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Using
Statistical Regression to Investigate Track and Field Records |
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Part
1: The Women's 800-meter Run (1950-1985)
1. Below is a table showing the world record for the women’s 800-meter run at different points in history.
2. Enter the data values from the table into lists on your calculator. 3. Using the calculator's statistical plotting features graph the data. You may need to set an appropriate viewing window for your calculator. 4.
What characteristics does the scatterplot have?
Does a linear function ‘fit’ the data points? 5.
Find the linear regression equation for the data
points. 6.
Graph the regression equation on the same axis as the scatterplot. 7. Does this regression equation ‘fit’ the data well? Are all the points close to the regression line? 8. Use the trace feature to find the regression equation’s estimate of the world record in 1977. Is this estimate close to the actual world record of 117.5 seconds set by Svetla Zlateva on 8/24/1973? 9.
What would be a good estimate, based on the regression equation,
for the world record next year? 10. What would be the world record in the year 2224 based on the regression equation? Does this make sense? Why? 11. Add the following data to your lists. Once added, sort your data so that the years are in ascending order. Make sure that the times are also sorted to retain the proper (year, time) relationship.
12. Recalculate the linear regression equation for this amended set of data. Graph the new regression equation. Does the regression equation fit this set of data points well? 13. Find the quartic regression equation for this set of data. Graph the quartic regression equation on the same axis as the linear regression equation. 14. Does the quartic regression equation model the data better than the linear regression equation? Justify your answer. 15. Are there years where the linear regression equation approximates the data better than the quartic regression equation? When? 16.
If you were to pick one of these two equations to predict the world
record time in the year 2002, which one would you choose?
Justify your answer. 17.
Find the estimated world record time in the year 2002 for each
regression equation. Do both
estimates make sense? Explain
your answer. 18.
Zoom out one time and look at the equations.
Would either equation work well for estimating world record times
in the distant future? Why?
Part 2: Beating the Calculator 19. Below are some general forms of different classes of functions. Which class of functions could be transformed to most resembles the world record data? Why?
20.
Using your response
from task 19, build an equation to model the data. Utilize shifting, scaling, and other techniques of
transforming equations. 21. Graph your model along with the quartic regression equation and the linear regression equation. Describe the strengths and weaknesses of your model compared to the other two models. Comment on the appropriateness of this model. 22. What happens to your model when estimating world records in the distant future? Does your model estimate future world records that are physically possible? 23. Graph the equation y = -5ln(x-1940) on the same axis as the other regression equations. How does this equation compare to the other three regression equations? What are some strengths and weaknesses of this equation? 24. What happens to this model when estimating world records in the distant future? Does this model estimate future world records that are physically possible? Comment on the appropriateness of this model. 25. Below is a table showing the world records for the men’s pole vault at different points in history. Place the table in your calculator.
26. Generate a scatterplot from the data. What are some characteristics of the scatterplot? 27. Using the scatterplot predict the world record in the year 1956. How did you arrive at that prediction? 28. Find a regression equation that models this data. You can use an equation from the calculator or construct one of your own. Justify why you chose this equation. 29. Use the equation from task 28 to estimate the world record in 1956. Does this estimation closely match your prediction from task 27? Which one do you believe is more reliable? Why? 30. In 1956 the world record was still 477cm. How would your current scatterplot look if it were extended to the year 1956? Why? 31. In 1956 Cornelius Warmerdam still held the world record. In 1957 Robert Gutowski broke the world record by vaulting 478cm. Which person was the better pole-vaulter? Why? Look at all the information concerning Warmerdam and Gutowski. Does any of this information change your mind about which person may have been the better pole-vaulter? Why? 33. Add the following data to your lists. Once added, sort your data so that the years are in ascending order. Make sure that the heights are also sorted to retain the proper (year, height) relationship. .
34.
Look at the amended scatterplot. Describe some the characteristics of this scatterplot. 35.
Return to the website
listed above. If you were
going to construct an equation to predict future world record, would you
include all the data points included in this website?
Which data points would you choose?
Justify your answer. 36.
Restrict your
scatterplot to only those data points you chose in task 35.
Construct a regression equation that fits this data set.
You can use a calculator regression equation or construct one of
your own. Why did you choose this equation? What are some strengths and weaknesses of this equation? 37. Using your equation from task 36, predict the world record next year. Predict the world record 10 years from now. Do these predictions make sense physically? Interpreting
the Spread of AIDS in the US |
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