and References.
1.2. Objective
The main objective of this study is to analyze which factors that might be
affecting measure of downward speed of the paper helicopter. There are several
objectives of this experiment, which are:
1. To conduct hypothesis testing
2. To analyze the residual and main effect plot between factors
3. To analyze ANOVA between factors using Minitab
4. To create regression model
1.3. Tools and Equipment
There are several tools and equipment that are being used for this analysis.
The main tool is Minitab Software. Minitab is being used to analyze the data that
is given. The others software are Ms. Office and Ms. Excel. These tools are being
used to do administration thing.
1.4. Steps
Minitab is being used to solve the problem. Generally, the Factorial
Design Analysis is used to solve all those problems. The steps of to analyze the
problem is clearly seen for each problem below.
The steps are:
1. Open the Minitab software → Click Stat on menu bar → DOE →
Factorial → Create Factorial Design as shown in Figure 1.1.
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Figure 1.1 Create Factorial Design
2. Choose 2-level Factorial (default Generators) → Number of Factor = 4 →
Change the Factor name (Factor A = Paper Clip, Factor B = Tail
Length, ,Factor C = Tail Width and Factor D = Body Length) →
Determine the Number of Levels for each Factor (Factor A = 2 Level,
Factor B = 2 Levels, Factor C = 2 Levels and Factor D = 2 Levels) →
Determine the Number of Replicates = 3 → OK. These steps are shown in
Figure 1.2.
Figure 1.2 Determinations of Factors and Replications
3. Click Factors → Determine the Level Values for the Factors (Ascending
number is required or from Low to High) → OK. These steps are shown in
Figure 1.3.
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Figure 1.3 Determination of Level Value
4. The Sessions Panel and Worksheet panel will appear as shown in Figure
1.4. Create a new column which is Response Column. Fill in the response
value based on level of factos.
Figure 1.4 Response Columns on Worksheet Panel
5. To analyze the data, click stat on the Menu Bar → DOE → Factorial →
Analyze Factorial Design, then the panel will appear as shown in Figure
1.5. Fill the Response box with C9 downward speed of the paper
helicopter (Response column) → Click Graph → Choose “Four in One”
on Residual Plots. This aims to shows all plots (Histogram, Normal plot,
Residual vs fits, and Residual vs order) into one panel.
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Figure 1.5 Analyze Factorial Design
6. Figure 1.6 shows the analysis of factorial design in Session panel and the
Residual Plots for Response in one panel. From this, the deeper analysis
can be conducted.
Figure 1.6 Result of Factorial Design Analysis
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7. To show the effect plot, click stat on the Menu Bar → DOE → Factorial
→ Factorial Plot, then the panel will appear as shown in Figure 1.7. Click
Graph → Check Main Effect Plot and Interaction Plot → OK
Figure 1.7 Factorial Plots Graphs
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CHAPTER II
LITERATURE STUDY
Factorial Experiment
Factorial experiment is experiments that investigate the effects of two or
more factors or input parameters on the output response of a process. Other
definition is about factorial experiments is a treatment arrangement in which the
treatments that are consist of all combinations of all levels of two or more factors.
Usually factorial experiment is called simply factorial design because it is a
systematic method for formulating the step needed to successfully implement a
factorial experiment and estimating the effect of various factors on the output of a
process with a minimal number of observations that function to optimize the
output of the process.
In a factorial experiment, the effects of varying the levels of the various
factors affecting the process output are investigated. Each complete trial or
replication of the experiment takes into account all the possible combinations of
the varying levels of these factors. Then for effective factorial design can ensures
that the least number of experiment runs are conducted to generate the maximum
amount of information about how to input the variable affect the output of the
process.
There are some advantages and disadvantages of Factorial experiment, which are:
Advantages
1. More precision on each factor than with single factor experimentation.
2. Broadening the scope of an experiment.
3. Possible to estimate the interaction effect.
4. Good for exploratory work where it wish to find the most important factor
or the optimal level of a factor.
Disadvantages
1. Some people says it`s complex, but in the reality it is all not complex and
it`s the phenomenon which is complex.
2. With a number of factors each for several levels, the experiment may be
become very large.
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Interaction is the failure of the differences in response to changes in levels
of one factor, to retain the same order and magnitude of performance through all
the levels of other factor OR the factors are said to interact if the effect of one
factor changes as the levels of the other factors changes.
For the running of factorial combinations and mathematical interpretation
of the output responses of the process combinations because it is the essence of
the factorial experiments and it allows to understand which factor the process that
improvement or corrective actions may be geared towards these. The experiments
in which numbers of levels of all the factors are equal are called symmetrical
factorial experiments and the experiments in at least two are different are called as
asymmetrical factorial experiments.
Factorial also provides an opportunity to study not only the individual
effects of each factor but also their interactions. It have the further advantages of
economizing on experimental resources and the experiments are conducted factor
by much more resources are required for the same precision than when there are
tried in factorial experiments.
Experiments with Factor Each at Two levels
The simple of the symmetrical factorial experiments are with each of the
factors at 2 levels. If there are “n” factors each at 2 levels it called as 2 n factorial
where the power stands for the number of factors and the base the level of each
factor. For make it simple the symmetrical factorial experiments is the 22 factorial
experiment where i.e. 2 factor are A and B, A and B have two levels lower (0) and
High (1). In a 22 factorial experiment has r or replicates were run for each
combination treatment, the main and interactive effect of A and B on the output
may be mathematically expresses such as:
A= [ab+a-b-(1)] / 2r (main effect for factor A) (2-1)
B= [ab+b-a-(1)] / 2r (main effect for factor B) (2.2)
AB= [ab+(1)-b-a] / 2r (main effect for factor AB) (2-3)
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Where r is the number of replicates per treatment combination and A is the
total of outputs of each of the r replicates of the treatment combination A because
A is high and B is low. For B is the total output for then n replicates of the
treatment B because B is high and A is low. And then for AB, it is the total output
for the r replicates of the treatment combination AB where both A and B are high
and the last is (1), it is the total output for the r replicates of the treatment
combination (1) where A and B are low.
Two factor had been independent because [ab+(1)-a –b ] /2n will be of
the order of zero. If not then it will give an estimate of interdependence of the two
factors and it is called the interaction between A and B. it is easy to verify because
the interaction of the factor B with factor A is BA which will be same as the
interaction AB and hence the interaction does not depend on the order of the
factors. It`s also easy to verify the main effect of factor B because a contrast of the
treatment total is orthogonal to each of A and AB.
There are several steps for analyze experiments with factor each at two
levels:
Step 1 : Calculating the Sum of Squares or SS due to the SS treatment, SS rows
and columns, SS error and the last SS total.
Step 2 : Calculating the DF between treatment, rows and columns, error and total.
Step 3 : After calculating SS and DF it can find to calculate the Mean Square
(MS), formulate to calculate MS is SS / DF.
Step 4 : The last is calculating the F value. Formulate F value is MS/ MSerror.
For example calculating F value is MSa /MSerror.
Step 5 : After calculating all of F value, after that analyze the hypothesis all of F-
value if the F value > α it means reject H0 or Accept H1 but if F value < α it means
do not reject H0 or accept H0.
Step 6 : Calculating the standard errors for main effect and two factor
interactions.
SE of difference between main effect means = √ 2 MSEr .2n−1 (2-4)
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SE of difference between A means at the same level of B= SE of difference
between B means at same level of A = √ 2 MSEr . . 2n−2
So, for general SE for testing the difference between means of r-factor
interactions is
√ 2 MSEr .2n− y
The table below has shown the sources of variation for solving with
ANOVA for 2 factors.
Table 2.1 Sources of Variation Is About 2 Factors (A and B)
Sources of variation DF SS MS F-value
Between
replicationsr-1 SSR
MSR = SSR /
DFreplicationMSR / MSE
Between Treatments 22-1 = 3 SSTMST = SST /
DFtreatmentMST / MSE
A 1 SSA = [A]2 / 4rMSA =
SSA / DFaMSA / MSE
B 1 SSB = [B]2 / 4rMSB = SSB /
DFbMSB / MSE
AB 1 SSAB = [AB]2 /4rMSAB =
SSAB / DFab
MSAB /
MSE
Error
(r-1) (22 -
1) = 3 (r-
1)
SSEMSE = SSE /
DFerror
Totalr. 22 -1 =
4r-1TSS
The table below has shown the sources of variation for solving with
ANOVA for 3 factors.
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Table 2.2 Sources of Variation Is About 3 Factors (A, B and C)
Sources of
variationDF SS MS F-value
Between
replicationsr-1 SSR
MSR = SSR /
DFreplicationMSR / MSE
Between
Treatments22-1 = 3 SST
MST = SST /
DFtreatmentMST / MSE
A 1 SSA = [A]2 / 4rMSA = SSA /
DFaMSA / MSE
B 1 SSB = [B]2 / 4rMSB = SSB /
DFbMSB / MSE
C 1 SSC = [C]2 / 4rMSC = SSC /
DFcMSC / MSE
AB 1SSAB = [AB]2
/4r
MSAB =
SSAB / DFabMSAB / MSE
AC 1SSAC = [AC]2
/4r
MSAC =
SSAC / DFacMSAC / MSE
BC 1SSBC = [BC]2
/4r
MSBC =
SSBC / DFbcMSBC / MSE
ABC 1SSABC =
[ABC]2 /4r
MSABC =
SSABC /
DFabc
MSABC /
MSE
Error(r-1) (23 -1) = 7
(r-1)SSE
MSE = SSE /
DFerror
Total r. 23 -1 = 8r-1 TSS
Experiments with Factor 2k Designs
The factorial experiments, where all combination of the levels of the
factors is run, are usually referred to as full factorial experiments. Factorial two
level experiments are also referred to as 2k designs where k is the number of
factors being investigated in the experiment. A full factorial two level design
with k factors requires 2k runs for a single replicate. For example, a two level
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experiment with three factors will require 2 x 2 x 2 = 23 = 8 runs. The choice of
the two levels of factors used in two level experiments depends on the factor;
some factors naturally have two levels. For example, if gender is a factor, then
male and female are the two levels. For other factors, the limits of the range of
interest are usually used.
The two levels of the factor in the 2k design are usually represented as -
1 (for the first level) and 1 (for the second level). For note about the representation
is reversed from the coding used in General Full Factorial Designs for the
indicator variables that represent two level factors in ANOVA models. For
ANOVA models, the first level of the factor were represented using a value
of 1 for the indicator variable, while the second level was represented using a
value of -1.
Experiments with Factor 22 Design
The simpler of the two level factorial experiments is the 22 design where
two factors (say factor A and factor B) are investigated at two levels. A single
replicate of this design will require four runs (2 x 2 = 22 = 4). The effects
investigated by this design are the two main effects, A and B and the interaction
effect AB. The presence of a letter indicates the high level of the corresponding
factor and the absence indicates the low level.
Table 2.3 Experiments with Factor 22 Design
Treatment NameFactor
A B
(1) -1 -1
A 1 -1
B -1 1
Ab 1 1
For example, the first is represents the treatment combination where all factors
involved are at the low level or the level represented by -1 and α represents the
treatment combination where factor A is at the high level or the level of 1, while
the remaining factors in this case, factor B are at the low level or the level of -1.
Similarly, b represents the treatment combination where factor B is at the high
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level or the level of 1, while factor a, is at the low level and AB represents the
treatment combination where factors A and B are at the high level or the level of
the 1.
Experiments with Factor 23 Design
The 23 design is a two level factorial experiment design with three factors
(factors A, B and C) and this design will design tests three where k = 3 and also
main effects, A, B and C , two factor interaction effects, AB, BC, AC ; and one
three factor interaction effect is ABC. The design requires eight runs per replicate.
The eight treatment combinations corresponding to these runs are 1,a , b, c, ab ,ac
, bc and abc. The treatment combinations are written in such an order that factors
are introduced one by one with each new factor being combined with the
preceding terms and also in this order of writing the treatments is called
the standard order or Yates order. Table 2.4 is the example of 23 designs or called
3 factors.
Table 2.4 Experiments with Factor 23 Designs
Treatment NameFactor
A B C
(1) -1 -1 -1
A 1 -1 -1
B -1 1 -1
C 1 1 -1
AB -1 -1 1
AC 1 -1 1
BC -1 1 1
ABC 1 1 1
Response Surface Methodology or RSCM :
Response Surface methodology is a collection of mathematical and
statistical techniques that are useful for modeling and analysis of problem in
which a response of interest is influenced by several variables and the objective is
to optimize this response. If we denote the expected response by E (y) = f(x1,x2) =
Helicopter Project | Design of ExperimentPage 14Industrial Engineering 2012 | President University
h , then the surface represented by h = f(x1,x2) is called a response surface. For
example, suppose that a chemical engineer wishes to find the levels of
temperature (x1) and pressure (x2) that maximize the yield (y) of a process.
The processes function of yield:
y = f(x1,x2) + e (2-5)
The Steepest Ascent Method
The method of steepest ascent is a procedure for moving sequentially
along the path of steepest ascent, that is, in the direction of the maximum increase
in the response. If minimization is desired, then it call is technique the method of
steepest descent.
Experiments are conducted along the path of steepest ascent until no
further increase is response is observed. Then a new first-order model may be fit,
a new path of steepest ascent determined, and the procedure continued.
Eventually, the experimenter will arrive in the vicinity of the optimum. This is
usually indicated by lack of fit of a first-order model. At that time additional
experiments are conducted to obtain a more precise estimate of the optimum.
There are several steps of steepest ascent:
1. Choose a step size in one of the process variables, say Dxj. Usually, it
would select the variable it is know the most about, or it would select the
variable that has the largest absolute regression coefficient |bj|.
2. The step size in the other variables is
∆ x i=β̂ i
β̂ j
∆ x j
(2-6)
3. Convert the Dxi from coded variables to the natural variables.
Center Points to the 2k Design
A potential concern in the use of two-level factorial design is the
assumption of linearity in the factor effects and the perfect linearity is unnecessary
and the 2k system will work quite well even when the linearity assumption holds
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only very approximately. There are two purposes why the center point runs
interspersed among the experimental setting runs for two purposes:
1. To provide a measure of process stability and inherent variability
2. To check for curvature.
Based on the idea of some replication in a factorial design, runs at the center
provide an estimate of error and allow the experimenter to distinguish between
two possible models:
First order model
Consider the following first-order model in k variables for fitting
y=β0+∑i=1
k
β i xi+∑i=1
k
∑j>i
k
βij xi x j+¿ ε ¿ (2-7)
There is a unique class of designs that minimize the variance of the
regression coefficients β1. These are the orthogonal first order designs. A first
order design is orthogonal if the off diagonal elements of the (X`X) matrix are all
zero. This implies that the cross products of the columns of the X matrix sum to
zero. The 2k factorial and fractions of the 2k series in which main effects are not
aliased with each other belongs to the class of orthogonal first order design and
assume the low and high level of the k factors are coded -1 and 1 levels to used in
design.
Figure 2.1 Surface Graph and Contour Map
Second order model
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Series1Series8
Series15Series22Series29
Series36
Series43
Series50
-10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0
1471013161922252831343740434649
The central composite design or CCD is used for fitting a second-order model.
The CCD consists of a 2kfactorial with nf runs, 2kaxial or star runs, and nc center
runs. Following figure shows the CCD for k = 2 and k = 3 factors.
y=β0+∑i=1
k
β i xi+∑i=1
k
∑j>i
k
βij xi x j+¿∑i=1
k
β ii x2
i+ε ¿ (2-8)
The CCD is developed through sequential experimentation. Suppose a 2k
is used to fit a first order model and suppose this model exhibits lack off it. Then
axial runs are added to allow the quadratic terms to be incorporated in to the
model. The CCD is a very efficient design for fitting the second order model.
There are two parameters in the design that must be specified:
The distance α of the axial runs from the design center
The number of center points nc.
Figure 2.2 CCD
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CHAPTER III
DATA COLLECTION
3.1. Experiment Procedure
Some tools and ingredients are necessary to conduct the experiment Paper,
Scissors, Ruler, Pencil, and Operator are the main ingredient and tools to conduct
the experiment. Basically, paper, scissor, ruler, pencil are used to draw the
helicopter paper.
There are 16 combinations and 3 replications, so the total experiment is 48
experiments. There are three factors or variables that might be considered for
effecting the measure downward speed of the paper helicopter. Those factors are:
1) Paper Clip
Small Size
Large Size
2) Tail Lenght
Small Size = the tail lenght size is 10 cm
Large Size = the tail lenght size is 15 cm
3) Tail Widht
Small size = the tail widht size is 3 cm
Large size = the tail widht size is 4.5 cm
4) Body Lenght.
Small Size = the lenght size is 5 cm
Large Size = the lenght size is 7.5 cm
5) Controllable Factor
Type of Paper
Height of Experiment = 3 meters
Type of Paper Clip
Tool of Time Measurement
Among those variable, the experiment can be made into the 16
combinations with 3 replications, so the total paper helicopters are 48
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experiments. Table 3.1 shows the paper helicopeter combination with single
replication. In order to make the monitoring process easier, the levels of each
factor are symbolized shown in Table 3.1.
Table 3.1paper helicopter Combination
Number of Experiment
Body Length Tail Widht Tail Lenght Paper Clip
1 -1 -1 -1 -12 -1 -1 -1 13 -1 -1 1 -14 -1 -1 1 15 -1 1 -1 -16 -1 1 -1 17 -1 1 1 -18 -1 1 1 19 1 -1 -1 -110 1 -1 -1 111 1 -1 1 -112 1 -1 1 113 1 1 -1 -114 1 1 -1 115 1 1 1 -116 1 1 1 1
Source: Self-constructed by experimenters
Table 3.2 Center Point
No 1 2 3 4 5 6 7 8 9 10 11 12Coded -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
Center point3.26
3.28 3.1 3.42 3.07 3.42 3.28 3.64 3.42 2.88 3.28 3.28
Source: Self-constructed by experimenters
3.2. Response Measurement
After did an experiments, the downward speed of paper helicopter was
measured and the result could be obtained by using stopwatch. The table below
shows the result of response measurement. For example experiment number 1, the
first replication shows the time response is 3.42 second, the second replication is 3.28
second and third response is 3.01 second. This can be explained because of
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uncontrollable variables. The researcher assumed there are several uncontrollable
factors that caused this observation. One hypothesis is because of the air
distraction towards several paper helicopters.
Table 3.3 Response Measurement Result
NoBody
Length (BL)
Tail Width (TW)
Tail Length
(Ti)
Paper Clip
Responses
1 2 3
1 S S S S 3,6 3,55 3,422 S S S L 3,7 3,46 3,63 S S L S 3,73 3,78 3,64 S S L L 3,64 3,42 3,735 S L S S 3,51 3,28 3,016 S L S L 3,37 3,64 3,567 S L L S 3,69 3,62 3,918 S L L L 3,69 3,28 3,429 L S S S 3,42 3,64 3,2810 L S S L 3,69 3,37 3,7311 L S L S 3,73 3,78 3,8212 L S L L 3,51 3,6 3,4613 L L S S 3,42 3,28 3,4214 L L S L 3,24 3,24 3,3315 L L L S 3,96 3,42 3,5516 L L L L 3,64 3,1 3,55
Source: Self-constructed by experimenters
It is clearly seen on the table above, the downward speed rate between
paper helicopter that used small body lenght, large tail width, small tail lenght and
small paper clip has higher result rather than the another paper helicopter.Thus, it
can be assumed that body length size, tail widht size, tail lenght size and size of
paper clip has an effect on the downward speed rate. This assumption can be
tested later in the hypothesis testing.
Those three factors will be analyzed by using several methods to
determine whether or not those factors have significant effect towards response
(downward speed rate), which are: ANOVA test, residual plot, interaction plot,
and regression model. ANOVA test is being used to determine the effect of
factors towards speed rate. Residual plot is used to determine the goodness of
model of the experiment. Interaction plot is used to determine whether or not the
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factors have interaction with another factor. Later, regression model is used to
predict the future experiment with different input towards downward paper
helicopter speed rate.
3.3. Experiment Hypothesis
There are fifth models of hypothesis that is going to be tested, which are
Linear, two-way interaction, three way interaction, fourth way interaction and
interaction effect. The hypotheses are:
Linear:
1. H0A: There is no significant effect of Factor A (paper clip) to the response
(downward speed the paper helicopter).
H1A: There is a significant effect of Factor A (paper clip) to the response
(downward speed the paper helicopter).
2. H0B: There is no significant effect of Factor B (tail length) to the response
(downward speed the paper helicopter).
H1B: There is a significant effect of Factor B (tail length) to the response
(downward speed the paper helicopter).
3. H0C: There is no significant effect of Factor C (tail width) to the response
(downward speed the paper helicopter).
H1C: There is a significant effect of Factor C (tail width) to the response
(downward speed the paper helicopter).
4. H0D: There is no significant effect of Factor D (body length) to the
response (downward speed the paper helicopter).
H1D: There is a significant effect of Factor D (body length) to the response
(downward speed the paper helicopter).
Two-way Interaction(s):
5. H0AB: There is no interaction between Factor A (paper clip) and Factor B
(tail length).
H1AB: There is an interaction between Factor A (paper clip) and Factor B
(tail length).
6. H0AC: There is no interaction between Factor A (paper clip) and Factor C
(tail width).
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H1AC: There is an interaction between Factor A (paper clip) and Factor C
(tail width).
7. H0AD: There is no interaction between Factor A (paper clip) and Factor D
(body length).
H1AD: There is an interaction between Factor A (paper clip) and Factor D
(body length).
8. H0BC: There is no interaction between Factor B (tail length) and Factor C
(tail width).
H1BC: There is an interaction between Factor B (tail length) and Factor C
(tail width).
9. H0BD: There is no interaction between Factor B (tail length) and Factor D
(body length).
H1BD: There is an interaction between Factor B (tail length) and Factor D
(body length).
10. H0CD: There is no interaction between Factor C (tail width) and Factor D
(body length).
H1CD: There is an interaction between Factor C (tail width) and Factor D
(body length).
Three-way Interaction:
11. H0ABC: There is no interaction between Factor A (paper clip), Factor B (tail
length), and Factor C (tail width).
H1ABC: There is an interaction between Factor A (paper clip), Factor B (tail
length), and Factor C (tail width).
12. H0ABD: There is no interaction between Factor A (paper clip), Factor B (tail
length), and Factor D (body length).
H1ABD: There is an interaction between Factor A (paper clip), Factor B (tail
length), and Factor D (body length).
13. H0ACD: There is no interaction between Factor A (paper clip), Factor C (tail
width), and Factor D (body length).
H1ACD: There is an interaction between Factor A (paper clip), Factor C (tail
width), and Factor D (body length).
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14. H0BCD: There is no interaction between Factor B (tail length), Factor C (tail
width), and Factor D (body length).
H1BCD: There is an interaction between Factor B (tail length), Factor C (tail
width), and Factor D (body length).
Fourth-way Interaction:
15. H0ABCD: There is no interaction between Factor A (paper clip), Factor B
(tail length), Factor C (tail width) and Factor D (body length).
H1ABCD: There is an interaction between Factor A (paper clip), Factor B
(tail length), Factor C (tail width) and Factor D (body length).
Interaction effects:
16. H0AB: There is no interaction between Factor A (paper clip) and Factor B
(tail length) towards response (downward speed the paper helicopter).
H1AB: There is an interaction between Factor A (paper clip) and Factor B
(tail length) towards response (downward speed the paper helicopter).
17. H0AC: There is no interaction between Factor A (paper clip) and Factor C
(tail width) towards response (downward speed the paper helicopter).
H1AC: There is an interaction between Factor A (paper clip) and Factor C
(tail width) towards response (downward speed the paper helicopter).
18. H0AD: There is no interaction between Factor A (paper clip) and Factor D
(body length) towards response (downward speed the paper helicopter).
H1AD: There is an interaction between Factor A (paper clip) and Factor D
(body length) towards response (downward speed the paper helicopter).
19. H0BC: There is no interaction between Factor B (tail length) and Factor C
(tail width) towards response (downward speed the paper helicopter).
H1BC: There is an interaction between Factor B (tail length) and Factor C
(tail width) towards response (downward speed the paper helicopter).
20. H0BD: There is no interaction between Factor B (tail length) and Factor D
(body length) towards response (downward speed the paper helicopter).
H1BD: There is an interaction between Factor B (tail length) and Factor D
(body length) towards response (downward speed the paper helicopter).
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21. H0CD: There is no interaction between Factor C (tail width) and Factor D
(body length) towards response (downward speed the paper helicopter).
H1CD: There is an interaction between Factor C (tail width) and Factor D
(body length) towards response (downward speed the paper helicopter).
22. H0ABC: There is no interaction between Factor A (paper clip), Factor B (tail
length), and Factor C (tail width) towards response (downward speed the
paper helicopter).
H1ABC: There is an interaction between Factor A (paper clip), Factor B
(tail length), and Factor C (tail width) towards response (downward speed
the paper helicopter).
23. H0ABD: There is no interaction between Factor A (paper clip), Factor B (tail
length), and Factor D (body length) towards response (downward speed
the paper helicopter).
H1ABD: There is an interaction between Factor A (paper clip), Factor B (tail
length), and Factor D (body length) towards response (downward speed
the paper helicopter).
24. H0ACD: There is no interaction between Factor A (paper clip), Factor C (tail
width), and Factor D (body length) towards response (downward speed the
paper helicopter).
H1ACD: There is an interaction between Factor A (paper clip), Factor C (tail
width), and Factor D (body length) towards response (downward speed the
paper helicopter).
25. H0BCD: There is no interaction between Factor B (tail length), Factor C (tail
width), and Factor D (body length) towards response (downward speed the
paper helicopter).
H1BCD: There is an interaction between Factor B (tail length), Factor C (tail
width), and Factor D (body length) towards response (downward speed the
paper helicopter).
26. H0ABCD: There is no interaction between Factor A (paper clip), Factor B
(tail length), Factor C (tail width) and Factor D (body length) towards
response (downward speed the paper helicopter).
Helicopter Project | Design of ExperimentPage 24Industrial Engineering 2012 | President University
H1ABCD: There is an interaction between Factor A (paper clip), Factor B
(tail length), Factor C (tail width) and Factor D (body length) towards
response (downward speed the paper helicopter).
Helicopter Project | Design of ExperimentPage 25Industrial Engineering 2012 | President University
CHAPTER IV
DATA ANALYSIS
4.1. Pre Test
Fourth Factors are being considered as the Independent Variables that will
be examined whether or not the factor influenced (or has significant effect) to the
response as the Dependent Variables (downward speed of paper helicopter).
Those fourth factors are: body length, tail width, tail length and paper clip; each of
it have same levels. The first factor, paper clip, is the “categorical” factor with two
levels which are large and small. The second factor, tail length, is the “numerical”
factor with two levels of factor which are (10 cm) small and (15cm) large. The
third factor, tail width, is the “numerical” factor with two levels of factor which
are (3 cm) small and (4.5 cm) large. The last factor, body length, is clearly the
“numerical” factor with two levels of factor which are 5 cm (small) and (7.5 cm)
large.
The symbol of minus (-) and plus (+) means a low and high level
respectively. It is perfectly indicates for level of downward speed of paper
helicopter factor can be assumed which one indicates the low or high level. For
factor with 2 levels, the level can be obtained by -1 and +1. In this case, the fourth
levels factors are 3 numerical and 1 categorical, so it can be assumed at any level.
This case indicates the 4 factors and same levels with 2 of Factorial Design or
simply called by 2k Level Factorial Design. Three Replications is being observed
in order to accurate the data experiment.
The run number test is shown from the total combination of the factorial
design. The total run number is 48 combinations (2 level * 2 level * 2 level * 2 level *
3 replication = 48 combinations). The run number is obtained by using Minitab
Software. The order of the run number is shown in Table 4.1.
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Table 4.1 Run Number
Std Order
Run Order
Center Pt
Paper Clip
Tail Length
Tail Width
Body Length
Response
1 1 1 -1 -1 -1 -1 3.602 2 1 1 -1 -1 -1 3.703 3 1 -1 1 -1 -1 3.734 4 1 1 1 -1 -1 3.645 5 1 -1 -1 1 -1 3.516 6 1 1 -1 1 -1 3.377 7 1 -1 1 1 -1 3.698 8 1 1 1 1 -1 3.699 9 1 -1 -1 -1 1 3.4210 10 1 1 -1 -1 1 3.6911 11 1 -1 1 -1 1 3.7312 12 1 1 1 -1 1 3.5113 13 1 -1 -1 1 1 3.4214 14 1 1 -1 1 1 3.2415 15 1 -1 1 1 1 3.9616 16 1 1 1 1 1 3.6417 17 1 -1 -1 -1 -1 3.5518 18 1 1 -1 -1 -1 3.4619 19 1 -1 1 -1 -1 3.7820 20 1 1 1 -1 -1 3.4221 21 1 -1 -1 1 -1 3.2822 22 1 1 -1 1 -1 3.6423 23 1 -1 1 1 -1 3.6224 24 1 1 1 1 -1 3.2825 25 1 -1 -1 -1 1 3.6426 26 1 1 -1 -1 1 3.3727 27 1 -1 1 -1 1 3.7828 28 1 1 1 -1 1 3.6029 29 1 -1 -1 1 1 3.2830 30 1 1 -1 1 1 3.2431 31 1 -1 1 1 1 3.4232 32 1 1 1 1 1 3.1033 33 1 -1 -1 -1 -1 3.4234 34 1 1 -1 -1 -1 3.6035 35 1 -1 1 -1 -1 3.6036 36 1 1 1 -1 -1 3.7337 37 1 -1 -1 1 -1 3.0138 38 1 1 -1 1 -1 3.5639 39 1 -1 1 1 -1 3.9140 40 1 1 1 1 -1 3.4241 41 1 -1 -1 -1 1 3.2842 42 1 1 -1 -1 1 3.7343 43 1 -1 1 -1 1 3.8244 44 1 1 1 -1 1 3.4645 45 1 -1 -1 1 1 3.4246 46 1 1 -1 1 1 3.3347 47 1 -1 1 1 1 3.5548 48 1 1 1 1 1 3.55
Source: Primary Data by Minitab 17
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4.2. Estimating the Factor Effect
The first step of experimental design is to estimate the factor effect. The
factor effect could give the information of important design factor and interaction
as well as its signs and magnitudes. This step involves main effect plot and
interaction between factors supported by normal probability plot, half normal
probability plot, and Pareto charts of the standardize effect.
When performing a statistical analysis, one of the simplest graphical tools
is a Main Effects Plot. This plot shows the average outcome for each value
(response) of each variable (factor), combining the effects of the other variables as
if all variables were independent.
1-1
3.625
3.600
3.575
3.550
3.525
3.500
3.475
3.450
1-1 1-1 1-1
Paper Clip
Mea
n
Tail Length Tail Width Body Length
Main Effects Plot for ResponseData Means
Figure 4.1 Main Effect Plot for Response
Source: Primary Data by Minitab 17
Figure 4.1 shows the Main Effect Plot for the Response for each Factor.
First, the average (mean) of response for Factor A (Paper Clip) indicates the effect
of small (-1) level is the greater than the large level (+1). Second, the average
(mean) of response for Factor B (Tail Length) indicates the effect of 15 cm (+1)
level is extremely greater than 10 cm level (-1). Third, the average (mean) of
response for Factor C (Tail Width) indicates the effect of 3 cm (-1) level is the
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greater than 4.5 cm (+1). Fourth, the average (mean) of response for factor D
(Body Length) indicates the effect of 5 cm (-1) level greater than 7.5 cm (+1)
level. By this graph, it can be concluded that the lower level of three factors has
greater mean rather than the higher level.
Interaction Plot
Another graphic statistical tool is called an Interaction Plot. This type of
chart illustrates the effects between variables which are not independent. If there
is any intersection between factors, means the factor has interaction with another
factor. Figure 4.2 shows the Interaction Plot for data means
1-1 1-1 1-13.75
3.60
3.45
3.75
3.60
3.45
3.75
3.60
3.45
Paper Clip
Tail Length
Tail Width
Body Length
-11
ClipPaper
-11
LengthTail
-11
WidthTail
Interaction Plot for ResponseData Means
Figure 4.2 Interaction Plot for Response
Source: Primary Data by Minitab 17
Based on Figure 4.2, it is shown that there is an interaction between Paper
Clip and Tail. Besides that, it is shown that there is no interaction between all of
it; paper clip-tail width, tail length-tail width, paper clip-body length, tail length-
body length, and tail width-body length.
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Coded Coefficients
Term Effect Coef SE Coef T-Value P-ValueConstant 3.5290 0.0244 144.71 0.000Paper Clip -0.0604 -0.0302 0.0244 -1.24 0.024Tail Length 0.1612 0.0806 0.0244 3.31 0.002Tail Width -0.1304 -0.0652 0.0244 -2.67 0.012Body Length -0.0429 -0.0215 0.0244 -0.88 0.385Paper Clip*Tail Length -0.1521 -0.0760 0.0244 -3.12 0.004Paper Clip*Tail Width -0.0237 -0.0119 0.0244 -0.49 0.630Paper Clip*Body Length -0.0446 -0.0223 0.0244 -0.91 0.368Tail Length*Tail Width 0.0496 0.0248 0.0244 1.02 0.317Tail Length*Body Length 0.0104 0.0052 0.0244 0.21 0.832Tail Width*Body Length -0.0262 -0.0131 0.0244 -0.54 0.594Paper Clip*Tail Length*Tail Width
-0.0088 -0.0044 0.0244 -0.18 0.859Paper Clip*Tail Length*Body Length
0.0238 0.0119 0.0244 0.49 0.630Paper Clip*Tail Width*Body Length
-0.0296 -0.0148 0.0244 -0.61 0.548Tail Length*Tail Width*Body Length
-0.0063 -0.0031 0.0244 -0.13 0.899Paper Clip*Tail Length*Tail Width*Body Length
0.0821 0.0410 0.0244 1.68 0.102
Term VIFConstantPaper Clip 1.00Tail Length 1.00Tail Width 1.00Body Length 1.00Paper Clip*Tail Length 1.00Paper Clip*Tail Width 1.00Paper Clip*Body Length 1.00Tail Length*Tail Width 1.00Tail Length*Body Length 1.00Tail Width*Body Length 1.00Paper Clip*Tail Length*Tail Width 1.00Paper Clip*Tail Length*Body Length 1.00Paper Clip*Tail Width*Body Length 1.00Tail Length*Tail Width*Body Length 1.00Paper Clip*Tail Length*Tail Width*Body Length 1.00
Figure 4.3 Estimated Effects and Coefficients for Response (Full Model)
Source: Primary Data by Minitab 17
Graphical plot is necessary to estimate the factorial effect, however, it
cannot predict accurately. Then, numerical statistic analysis is being used to
analyze the factorial effect accurately based on numerical value that is obtained by
statistical software, Minitab 17. Based on Figure 4.3, there are three factors that
significantly affect the response, which are: Paper Clip, Tail Length, and Tail
Width. In addition, the interaction plot between Paper Clip and Tail Length is
Mung Bean Project | Design of ExperimentPage 30Industrial Engineering 2012 | President University
significant. The p-value of those factors and interactions are lower than
significance levels (P-value ≤ α = 0.05).
Figure 4.4 showed the normal plot of the standardized effects on response.
Based on that plot, it is shown that the red point is significant with α =0.05, which
are: Factor B (Tail Width), Factor C (Tail Length), and Interaction AB (Paper
Clip and Tail Length). It shows the Factor B (Tail Width) has significant positive
effects on response because it is located at the right side of line. Otherwise, Factor
C and Interaction AB have significant negative effects on responses.
43210-1-2-3
99
95
90
80
70
60
5040
30
20
10
5
1
A Paper ClipB Tail LengthC Tail WidthD Body Length
Factor Name
Standardized Effect
Perc
ent
Not SignificantSignificant
Effect Type
AB
C
B
Normal Plot of the Standardized Effects(response is Response, α = 0.05)
Figure 4.4 Normal Plot of Standardized Effects on Response (Full Model)
Source: Primary Data by Minitab 17
Meanwhile, the half normal plot shows the absolute standardized effects to
compare their relative magnitudes. Since the point of factor B is the furthest to the
right means the effect is most highly significant to the response, followed by
Interaction AB and Factor C respectively.
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3.53.02.52.01.51.00.50.0
98
95
90
85
80
70
60
50
40
3020100
A Paper ClipB Tail LengthC Tail WidthD Body Length
Factor Name
Absolute Standardized Effect
Perc
ent
Not SignificantSignificant
Effect Type
AB
C
B
Half Normal Plot of the Standardized Effects(response is Response, α = 0.05)
Figure 4.5 Half Normal Plot of Standardized Effects on Response (Full Model)
Source: Primary Data by Minitab 17
Pareto Chart of the Standardized Effects helps to determine the
magnitudes as well as the significant of this effect. The effect that exceeds the red
line is statistically important or significant. It is shown that the Factor A,
Interaction AB, and Factor C are passing the reference line at the level of
significance of 5%.
Based on previous statistical software output, it is shown that there is the
difference result by numerical output and categorical output.
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Term
BCD
ABC
BD
AC
ABD
CD
ACD
D
AD
BC
A
ABCD
C
AB
B
3.53.02.52.01.51.00.50.0
A Paper ClipB Tail LengthC Tail WidthD Body Length
Factor Name
Standardized Effect
2.037
Pareto Chart of the Standardized Effects(response is Response, α = 0.05)
Figure 4.6 Pareto Chart of Standardized Effects on Response (Full Model)
Source: Primary Data by Minitab 17
Form Initial Model
The initial Full model including all terms in coded units by using the coefficients
presented in Figure 4.3 is:
4.4. ANOVA Test
The main effects plot and interaction plot do not provide a great deal of
information. Showing just the main effects and interaction of each factor level
without accounting for the levels of other factors is simplistic and could be
misleading. The ANOVA test is being used to determine the effect of the factors
and/or interaction towards the response in the numerical model.
Figure 4.4 shows the p-value of each factor and interaction between
factors that are obtained from Minitab. The rejection criterion for p-value shows if
the p-value < than α (α = 0.05) means to reject H0. Based on ANOVA test on
Figure 4.4, it can be concluded that tail widht and tail lenght that have an effect to
the response (downward speed of paper helicopter), the p-value of tail width is
0.003 and p-value of tail lenght is 0.015. Another factor which is body length and
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paper clip have no significant effect to the response because the p-value is greater
than α, the p-value of body lenght is 0.409 and p-value of paper clip is 0.500. In
addition, there is no significant effect for interaction between factors (two-way
interaction, three-way interaction and four way intercactions) towards response.
Analysis of Variance
Source DF Adj SS AdjMS F-Value P-ValueModel 16 1.60673 0.100420 3.16 0.001 Linear 4 0.55295 0.138237 4.35 0.005 Paper Clip 1 0.01473 0.014727 0.46 0.500 Tail Length 1 0.31202 0.312019 9.81 0.003 Tail Width 1 0.20410 0.204102 6.42 0.015 Body Length 1 0.02210 0.022102 0.69 0.409 2-Way Interactions 6 0.34725 0.057874 1.82 0.118 Paper Clip*Tail Length 1 0.27755 0.277552 8.73 0.005 Paper Clip*Tail Width 1 0.00677 0.006769 0.21 0.647 Paper Clip*Body Length 1 0.02385 0.023852 0.75 0.391 Tail Length*Tail Width 1 0.02950 0.029502 0.93 0.341 Tail Length*Body Length 1 0.00130 0.001302 0.04 0.841 Tail Width*Body Length 1 0.00827 0.008269 0.26 0.613 3-Way Interactions 4 0.01866 0.004665 0.15 0.964 Paper Clip*Tail Length*Tail Width 1 0.00092 0.000919 0.03 0.866 Paper Clip*Tail Length*Body Length 1 0.00677 0.006769 0.21 0.647 Paper Clip*Tail Width*Body Length 1 0.01050 0.010502 0.33 0.569 Tail Length*Tail Width*Body Length 1 0.00047 0.000469 0.01 0.904 4-Way Interactions 1 0.08085 0.080852 2.54 0.118 Paper Clip*Tail Length*Tail Width*Body Length 1 0.08085 0.080852 2.54 0.118 Curvature 1 0.60702 0.607020 19.08 0.000Error 43 1.36777 0.031809 Lack-of-Fit 1 0.05075 0.050750 1.62 0.210 Pure Error 42 1.31702 0.031358Total 59 2.97449
Figure 4.3 P-value of ANOVA
4.5. Residual Plot and Model Adequacy
The normal probability plot is a graphical technique for assessing whether
or not a data set is approximately normally distributed (Chambers et al., 1983).
The data are plotted against a theoretical normal distribution in such a way that
the points should form an approximate straight line. Departures from this straight
line indicate departures from normality.
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0.500.250.00-0.25-0.50
99.9
99
90
50
10
1
0.1
Residual
Perc
ent
3.73.63.53.43.3
0.4
0.2
0.0
-0.2
-0.4
Fitted Value
Res
idua
l
0.320.160.00-0.16-0.32
16
12
8
4
0
Residual
Freq
uenc
y
605550454035302520151051
0.4
0.2
0.0
-0.2
-0.4
Observation Order
Res
idua
l
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for Response
Figure 4.4 Residual Plots for Response
Based on Figure 4.4, the points on this plot are distributed because it is the
straight line, which indicates that the model is normal distributed. The plot shows
that it is light-tailed distribution. Histogram Chart shows this model is also
normally distributed, it can be shown that the chart is bell shaped. Based on
normal probability plot and histogram, it can be concluded that the model is
normally distributed.
Figure 4.4 shows the Residual Plots for Response obtained by Minitab.
The residual plot (versus fits) shows the variance is an increase function of y
(response or growth rate). The residual plot (versus order) shows that is negative
autocorrelation.
4.6. Hypothesis Testing
Based on Effect Test, Interaction Plot, Residual Plot, and ANOVA test;
the hypothesis testing can be done based on those analyses. The following
Hypotheses Testing is shown on Table 4.5.
Table 4.4 Hypotheses Testing for Problem 6.20
Hypotheses H0 H1 DecisionLinear
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Factor A There is no significant effect of Factor A (paper clip) to the response (downward speed rate of paper helicopter)
There is a significant effect of Factor A (paper clip) to the response (downward speed rate of paper helicopter)
Reject H0
Factor B There is no significant effect of Factor B (tail lenght) to the response (downward speed rate of paper helicopter)
There is a significant effect of Factor B (tail lenght) to the response (downward speed rate of paper helicopter)
Reject H0
Factor C There is no significant effect of Factor C (tail width ) to the response (downward speed rate of paper helicopter)
There is a significant effect of Factor C (tail width) to the response (downward speed rate of paper helicopter)
Do not Reject H0
Factor D There is no significant effect of Factor D (body length) to the response (downward speed rate of paper helicopter)
There is a significant effect of Factor D (body length) to the response (downward speed rate of paper helicopter)
Do not Reject H0
Two-way InteractionsFactor A & B There is no interaction between
Factor A (paper clip) and Factor B (tail lenght)
There is an interaction between Factor A (paper clip) and Factor B (tail lenght)
Do not Reject H0
Factor A & C There is no interaction between Factor A (paper clip) and Factor C (tail width)
There is an interaction between Factor A (paper clip) and Factor C (tail width)
Do not Reject H0
Factor A & D There is no interaction between Factor A (paper clip) and Factor D (body length)
There is an interaction between Factor A (paper clip) and Factor D (body length)
Do not Reject H0
Factor B & C There is no interaction between Factor B (tail lenght) and Factor C (tail width)
There is an interaction between Factor B (tail lenght) and Factor C (tail width)
Do not Reject H0
Factor B & D There is no interaction between Factor B (tail lenght) and Factor D (body length)
There is an interaction between Factor B (tail lenght) and Factor D (body length)
Do not Reject H0
Factor C & D There is no interaction between Factor C (tail width) and Factor D (body length)
There is an interaction between Factor C (tail width) and Factor D (body length)
Do not Reject H0
Three-way InteractionsFactor A-B-C There is no interaction between
Factor A (paper clip), Factor B (tail lenght), and Factor C (tail width)
There is an interaction between Factor A (paper clip), Factor B (tail lenght), and Factor C (tail width)
Do not Reject H0
Factor A-B-D There is no interaction between Factor A (paper clip), Factor B (tail lenght), and Factor D (body length)
There is an interaction between Factor A (paper clip), Factor B (tail lenght), and Factor D (body length)
Do not Reject H0
Factor A-C-D There is no interaction between Factor A (paper clip), Factor C (tail width), and Factor D (body length)
There is an interaction between Factor A (paper clip), Factor C (tail width), and Factor D (body length)
Do not Reject H0
Factor B-C-D There is no interaction between Factor B (tail lenght), Factor C (tail width), and Factor D (body length)
There is an interaction between Factor B (tail lenght), Factor C (tail width), and Factor D (body length)
Do not Reject H0
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Four-way InteractionsFactor A-B-C-D
There is no interaction between Factor A (paper clip), Factor B (tail lenght), Factor C (tail width), and Factor D (body length)
There is an interaction between Factor A (paper clip), Factor B (tail lenght), Factor C (tail width), and Factor D (body length)
Do not Reject H0
The decision for reject or do not reject H0 is based on ANOVA test. The p-
value indicates the effect on the factor. If p-value is greater than α (α = 0.05), do
not reject H0, or vice versa. The p-value of Factor B is 0.03 and the p-value of
Factor C is 0.015 which are less than α (α = 0.05), which means those Factors are
significantly has effect on the response.
Based on Table 4.5, it can be concluded the Factor B (tail lenght) and Factor C
(tail width) has significant effect towards Response (downward speed rate of
paper helicopter) independently. There is no interaction between Factor B and
Factor C. Thus, the others H0 on hypothesis should not be rejected. Factor A
(paper clip) and Factor D (body length) are not significantly effect to the
Response (downward speed rate of paper helicopter).
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4.7. Regression Model
Multiple regression analysis is a statistical technique to predict the
variance in the dependent variable by regressing the independent variable against
it. Multiple regression analysis used in situation where two or more independent
variables are hypothesized to affect one dependent variable. Based on ANOVA
test, the regression model can be obtained by one factor only, which is: Factor B
(Plant Food). Then the model equation used in this case can be explained as
follows:
Y = β0 + β2 X2 + e
Where:
Y = Growth Rate
β0 = Constant
β2 = X2 Regression coefficient
X2 = Factor B (Plant Food)
e = random error term/residuals
Regression EquationGrowth Rate = 3.561 - 2.089 Food_No + 2.089 Food_Yes
Figure 4.5 Regression Model from Minitab
According to the result of multiple regression analysis tests that has been
done by Minitab; the regression model is clearly shown in Figure 4.5. The general
equation of regression model is:
Y = 3.561 ± 2.089 X2 + e
From the regression linear above, the conclusions are as follow:
1. The equation has a Constant of 12.295 which means that if Factor B (Plant
Food) is assumed being zero, the response (growth rate) is 3.561.
2. The coefficient regression of Factor B (Plant Food) is 2.089 which means
every 100% improvement in variable of Factor B (Plant Food) will
increase (+) the response (Growth Rate) for 208.9% if the plant using
Plant Food, otherwise Factor B (Plant Food) will decrease (-) the response
(growth rate) for 208.9% if the plant is not using Plant Food.
From Regression Analysis, it can be conclude that the Plant Food has
significant effect to the Growth Rate of Mung Bean Plant. The Plant Food has
influenced about 208.9% towards Growth Rate.
Mung Bean Project | Design of ExperimentPage 38Industrial Engineering 2012 | President University
CHAPTER V
CONCLUSION
The analyses of problems are obtained using Minitab Software. Mung
Bean sprout is being used as the experimental design. Frequency of watering,
Plant Food usage, and Volume of water are the factors that might be affecting the
response, which is Growth Rate. 18 combinations are being observed with 2
replications each. The total run number is 36 combinations. Main Effect Plot,
Interaction Plot, ANOVA test, and Residual Plot are being used to analyze the
experiment.
Based on Main Effect Plot, the average (mean) of response for Factor A
(Frequency) indicates the effect of Three a day Level is the greatest followed by
Twice and Once a day. Second, the average (mean) of response for Factor B
(Plant Food) indicates the effect of Yes Level is extremely greater than No Level.
Third, the average (mean) of response for Factor C (Volume) indicates the effect
of 2 squirts Level is the greatest followed by 3 squirts and 1 squirt.
Based on Interaction Plot, it is shown that there is no interaction between
Factor A (frequency) and Factor B (plant food). Also, there is an interaction
between Factor A (frequency) and Factor C (volume). Last, there is no interaction
between Factor B (plant food) and Factor C (volume).
Based on ANOVA Test, only Factor B (plant food) that has an effect to
the response (growth rate). Another factors, Factor A (frequency) and Factor C
(volume) has no significant effect to the response. In addition, there is no
significant effect for interaction between factors (two-way interaction and three-
way interaction) towards response.
Based on Residual Plot, the points on this plot are not distributed closed to
the straight line, which indicates that the model is not normal distributed. The plot
shows that it is light-tailed distribution. Histogram Chart shows this model is not
normally distributed, it can be shown that the chart is not bell shaped. The residual
plot (versus fits) shows the variance is an increase function of y (response or
growth rate). The residual plot (versus order) shows that is negative
autocorrelation.
Mung Bean Project | Design of ExperimentPage 39Industrial Engineering 2012 | President University
REFERENCE
Haryadi. 2012. Perencanaan dan Analisis Experimen dengan Minitab.
Palangkaraya : Karya Ilmiah Pengabdian pada Masyarakat.
Montgomery, Douglas C. 2009. Design and Analysis of Experiments 7th
Edition. Asia : John Wiley and Sons Pte Ltd
Pan,Jianbiao . Minitab Tutorials for Design and Analysis of Experiments
pdf : Accessed from www.google.co.id. (21 January 2015)
Mung Bean Project | Design of ExperimentPage 40Industrial Engineering 2012 | President University