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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 w hich athletes participate in different competitions. In ancient Greece the Olympic G ames were athletic competitions held in honor of Zeus. Since the Olympic Games began they have been the competition grounds for the worlds greatest athletes. First place obtaining g old; 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 w inners 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 C ountries
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Country The United States Soviet Union Great Britain France Germany Italy Sweden East Germany Hungary Finland
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 Height (cm)
1932 197
1936 203
1948 198
1952 204
1956 212
1960 216
1964 218
1968 224
1972 223
1976 225
1980 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 pa ttern the pattern would become more apparent. On top of this there is a gap between 1936 and 1948 whic h 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 Gam es 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 d ata. 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)
Year Of Olympics VS. Gold Medal
Heights
240 235 230 ) 225 m c ( 220 t h g 215 i e H
Initial values
210 205 200 195 1920
1930
1940
1950
1960
1970
1980
1990
Year
Analysis and Model Construction 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 didnt it looks like a third degree function would fit well.
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 dont 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 wasnt 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 General form for a linear equation:
When 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.
To find the value of a, we can subtract the two equations. This will essentially cancel out b.
We can now find m by dividing both sides by 12 so we get:
The value for b can now be found by substituting to any of the equations
Quadratic Model The general form for an quadratic equation is . With 3 points on the graph an equation can be formulated in the form . 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 . by substituting x and y values we get: or
The equation will now be:
Now I will subtract the second and third equation to get two new equations to solve for a and b.
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 By since . By multiplyin by we get . then we add the equations:
Now we can solve for a and we get . Now we can substitute a into one of the equations with two variables to solve for . I will be using the equation .by substituting a we get:
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.
Now we have found all variables and we can write our equation. And we get:
The reason why I chose a linear model is because despite some points the data i n 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:
Year Of Olympics VS. Gold Medal Heights 240 235 230 ) 225 m c ( 220 t h g 215 i e H
Initial values
210
Linear Equation
205 200 195 1920
1930
1940
1950
1960
1970
1980
1990
Year
Initial values
Years
Linear Equation
1932
197
199.67
1936
203
202.0033333
1948
198
209.0033333
1952
204
211.3366667
1956
212
213.67
1960
216
216.0033333
1964
218
218.3366667
1968
224
220.67
1972
223
223.0033333
1976
225
225.3366667
1980
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.
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 quick er than the actual values. This causes the data to
1990
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.
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.
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.
Year Of Olympics VS. Gold Medal
Heights
240 235 230 225
) m c ( 220 t h g i 215 e H
Initial values
210
Quadratic Regression
205 200 195 1920
1930
1940
1950
1960
1970
1980
1990
Year
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 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.
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
Year Of Olympics VS. Gold Medal Heights 350 300 250 ) m 200 c ( t h g i e 150
Quadratic Regression
H
100 50 0 1920
1930
1940
1950
1960
1970
1980
Year
1990
2000
2010
2020
2030
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
Heig ht (cm)
190
180
191
193
193
194
235
238
234
239
235
236
236
Year Of Olympics VS. Gold Medal
Heights
300 250 ) 200 m c ( t h 150 g i e H
Initial values
100
Quadratic Regression
50 0 1880
1900
1920
1940
1960
Year
1980
2000
2020
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 Height (cm) Year Height (cm) Year
1896 190
1904 180
1908 191
1912 193
1920 193
1928 194
1932 197
1936 203
1948 198
1952 204
1956 212
1960 216
1964 218
1968 224
1972 223
1976 225
1980 236
1984 235
1988 238
1992 234
1996 239
2000 235
2004
2008
Year Of Olympics VS. Gold Medal Heights 300 250 ) 200 m c ( t h 150 g i e H
Initial values
100
Quadratic Regression
50 0 1880
1900
1920
1940
1960
1980
2000
2020
Year
Height (cm)
236
236
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 reg ression 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.
Initial values
X-Values
Quadratic Regression
1896 1904
190 180
204.878198 200.8182743
1908 1912
191 193
199.3305388 198.2042877
1920
193
197.0362383
1928
194
197.3141261
1932
197
197.9952965
1936
203
199.0379511
1948
198
204.3348209
1952
204
206.8234127
1956
212
209.6734888
1960
216
212.8850493
1964
218
216.458094
1968
224
220.392623
1972
223
224.6886364
1976
225
229.346134
1980
236
234.3651159
1984
235
239.7455821
1988
238
245.4875327
1992
234
251.5909675
1996 2000
239 235
258.0558866 264.88229
2004 2008
236 236
272.0701777 279.6195497
Further
testing and application
The patterns that showed up in Mens High Jump also show up in other sports in the Olympics. For example the Olympic records for mens 100 m free style shows this same pattern.
Year Time ( seconds) Year Time ( seconds) Year Height (cm)
1896 82.2
1904 62.8
1908 65.6
1912 63.4
1920 61.4
1928 58.6
1932 58.2
1936 57.6
1948 57.3
1952 1956 57.4 55.4
1960 55.2
1964 53.4
1968 52.2
1972 51.2
1976 49.9
1980 50.4
1984 49.8
1988 48.6
1992 49.0
1996 2000 48.7 48.3
2004 48.2
2008 47.2
When the data is graphed it looks like this:
Year Of Olympics VS. Gold Medal Heights 90 80 70 ) 60 m c ( 50 t h g i 40 e H
Initial values
30 20 10 0 1880
1900
1920
1940
1960
1980
2000
2020
Year
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 tim e since the early 1900s. The gold medalists results were becoming st
shorter and shorter over the year until towards the start of the 21 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.
Year Of Olympics VS. Time of 100m FreeStyle 90 80 70 60
Year Of Olympics VS. Time of 100m Freestyle 90 80 70 ) 60 m c ( 50 t h g i 40 e H
30 20 10 0 1880
1900
1920
1940
1960
Year
1980
2000
2020
Year Of Olympics VS. Gold Medal Heights 300 250 ) 200 m c ( t h 150 g i e H
100 50 0 1880
1900
1920
1940
1960
1980
2000
Year
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 mens freestyle is concave down. The gold medal heights graph is increasing throughout the data and the 100 m mens freestyle is decreasing throughout the data. They are both similar in the way that they dont 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 l imits. 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 mens freestyle seems to fi t the data better because towards recent years the times have been very close. Over all we can see that the 100m mens freestyle quadratic regression model fits its data better than the gold medal heights.
Conclusion a) The Mens 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 mens 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 w as also a quadratic function. This modeled the data almost pe rfectly
2020
Bibliography
Swim-City. "Swim-City.com - Record History Olympic Records Men." Swim-City.com - Swimming Metropolis. Swim-City, 2011. Web. 20 Jan. 2012.
city.com/recordhistorie.php3?record=orh>.
