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Gold Metal Heights in High Jump.

Extracts from this document...

Introduction

 Hannah Krohn

Gold Metal Heights

Aim: The aim of this task is to consider the winning height for the men’s high jump in the Olympic games.

The graph below gives the Winning height (in centimeters) by the gold medalists at various Olympic games form 1932 to 1980 with the exception of 1940 and 1944.

Constraints to this graph: Since there were no Olympic games in 1940 and 1944 the slope is from 1936 to 1948 is not defined on the same domain as the other data points. This is limiting because the amount of data points vs. time is not consistent throughout the whole graph so it will be more difficult to create an accurate model.

The function that is modeled after the data points and graph is a linear graph seen below:

yimage00.png

The following variables on the next page are the variables that are used in the linear equation. To achieve these numbers plug in the data in the form of linear regression, where the height (y-value) is the dependent variable while the year (x-value) an explanatory value or independent.

...read more.

Middle

Using technology (Microsoft Word>charts) I was able to increase the order until the image01.png

 was extremely close to 1 in value, therefore the goodness of fit is as good as it could be for a model.

This is the model that occurred after trying to find the best fit. This equation had the closest value to r² out of all the order possibilities.

image08.png

image03.png

 0.99099

The lower r2 value for the linear regression indicates that the polynomial regression is a better fit to the data.

Here is the graph that shows that the polynomial better is a better fid for the data then the linear model.

Polynomial Model

Linear Model

image09.png

image10.png

image03.png

 0.99099

image05.png

image04.png

Estimating the winning heights of 1940 and 1944 would be done in the following way:

Look at the Polynomial Model since it has a better goodness of fit.

...read more.

Conclusion

The following table gives height for all the Olympic Games since 1896 to 2008

image16.png

image03.png

 0.97663

Since the original polynomial model changed with the additional data, it did not fit the datat well, but with the new data the r² is less than the original polynomial graph which had a r² of r² image03.png

 0.99099. Still better than the linear graph which had a r² image03.png

image17.png

Here is the data in a table to compare graphs

1st Polynomial Model

2nd Polynomial Model

Linear Model

image18.png

image19.png

image20.png

image21.png

image03.png

 0.99099

image22.png

image23.png

image24.png

image25.png

image03.png

 0.97663

image05.png

image04.png

The trend from 1896 to 2008 is relatively stable slope until 1952 where the slope increases in a (if it was a double derivative) concave down manner until 1996, where it levels off.

Modifications to the data would be to interpolate the data, which is the process one uses to determine a value on the line of best fit within the cluster of scatter plot data.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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