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Introduction
a) The Olympic
Games is aninternational
event featuringsummer and
winter sports,in which
athletesparticip
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ate in different
competitions.
In ancientGreece the
Olympic Gameswere athletic
competitionsheld in honor of
Zeus. Since the
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Olympic Games
began they have
been thecompetition
grounds for
greatestathletes. First
place obtaining
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gold; second
silver and third
bronze. TheOlympic
medalsrepresent the hardship
of what thecompetitors of
the Olympics
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have done in
order to obtain
the medal.Onone side the
Olympic medalhas Nike the
goddess ofvictory holding
a palm and a
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winners crown
andon the other
side the medalhas a different
label for eachOlympiad
reflecting thehost of the
games.Olympic
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medals could be
used as a unit
of measure ofathleticism.Top
10 OlympicMedal- winning
CountriesCountry Medals won1.
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The
UnitedStates24
042.
Soviet Union
12043.
Great Britain
6894.
France 6795.Germany 6486.
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Italy 5957.Sweden 5888.East Germany5199.Hungary 45410.
Finland 446This
is a tableshowing the top
10 Olympic
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Medal- Winning
Countries and
definitelyshows how
theOlympicscan be seen as a
standardizedunit of
athleticism
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Dataa)Height (in
centimeters)
achieved by the
gold medalist at
various Olympicgames.Year
1932 1936 1948
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1952 1956 1960
1964 1968 1972
19761980Height(cm)
197 203 198204 212 216
218 224 223225 236
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Variables andConstraintsa) Thedependent
variable forthis data set is
the OlympicGold Medalist
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Heights. The
independentvari
able for thisdata set is the
years in whichthe summer
Olympic Gamesoccurred in. A
constraint of
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this data set is
the limited
amount of datathat is
available. Thedata available is
only between1932 and
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19520020521021
5220225230235
2401 9 2
0 1 9 3 0
1 9 4 0 1
9 5 0 1 96 0 1 9 7
0 1 9 8 01 9 9 0 H
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e i g h t ( c m )YearYear Of
OlympicsVS. Gold
Medal
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H
eightsInitial values1980. If there
were more data
and if there is a
pattern the
pattern would
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become more
apparent. Ontop
of this there isa gap between
1936 and 1948which is a 12
year chunk ofdata missing.
And since
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theOlympics
are held every
4 years that is3 Olympic
competitionsmissing from
the data.b) In amath textbook
the variables
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and constraints
could be seen
as y= OlympicGold Medalist
Heightsand x=years in which
the summerOlympic games
occurred in.c)
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In the context
of this problem
is that the xaxis would be
used to showthe Year of the
OlympicGamesand y
would be used
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to show the
height of the
gold medalist.d)This data set is
continuousbecause it is
associated witha measurement
and its possible
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to have
thesame y value
for different xvalues. And
since the datais measuring
height adecimal answer
is possible.A
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function that
would fit most
of the datawould be a
quadraticfunction. The
constraint thatthere is a12
year gap
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between 1936
and 1948 could
skew the data.The data that
we are missingcould of
showedus amuch clearer
model like a
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linear or could
of reassured us
of a quadraticmodel.Graph ofInitialD
ataa)
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Analysis andModel
ConstructionBased on thedata the typeof curve that
might beexpected is
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quadratic and
maybe even a
thirddegreefunction.
The expectedshape would be
quadratic if wedisregarded the
1936 value and
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itlooks like a
third degreefunction would
fit well.
a)Some generalformulas that
the data could
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fit can be
quadratic
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f(x),linear
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, or exponential
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.Quadratic
possible values
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:X Y1932 197
begin_of_the_s
kype_highlighting 1932 197
end_of_the_skype_highlightin
g.991936199.041948
204.331952
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206.821956
209.671960
212.881964216.461968
220.391972224.691976
229.341980234.36Linear
Possible values
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X Y1932 194
begin_of_the_s
kype_highlighting 1932 194
end_of_the_skype_highlightin
g.141936197.161948
206.221952
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209.241956
212.261960
215.281964218.31968
221.321972224.341976
227.361980230.38Exponen
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tial Possible
Values
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X Y1932 194
begin_of_the_skype_highlighti
ng 1932 194end_of_the_sk
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ype_highlightin
g.721936
197.491948206.041952
208.971956211.951960 214begin_of_the_s
kype_highlighti
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ng 1960 214
end_of_the_sk
ype_highlighting.961964
218.021968221.121972
224.271976227.461980
230.69All of
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the formulas
used to fit the
data areincreasing and
all have a min (when x=0) for
thequadraticfit the minimum
is 41946.847 of
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the linear fit
the minimum is
-1264.6484 andfor
theexponentialfit the min is
0.21211804. Allof the formulas
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maximum.For
the model to
become muchmore realistic
the logical fitwould be a
sinusoidalcurve.
Theconstraints
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on a sinusoidal
curve would be
that the curvewould have to
be half a cycle.The curve has
tobe half acycle because it
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curve would
show that the
heightsfluctuate from
OlympicstoOlympics and
this would notrepresent our
data. But by
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limiting the
curve to half a
cycle then thecurvewould fit
the data.b)
Linear Model
QuadraticModel
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General formfor a linear
equation:
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When using twopoints from thedata
andsubstituting
them into the
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values for x
andy the values
for m and b canbe found.
Thepoints thatare going to be
used aregoingto be
(1972,223) and
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(1960,216).
Thesepoints
were chosenbecause they
look likea niceline can be
drawn betweenthemwithout
too much
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differentiation
betweenthe
other points.
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To find the
value of a, we
can subtractthetwo
equations. Thiswill
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We can now
find m by
dividing bothsidesby 12 so
we get:
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The value for b
can now be
foundbysubstituting
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to any of theequations
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The general
form for an
quadraticequation is
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. With 3pointson the graph anequation can
beformulated in
the form
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. The points
that are going
to be usedaregoing to be
(1936,203),(1972,233)
and(1960,216)We can make
three equations
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using
thesethree
points and thegeneral formula
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. bysubstitutingxand y values
we get:
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or
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or
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or
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Then we cansubtract twoequationstogeth
er to eliminate
C. I chose to
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subtractthe
first and
second equation
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The equationwill now be:
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Now I willsubtract thesecond and
thirdequation
to get two new
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equations
tosolve for a
and b.
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With the two
equations now
we mustisolatea variable to
solve, I will bechosinga, so we
need toeliminate b. we
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can dothis by
multiplying
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Bysince
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. By multiplyinby
we get
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. then weadd
the equations:
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Now we can
solve for a and
we get
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. Now we can
substitute ainto
one of theequations with
twovariables tosolve for
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. I will be using
theequation
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.bysubstitutinga we get:
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we can nowsubstitute botha and b intothe
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original
equations and
solve for c. Iwillbe
substituting ininto the
secondequation.
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Now we havefound allvariables and
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wecan write our
equation. And
we get:
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Linear FunctionQuadraticfunction
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The reason whyI chose a linearmodel is
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because despite
some points the
data in thegraph seemed
tobe modeledwell by a linear
model. I alsochose a
quadratic model
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because the
data points
seemed tobemodeled well
by a quadraticmodel.b)Linear Equation:
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1952002052102152202252302352401 9 2
0 1 9 3 01 9 4 0 1
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9 5 0 1 9
6 0 1 9 7
0 1 9 8 0
1 9 9 0 He i g h t ( c m )
Year
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Year Of
Olympics
VS. GoldMedalH
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eightsInitialvaluesLinearEquationYearsInitialvalues Linear
Equation1932197 199.671936
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203
202.003333319
48 198209.003333319
52 204211.336666719
56 212213.671960 216
216.003333319
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64 218
218.336666719
68 224220.671972
223223.003333319
76 225225.336666719
80 236
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227.67This
linear function
is not the bestfit for the
model becauseas we can see
from the graphthere are a
lotof points left
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out and that
are not even
close to thelinear equation.
Also in the datatable there
aresome yearswhere the
linear equation
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varies greatly
from the actual
seethat forsome years it is
close like for1936 but for
others it is
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much farther
away from the
linearequationlike in 1948.0501001502002503001 9
2 0 1 9
3 0 1 9
4 0 1 9
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5 0 1 9
6 0 1 9
7 0 1 9
8 0 1 9
9 0 He i g h t ( c m )Year
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Year Of
Olympics
VS. GoldMedalH
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eightsQuadraticEquation:
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Years Initialvalues
Quadratic
Equation1932
197
204.578947419
36 203
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2031948 198
204.157894719
52 204206.508771919
56 212209.842105319
60 216214.157894719
64 218
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219.456140419
68 224
225.73684211972 223
2331976 225241.245614198
0 236250.4736842T
he quadratic
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equation that I
came up with is
a reasonable fitbut still not the
best fit. As wecan seefrom
the graph thequadratic
equation
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increases
quicker than
the actualvalues. This
causes the datato0501001502002501 9 2 0
1 9 3 0 1
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9 4 0 1 9
5 0 1 9 6
0 1 9 7 0
1 9 8 0 1
9 9 0 He i g h t ( c m )Year
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Year Of
Olympics
VS. GoldMedalH
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eightsInitialvaluesLinearRegressionbe way off forthe final two
years in ourdata. We can
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see this from
the data table
as 1976 and1980 arenot as
close are theother actual
values to thevalues from the
quadratic
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equation.Linear
Regression
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Years Initial
values Linear
Regression1932197
194.13824261936 203
197.15850481948 198
206.219291419
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52 204
209.239553619
56 212212.259815819
60 216215.280078196
4 218218.3003402
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19520020521021
5220225230235
2401 9 2 0
1 9 3 0 1
9 4 0 1 9
5 0 1 9 60 1 9 7 0
1 9 8 0 19 9 0 H
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e i g h t ( c m )YearYear Of
OlympicsVS. Gold
Medal
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H
eightsInitial
valuesQuadratic
Regression1968 224
221.32060241972 223
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224.340864619
76 225
227.36112681980 236
230.381389This is a better
linear model butit is still not
the best.
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Because it is
linear it
excludes somepoints like1948.
In 1948 we cansee that
differencebetween the
actual values
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and the linear
regression
models isquitesignificant.Qua
draticRegression
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Years Initial
values
QuadraticRegression1932
197197.995296519
36 203199.0379511194
8 198
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204.334820919
52 204
206.82341271956 212
209.67348881960 216
212.88504931964 218
216.458094
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19520020521021
5220225230235
2401 9 2 0
1 9 3 0 1
9 4 0 1 9
5 0 1 9 60 1 9 7 0
1 9 8 0 19 9 0 H
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e i g h t ( c m )YearYear Of
OlympicsVS. Gold
Medal
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H
eightsInitial
valuesQuadratic
Regression1968 224
220.3926231972 223
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224.688636419
76 225
229.3461341980 236
234.3651159This is the best
fit because aswe can see
from the graph
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it fits most
points without
leaving othertoo faroff.
From the datatable we can
see that this isthe only
function that
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does not vary
too wildly from
theactual value.
Proposed
modelThe quadraticregressionmodel is a
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reasonable fit
for the data
because theregression line
does not varyallthat much from
the actualvalues.As we
can see the
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quadratic
regression
model fits thegraph almost
perfectly. Theonly year that
thegraph doesnot fit all that
well is 1948.
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The data is
increasing for
the entiredomain but this
doesnotnecessarily
mean thatfuture years
will be
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accurate. This
is because
there arealmost
asymptotes inall of the
graphs. Therehas to be a limit
of how low the
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heights have to
be to qualify
and since theOlympiccompeti
tors are humanthere is a limit.
This makes allof the graphs
not a best fit
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for all of the
futureyears
that the eventwill be held and
all of the pastyears but this
quadratic modeldoes model the
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datagiven
accurately.Considerationof AccuracyLinear Equation:
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Years Initial
values Linear
EquationPercent
Error1932 197199.67
1.351936 203202.0033333
.491948 198
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209.0033333
5.551952 204
211.33666673.61956 212
213.67 .791960216
216.0033333.00151964 218
218.3366667
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.1541968 224
220.67
1.481972 223223.0033333
.00151976 225225.3366667
.151980 236227.67
3.53Average
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1.55Quadratic
Equation:
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X-Values Initialvalues
Quadratic
Equation Error
Percent1932
197 204
begin_of_the_s
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kype_highlighti
ng 1932 197
204end_of_the_sk
ype_highlighting.5789474
3.851936 203203 01948 198
204.1578947
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3.111952 204
206.5087719
1.231956 212209.8421053
1.021960 216214.1578947
.851964 218219.4561404
.671968 224
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225.7368421
.771972 223
233 01976 225241.245614
7.221980 236250.4736842
6.13Average2.26Linear
Regression:
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X-Values Initialvalues LinearRegression
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Error
Percent1932
197 194begin_of_the_s
kype_highlighting 1932 197
194end_of_the_sk
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ype_highlightin
g.1382426 1.451936 203 197begin_of_the_s
kype_highlighting 1936 203
197end_of_the_sk
ype_highlightin
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g.1585048
2.881948 198
206.21929144.151952 204
209.23955362.561956 212
212.2598158.1221960 216
215.280078
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.3331964 218
218.3003402
.141968 224221.3206024
1.21972 223224.3408646
.601976 225227.3611268
1.051980 236
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230.381389
2.38Averga
1.53QuadraticRegression:
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X-Values InitialvaluesQuadratic
Regression
Error
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Percent1932
197 197
begin_of_the_skype_highlighti
ng 1932 197197
end_of_the_skype_highlightin
g.9952965
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.501936 203
199.0379511
1.951948 198204.3348209
3.21952 204206.8234127
1.381956 212209.6734888
1.11960 216
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212.8850493
1.441964 218
216.458094.711968 224
220.3926231.611972 223
224.6886364.761976 225
229.346134
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1.931980 236
234.3651159
.69Average1.36From the
errorpercentages we
can see thatthe most
accurate is the
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quadratic
regression
model.Thequadratic
regressionmodel had an
average errorpercent of
about 1.36 while
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the others had
highererror.
The only modelthat came close
was the linearregression
model with anerror
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percentage of
1.53.Predictions
These
predictions aregoing to be with
the model thatI thought was
the best.
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Quadratic
Regressionmode
l. The equationis:
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1984
Predictions
2016PredictionsFirst
we mustsubstitute in
the year for xand thenwe can
find y which is
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0 1 9 3 0 1 9 4
0 1 9 5 0 1 9 6
0 1 9 7 0 1 9 8
0 1 9 9 0 2 0 0
0 2 0 1 0 2 0 2
0 2 0 3 0 He i g h t ( c m )Year
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Year Of
Olympics
VS. GoldMedalH
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eightsQuadraticRegression
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Height in cm =
239.75
295.8Table:X-ValuesInitialval
uesQuadraticRegression1932
197197.995296519
36 203
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199.0379511194
8 198
204.33482091952 204
206.82341271956 212
209.67348881960 216
212.885049319
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64 218
216.458094196
8 224220.392623197
2 223224.688636419
76 225229.346134198
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0 236
234.36511591984239.752016
295.8
Graph:0501001502002503001 8 8
0 1 9 0 0
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1 9 2 0 1
9 4 0 1 9
6 0 1 9 8
0 2 0 0 0
2 0 2 0 He i g h t ( c m )Year
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Year Of
Olympics
VS. GoldMedalH
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eightsInitialvaluesQuadraticRegressionMy answers for1984 make
sense becauseit is not too far
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from the other
values like that
of 1980. Ontheother hand
my value for2016 is high.
The mainproblem with
any of the
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models is that
they show
thatthe heightis continuously
rising. It is veryunlikely that
the resultswould keep
rising like the
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models
Isuggested.
Since 2016 farfrom the years
given we cansafely say that
neither of themodels
stated(linear or
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quadratic) are a
suitable
regression touse.Additional
Data
a)Year 1896
1904 1908 1912
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1920 1928 1984
1988 1992 1996
2000 20042008Height(cm
)190 180 191193 193 194
235 238 234239 235 236
236
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0501001502002
503001 8 8
0 1 9 0 0
1 9 2 0 1
9 4 0 1 9
6 0 1 9 80 2 0 0 0
2 0 2 0 He i g h t ( c m )
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YearYear Of
Olympics
VS. Gold
MedalH
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eightsInitialvaluesQuadraticRegressionb)As we can seethe model does
fit the newdata for the
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most part. The
only part that
we see notfittingthe data
is when thequadratic
regressionstarts
increasing from
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1984 and
beyond.Year
1896 1904 19081912 1920 1928
1932 1936 19481952
1956Height(cm)190 180 191 193
193 194 197
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203 198 204
212Year 1960
1964 1968 19721976 1980 1984
1988 1992 19962000Height(cm
)216 218 224223 225 236
235 238 234
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239 235Year
2004
2008Height(cm)236 236As we
can see fromthe graph the
quadraticregression
seems to follow
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the data
closely. There
are onlysomevalues at the
beginning and atthe end where
the quadraticregression is
different. From
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the tablebelow
we can see that
most of thevalues are
reasonablyclose to the
given actualvalue. A thing
thatcould be
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done to make
the data more
accurate ismake a new
quadraticregression line
for all ofthedata and not
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just the initial
data given.X-ValuesInitialval
uesQuadraticRegression1896
190204.878198190
4 180
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200.818274319
08 191
199.33053881912 193
198.20428771920 193
197.03623831928 194
197.3141261193
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2 197
197.995296519
36 203199.0379511194
8 198204.334820919
52 204206.823412719
56 212
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209.673488819
60 216
212.88504931964 218
216.4580941968 224
220.3926231972 223
224.688636419
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76 225
229.346134198
0 236234.365115919
84 235239.745582119
88 238245.487532719
92 234
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251.590967519
96 239
258.05588662000 235
264.882292004 236
272.07017772008 236
279.6195497
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Further testingand applicationThe patterns
that showed up
Jump also show
up in other
sports in the
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Olympics.
Forexample the
Olympicrecords for
free style
shows this samepattern.Year
1896 1904 1908
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1912 1920 1928
1932 1936 1948
1952 1956Time(seconds)82.2
62.8 65.6 63.461.4 58.6 58.2
57.6 57.3 57.455.4Year 1960
1964 1968 1972
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1976 1980 1984
1988 1992 1996
2000Time(seconds)55.2
53.4 52.2 51.249.9 50.4 49.8
48.6 49.0 48.748.3Year 2004
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2008Height(cm
)48.2 47.20102030405060
7080901 8
8 0 1 9 0
0 1 9 2 0
1 9 4 0 1
9 6 0 1 98 0 2 0
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0 0 2 0
2 0 He i g h t ( c m )YearYear Of
Olympics
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VS. Gold
Medal
HeightsInitial values
When the data
is graphed it
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looks like
this:The X
valuesrepresent the
years in whichthe Olympics
were hostedinThe Y values
represent the
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time of the gold
medalist
resultsThegraph shows a
decline in timesince the early
1900s. The gold
results were
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becomingshorte
r and shorter
over the yearuntil towards
the start of the21stcentury. Like on
the Gold
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medalistheights
this suggests a
type ofasymptote. If
we do aquadratic
regression forthis data we
get theequation
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f(x) =
.0017830124x^
2-7.173749547x+
7264.004005.When we graph
this equation weget
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thefollowing
graph.0102030405060
7080901 8
8 0 1 9
0 0 1 9
2 0 1 9
4 0 1 96 0 1 9
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8 0 2 0
0 0 2 0
2 0 He i g h t ( c m )Year
Year Of
Olympics
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VS. Time
of 100m
FreeStyleInitial
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X-Values Initial
values
QuadraticRegression1896
82.272.1763676119
04 62.868.9899481919
08 65.6
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67.4823230819
12 63.4
66.031754361920 61.4
63.301786121928 58.6
60.800043471932 58.2
59.6347567319
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36 57.6
58.5265264194
8 57.355.5441737719
52 57.454.6641690319
56 55.453.8412206719
60 55.2
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53.0753287219
64 53.4
52.366493161968 52.2
51.714714197251.2
51.119991241976 49.9
50.5823248719
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80 50.4
50.10171491984
49.849.6781613319
88 48.649.31166415199
2 4949.0022233719
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96 48.7
48.749838990102030405060
7080901 8
8 0 1 9
0 0 1 9
2 0 1 9
4 0 1 96 0 1 9
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8 0 2 0
0 0 2 0
2 0 He i g h t ( c m )Year
Year Of
Olympics
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VS. Time
of 100m
Freestyle2000 48.348.5545112004
48.2
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48.4162394120
08 47.2
48.33502422Compare and
contrast:
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0501001502002
503001 8
8 0 1 9
0 0 1 9
2 0 1 9
4 0 1 96 0 1 9
8 0 2 0
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0 0 2 0
2 0 He i g h t ( c m )YearYear Of
Olympics
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VS. Gold
Medal
HeightsFrom the twographs above
we can see that
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they are really
opposites of
each other. Thegold
medalheightsare a concave
up quadraticfunction while
the time for
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freestyle is
concavedown.The gold medal
heights graph isincreasing
throughout thedata and the
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freestyle
isdecreasing
throughout thedata. They are
both similar inthe way that
accurately
model
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everysingle
year that the
Olympics will beheld because
they will bothdecrease and
increase beyondtheactual
values. This is
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because humans
are competing
in these eventsand we all have
limits. This isalsoapparent in
the gold medalheights, as the
years went on
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the heights got
more and more
close toeachother
withoutsignificant
difference. The
freestyle seems
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to fit the data
better
becausetowardsrecent years
the times havebeen very close.
Over all we cansee that the
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freestylequadra
tic regression
model fits itsdata better
than the goldmedal heights.Conclusion
a)
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jump results
from 1896 to2008 Olympics
showed thatthe gold
medalistresultsfor high
jump steadily
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increased.
Towards the
end the heightsstarted to level
off because ofhuman limits.
The best modelthat was found
to model the
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data was
quadratic. This
is becausefrom896 towards
2008 the datawas steadily
increasing at aparabola like
shape. The
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m
Freestyle
results from1896 to 2008
showed thatthe results for
the 100 mfreestyle
weresteadily
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decreasing, and
towards the
end the resultsleveled off.
This is becauseof
humanlimitations. A good type
of model to
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model the 100m
freestyle data
from 1896 to2008 was alsoa
quadraticfunction. This
modeled thedata almost
perfectly
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BibliographySwim-City.
"Swim-City.com
- Record
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History Olympic
Records Men."Swim-City.com -SwimmingMetro
polis. Swim-City,
2011. Web. 20Jan. 2012.
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