Gold Medal Heights-SL Type 2

IB Pre-Calculus SL

Guillermo Esqueda Silva

1/30/2012

Introduction

a) The Olympic Games is an international event featuring summer and winter sports, in which athletes participate in different competitions. In ancient Greece the Olympic Games were athletic competitions held in honor of Zeus. Since the Olympic Games began they have been the competition grounds for the world’s greatest athletes. First place obtaining gold; second silver and third bronze. The Olympic medals represent the hardship of what the competitors of the Olympics have done in order to obtain the medal. On one side the Olympic medal has Nike the goddess of victory holding a palm and a winners crown and on the other side the medal has a different label for each Olympiad reflecting the host of the games. Olympic medals could be used as a unit of measure of athleticism.

Top 10 Olympic Medal- winning Countries

Country Medals won1. The United

States2404

2. Soviet Union 12043. Great Britain 6894. France 6795. Germany 6486. Italy 5957. Sweden 5888. East Germany 5199. Hungary 45410. Finland 446

This is a table showing the top 10 Olympic Medal- Winning Countries and definitely shows how the Olympics can be seen as a standardized unit of athleticism

Data

a)

Height (in centimeters) achieved by the gold medalist at various Olympic games.

Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980Height(cm)

197 203 198 204 212 216 218 224 223 225 236

Variables and Constraints

a) The dependent variable for this data set is the Olympic Gold Medalist Heights. The independent variable for this data set is the years in which the summer Olympic Games occurred in. A constraint of this data set is the limited amount of data that is available. The data available is only between 1932 and

1980. If there were more data and if there is a pattern the pattern would become more apparent. On top of this there is a gap between 1936 and 1948 which is a 12 year chunk of data missing. And since the Olympics are held every 4 years that is 3 Olympic competitions missing from the data.

b) In a math textbook the variables and constraints could be seen as y= Olympic Gold Medalist Heights and x= years in which the summer Olympic games occurred in.

c) In the context of this problem is that the x axis would be used to show the Year of the Olympic Games and y would be used to show the height of the gold medalist.

d) This data set is continuous because it is associated with a measurement and its possible to have the same y value for different x values. And since the data is measuring height a decimal answer is possible. A function that would fit most of the data would be a quadratic function. The constraint that there is a 12 year gap between 1936 and 1948 could skew the data. The data that we are missing could of showed us a much clearer model like a linear or could of reassured us of a quadratic model.

Graph of Initial Data

a)

Analysis and Model Construction

1920 1930 1940 1950 1960 1970 1980 1990170

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Year Of Olympics VS. Gold Medal Heights

Initial values

Year

He

igh

t (

cm)

Based on the data the type of curve that might be expected is quadratic and maybe even a third degree function. The expected shape would be quadratic if we disregarded the 1936 value and if we didn’t it looks like a third degree function would fit well.

a) Some general formulas that the data could fit can be quadraticf ( x )=a x2+bx+c f(x), linear

f ( x )=mx+b, or exponentialf (x)=abx.Quadratic possible values f (x)=0.011296384 x2−43.433751 x+41946.847:

X Y1932 197.991936 199.041948 204.331952 206.821956 209.671960 212.881964 216.461968 220.391972 224.691976 229.341980 234.36

Linear Possible values f (x)=0.75506555 x−1264.6484 :

X Y1932 194.141936 197.161948 206.221952 209.241956 212.261960 215.281964 218.31968 221.321972 224.341976 227.361980 230.38

Exponential Possible Values f (x)=0.21211804∗1.0035374x :

X Y1932 194.721936 197.491948 206.041952 208.971956 211.951960 214.961964 218.021968 221.121972 224.271976 227.461980 230.69

All of the formulas used to fit the data are increasing and all have a min ( when x=0) for the quadratic fit the minimum is 41946.847 of the linear fit the minimum is -1264.6484 and for the exponential fit the min is 0.21211804. All of the formulas don’t have a maximum.

For the model to become much more realistic the logical fit would be a sinusoidal curve. The constraints on a sinusoidal curve would be that the curve would have to be half a cycle. The curve has to be half a cycle because it wasn’t the curve would show that the heights fluctuate from Olympics to Olympics and this would not represent our data. But by limiting the curve to half a cycle then the curve would fit the data.

b)

Linear Model Quadratic Modela) General form for a linear equation: f (x)=mx+bWhen using two points from the data and substituting them into the values for x and y the values for m and b can be found. The points that are going to be used are going to be (1972,223) and (1960,216). These points were chosen because they look like a nice line can be drawn between them without too much differentiation between the other points.

223=1972m+b216=1960m+b

To find the value of a, we can subtract the two equations. This will essentially “cancel” out b.

223=1972m+b −(216=1960m+b)

The general form for an quadratic equation is f (x)=a x2+bx+c. With 3 points on the graph an equation can be formulated in the form (x )=a x2+bx+c . The points that are going to be used are going to be (1936,203),(1972,233) and (1960,216)We can make three equations using these three points and the general formula f (x)=a x2+bx+c. by substituting x and y values we get:203=a (1936 )2+b (1936)+c or 203=3748096a+1936b+c

233=a (1972 )2+b(1972)+c or 233=3888784a+1972b+c

216=a (1960 )2+b (1960)+cor

7=12mWe can now find m by dividing both sides

by 12 so we get: M=7 /12

The value for b can now be found by substituting m=7 /12 to any of the equations

223=1972(7/12)+b B=−927.33The equation will now be:f (x)=(7 /12) x−927.33

216=3841660a+1960b+c

Then we can subtract two equations together to eliminate C. I chose to subtract the first and second equation

203=3748096a+1936b+c−233=−3888784a−1972b−c−30=−140688a−36b

Now I will subtract the second and third equation to get two new equations to solve for a and b.

233=3888784a+1972b+c−216=−3841660a−1960b+c17=47124 a+12b

With the two equations now we must isolate a variable to solve, I will be chosing a, so we need to eliminate b. we can do this by multiplying 17=47124 a+12bBy 3since 12∗3=36. By multiplyin by 3we get 51=141372a+36 b. then we add the equations:

−30=−140688a−36b+51=141372a+36b21=684a

Now we can solve for a and we get a=21/684. Now we can substitute a into one of the equations with two variables to solve for b. I will be using the equation −30=−140688a−36b.by substituting a we get:

−30=−140688( 21684 )−36b

−30+( 2954448684 )=−36 b

(−20520684 )+( 2954448684 )=−36b

( 2933928684 )=−36b

b=(−293392824624 )we can now substitute both a and b into the original equations and solve for c. I will be substituting in into the second equation.

233=3888784( 21684 )+1972(−293392824624 )+c

233=( 81664464684 )−(578570601624624 )+c

233=( 293992070424624 )−(578570601624624 )+c

233=−( 284578531224624 )+cc=(2851522704 /24624)

Now we have found all variables and we can write our equation. And we get:

f ( x )=( 21684 )x2−(293392824624 )x+( 285152270424624 )

Linear Function Quadratic functionf (x)=(7 /12) x−927.33

f ( x )=( 21684 )x2−(293392824624 )x+( 285152270424624 )

The reason why I chose a linear model is because despite some points the data in the graph seemed to be modeled well by a linear model. I also chose a quadratic model because the data points seemed to be modeled well by a quadratic model.

b)

Linear Equation:f (x)=(7 /12) x−927.33

YearsInitial values Linear Equation

1932 197 199.671936 203 202.00333331948 198 209.00333331952 204 211.33666671956 212 213.671960 216 216.00333331964 218 218.33666671968 224 220.671972 223 223.00333331976 225 225.33666671980 236 227.67

This linear function is not the best fit for the model because as we can see from the graph there are a lot of points left out and that are not even close to the linear equation. Also in the data table there are some years where the linear equation varies greatly from the actual values “initial value”. We can see that for some years it is close like for 1936 but for others it is much farther away from the linear equation like in 1948.

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Year Of Olympics VS. Gold Medal Heights

Initial valuesLinear Equation

Year

Heig

ht (

cm)

Quadratic Equation:f ( x )=( 21684 )x2−(293392824624 )x+( 285152270424624 )

Years Initial values Quadratic Equation1932 197 204.57894741936 203 2031948 198 204.15789471952 204 206.50877191956 212 209.84210531960 216 214.15789471964 218 219.45614041968 224 225.73684211972 223 2331976 225 241.2456141980 236 250.4736842

1920 1930 1940 1950 1960 1970 1980 19900

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Year Of Olympics VS. Gold Medal Heights

Year

Heig

ht (

cm)

The quadratic equation that I came up with is a reasonable fit but still not the best fit. As we can see from the graph the quadratic equation increases quicker than the actual values. This causes the data to be way off for the final two years in our data. We can see this from the data table as 1976 and 1980 are not as close are the other actual values to the values from the quadratic equation.

Linear Regression f (x)=0.75506555 x−1264.6484 :

Years Initial values Linear Regression1932 197 194.13824261936 203 197.15850481948 198 206.21929141952 204 209.23955361956 212 212.2598158

1920 1930 1940 1950 1960 1970 1980 19900

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Year Of Olympics VS. Gold Medal Heights

Initial valuesLinear Regression

Year

Heig

ht (

cm)

1960 216 215.2800781964 218 218.30034021968 224 221.32060241972 223 224.34086461976 225 227.36112681980 236 230.381389

This is a better linear model but it is still not the best. Because it is linear it excludes some points like 1948. In 1948 we can see that difference between the actual values and the linear regression models is quite significant.

Quadratic Regression f ( x )= (0.0112963843 ) x2−(43.43375081)x+41946.84671

Years Initial values Quadratic Regression1932 197 197.99529651936 203 199.03795111948 198 204.33482091952 204 206.8234127

1920 1930 1940 1950 1960 1970 1980 1990170

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Year Of Olympics VS. Gold Medal Heights

Initial valuesQuadratic Regression

Year

Heig

ht (

cm)

1956 212 209.67348881960 216 212.88504931964 218 216.4580941968 224 220.3926231972 223 224.68863641976 225 229.3461341980 236 234.3651159

This is the best fit because as we can see from the graph it fits most points without leaving other too far off. From the data table we can see that this is the only function that does not vary too wildly from the actual value.

Proposed model

The quadratic regression model is a reasonable fit for the data because the regression line does not vary all that much from the actual values.

As we can see the quadratic regression model fits the graph almost perfectly. The only year that the graph does not fit all that well is 1948. The data is increasing for the entire domain but this does not necessarily mean that future years will be accurate. This is because there are almost asymptotes in all of

1920 1930 1940 1950 1960 1970 1980 1990170

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Year Of Olympics VS. Gold Medal Heights

Initial valuesQuadratic Regression

Year

Heig

ht (

cm)

the graphs. There has to be a limit of how low the heights have to be to qualify and since the Olympic competitors are human there is a limit. This makes all of the graphs not a best fit for all of the future years that the event will be held and all of the past years but this quadratic model does model the data given accurately.

Consideration of Accuracy

Linear Equation:f (x)=(7 /12) x−927.33

Years Initial values Linear Equation Percent Error1932 197 199.67 1.351936 203 202.0033333 .491948 198 209.0033333 5.551952 204 211.3366667 3.61956 212 213.67 .791960 216 216.0033333 .00151964 218 218.3366667 .1541968 224 220.67 1.481972 223 223.0033333 .00151976 225 225.3366667 .151980 236 227.67 3.53

Average 1.55

Quadratic Equation:f ( x )=( 21684 )x2−(293392824624 )x+( 285152270424624 )X-Values Initial values Quadratic Equation Error Percent

1932 197 204.5789474 3.851936 203 203 01948 198 204.1578947 3.111952 204 206.5087719 1.231956 212 209.8421053 1.021960 216 214.1578947 .851964 218 219.4561404 .671968 224 225.7368421 .771972 223 233 01976 225 241.245614 7.221980 236 250.4736842 6.13

Average 2.26

Linear Regression:f (x)=0.75506555 x−1264.6484

X-Values Initial values Linear Regression Error Percent1932 197 194.1382426 1.451936 203 197.1585048 2.881948 198 206.2192914 4.151952 204 209.2395536 2.561956 212 212.2598158 .1221960 216 215.280078 .3331964 218 218.3003402 .141968 224 221.3206024 1.21972 223 224.3408646 .601976 225 227.3611268 1.051980 236 230.381389 2.38

Averga 1.53

Quadratic Regression: f ( x )= (0.0112963843 ) x2−(43.43375081)x+41946.84671

X-Values Initial values Quadratic Regression Error Percent1932 197 197.9952965 .501936 203 199.0379511 1.951948 198 204.3348209 3.21952 204 206.8234127 1.381956 212 209.6734888 1.11960 216 212.8850493 1.441964 218 216.458094 .711968 224 220.392623 1.611972 223 224.6886364 .761976 225 229.346134 1.931980 236 234.3651159 .69

Average 1.36

From the error percentages we can see that the most accurate is the quadratic regression model. The quadratic regression model had an average error percent of about 1.36 while the others had higher error. The only model that came close was the linear regression model with an error percentage of 1.53.

Predictions

These predictions are going to be with the model that I thought was the best. Quadratic Regression model. The equation is: f ( x )= (0.0112963843 ) x2−(43.43375081)x+41946.84671

1984 Predictions 2016 PredictionsFirst we must substitute in the year for x and then we can find y which is the height.

First we must substitute in the year for x and then we can find y which is the height.

f (x)=(0.0112963843 )∗(1984 )2−(43.43375081)∗(1984 )+41946.84671f (x)=(0.0112963843)∗(2016 )2−(43.43375081)∗(2016)+41946.84671Height in cm = 239.75 295.8

Table:

X-ValuesInitial values

Quadratic Regression

1932 197 197.99529651936 203 199.03795111948 198 204.33482091952 204 206.82341271956 212 209.67348881960 216 212.88504931964 218 216.4580941968 224 220.3926231972 223 224.68863641976 225 229.3461341980 236 234.36511591984 239.752016 295.8

Graph:

My answers for 1984 make sense because it is not too far from the other values like that of 1980. On the other hand my value for 2016 is high. The main problem with any of the models is that they show that the height is continuously rising. It is very unlikely that the results would keep rising like the models I suggested. Since 2016 far from the years given we can safely say that neither of the models stated (linear or quadratic) are a suitable regression to use.

Additional Data

a)

Year 1896 1904 1908 1912 1920 1928 1984 1988 1992 1996 2000 2004 2008

Height

190 180 191 193 193 194 235 238 234 239 235 236 236

1920 1940 1960 1980 2000 2020 20400

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Year Of Olympics VS. Gold Medal Heights

Quadratic Regression

Year

Heig

ht (

cm)

(cm)

b)

As we can see the model does fit the new data for the most part. The only part that we see not fitting the data is when the quadratic regression starts increasing from 1984 and beyond.

Year 1896 1904 1908 1912 1920 1928 1932 1936 1948 1952 1956Height (cm)

190 180 191 193 193 194 197 203 198 204 212

Year 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000Height (cm)

216 218 224 223 225 236 235 238 234 239 235

Year 2004 2008Height (cm)

236 236

1880 1900 1920 1940 1960 1980 2000 20200

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Year Of Olympics VS. Gold Medal Heights

Initial valuesQuadratic Regression

Year

Heig

ht (

cm)

As we can see from the graph the quadratic regression seems to follow the data closely. There are only some values at the beginning and at the end where the quadratic regression is different. From the table below we can see that most of the values are reasonably close to the given actual value. A thing that could be done to make the data more accurate is make a new quadratic regression line for all of the data and not just the initial data given.

X-ValuesInitial values

Quadratic Regression

1896 190 204.8781981904 180 200.81827431908 191 199.33053881912 193 198.20428771920 193 197.03623831928 194 197.31412611932 197 197.99529651936 203 199.03795111948 198 204.33482091952 204 206.82341271956 212 209.67348881960 216 212.88504931964 218 216.4580941968 224 220.3926231972 223 224.68863641976 225 229.346134

1880 1900 1920 1940 1960 1980 2000 20200

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300

Year Of Olympics VS. Gold Medal Heights

Initial valuesQuadratic Regression

Year

Heig

ht (

cm)

1980 236 234.36511591984 235 239.74558211988 238 245.48753271992 234 251.59096751996 239 258.05588662000 235 264.882292004 236 272.07017772008 236 279.6195497

Further testing and application

The patterns that showed up in Men’s High Jump also show up in other sports in the Olympics. For example the Olympic records for men’s 100 m free style shows this same pattern.

Year 1896 1904 1908 1912 1920 1928 1932 1936 1948 1952 1956Time ( seconds)

82.2 62.8 65.6 63.4 61.4 58.6 58.2 57.6 57.3 57.4 55.4

Year 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000Time ( seconds)

55.2 53.4 52.2 51.2 49.9 50.4 49.8 48.6 49.0 48.7 48.3

Year 2004 2008Height (cm)

48.2 47.2

When the data is graphed it looks like this:

The X values represent the years in which the Olympics were hosted in

The Y values represent the time of the gold medalist results

The graph shows a decline in time since the early 1900s. The gold medalist’s results were becoming shorter and shorter over the year until towards the start of the 21st century. Like on the Gold medalist heights this suggests a type of asymptote. If we do a quadratic regression for this data we get the equation f(x) = .0017830124x^2-7.173749547x+7264.004005. When we graph this equation we get the following graph.

X-Values Initial values Quadratic Regression1896 82.2 72.176367611904 62.8 68.989948191908 65.6 67.482323081912 63.4 66.031754361920 61.4 63.301786121928 58.6 60.800043471932 58.2 59.634756731936 57.6 58.52652641948 57.3 55.544173771952 57.4 54.664169031956 55.4 53.841220671960 55.2 53.07532872

1880 1900 1920 1940 1960 1980 2000 20200

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Year Of Olympics VS. Gold Medal Heights

Initial values

Year

Heig

ht (

cm)

1964 53.4 52.366493161968 52.2 51.7147141972 51.2 51.119991241976 49.9 50.582324871980 50.4 50.10171491984 49.8 49.678161331988 48.6 49.311664151992 49 49.002223371996 48.7 48.749838992000 48.3 48.5545112004 48.2 48.416239412008 47.2 48.33502422

Compare and contrast:

1880 1900 1920 1940 1960 1980 2000 20200

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Year Of Olympics VS. Time of 100m FreeStyle

Initial values

Year

Heig

ht (

cm)

1880 1900 1920 1940 1960 1980 2000 20200

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Year Of Olympics VS. Time of 100m Freestyle

Year

Heig

ht (

cm)

From the two graphs above we can see that they are really opposites of each other. The gold medal heights are a concave up quadratic function while the time for the 100m men’s freestyle is concave down. The gold medal heights graph is increasing throughout the data and the 100 m men’s freestyle is decreasing throughout the data. They are both similar in the way that they don’t accurately model every single year that the Olympics will be held because they will both decrease and increase beyond the actual values. This is because humans are competing in these events and we all have limits. This is also apparent in the gold medal heights, as the years went on the heights got more and more close to each other without significant difference. The 100m men’s freestyle seems to fit the data better because towards recent years the times have been very close. Over all we can see that the 100m men’s freestyle quadratic regression model fits its data better than the gold medal heights.

Conclusion

a) The Men’s High jump results from 1896 to 2008 Olympics showed that the gold medalist results for high jump steadily increased. Towards the end the heights started to level off because of human limits. The best model that was found to model the data was quadratic. This is because from 896 towards 2008 the data was steadily increasing at a parabola like shape. The men’s 100m Freestyle results from 1896 to 2008 showed that the results for the 100 m freestyle were steadily decreasing, and towards the end the results leveled off. This is because of human limitations. A good type of model to model the 100m freestyle data from 1896 to 2008 was also a quadratic function. This modeled the data almost perfectly

1880 1900 1920 1940 1960 1980 2000 20200

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Year Of Olympics VS. Gold Medal Heights

Year

Heig

ht (

cm)

Bibliography

Swim-City. "Swim-City.com - Record History Olympic Records Men." Swim-City.com - Swimming Metropolis. Swim-City, 2011. Web. 20 Jan. 2012. <http://www.swim-city.com/recordhistorie.php3?record=orh>.