Gold Medal Height

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Ben Cosh
George Reuter
IB Math
20 January 2013
Gold Medal Heights

 
Introduction:
a) The Olympic Games is an international event featuring summer and winter sports, in which athletes participate in different competitions. 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. Olympic medals could be used as a unit of measure of athleticism.

Data Provided:
[pic]
Height (in centimeters) achieved by the gold medalist at various Olympic games.

The following is a graphic representation of the data given:

Given Data

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.

b) There were no Olympic Games in 1940 and1944 due to the war. This meant that the data is not consistent as we are missing 2 years of Gold Medalist heights. The graph also only shows the heights of the 1932 to 1980 Olympic Games which is a small section of the overall Olympic Games data, beginning in 1986.

c) In the context of this problem the x axis would be used to show the Year of the Olympic Games and y axis 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 a12 year gap between 1936 and 1948 could skew the data. The data that we are missing could show us a much clearer model.

e) The constraints of this task are that, when performing a regression analysis on the data, it would be difficult to find the exact equation that models the data perfectly as this data does have some outliers and does not completely follow a pattern. The data does not follow an absolute pattern such as the points in the equation y=x^2 and so it would be difficult to determine an exact value of future heights based on a regression equation modeled by the data

Developing a Function:
When examining the data which is given, a perfect function is not evident. In my opinion, when looking at the data, a linear function would fit the data well. The following work was done in order to develop a linear function for the data:

The following is the standard linear function:

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 (1972, 223) and (1960, 216). These points were chosen because there seems to be a nice line 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 the b.

Now we can divide both sides by 12 in order to get “m” by itself

To find “b”, we can substitute the value of “m” found into the equation.

The final equation found is:

The following is a graphic comparison of the linear function found when compared to the given data

The following is a table which represents the values of the given data comparing to the values of the linear function.

[pic]

This linear function is not the best fit for the model because as we can see from the graph there are many points left out and that are not close to the linear equation. Also in the data...
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