2. A few years ago some Exercise Physiologists at UCLA published a paper in NATURE wherein they predicted that by the year 2004, the women's world record in the marathon would be faster than the men's record. The mechanism for the improvement in performance is thought to be the improvement of training methods and the expansion of the talent pool. But the data was examined only to describe the trend, not to explain it.
This problem examines the winning Olympic times for the 100 m races for both Men and Women. As the years have gone by, the times have improved for both Men and Women. Below we present a table with the data for the winning times (in seconds)
|
Year |
Men's 100 |
time |
Women's 100 |
time |
|
1896 |
Burke |
12.0 |
||
|
1900 |
Jarvis |
11.0 |
||
|
1904 |
Hahn |
11.0 |
||
|
1906 |
Hahn |
11.2 |
||
|
1908 |
Walker |
10.8 |
||
|
1912 |
Craig |
10.8 |
||
|
1920 |
Paddock |
10.8 |
||
|
1924 |
Abrahams |
10.6 |
||
|
1928 |
Williams |
10.8 |
Robinson |
12.2 |
|
1932 |
Tolan |
10.3 |
Walasiewicz |
11.9 |
|
1936 |
Owens |
10.3 |
Stephens |
11.5 |
|
1948 |
Dillard |
10.3 |
Blankers-Koen |
11.9 |
|
1952 |
Remigino |
10.4 |
Jackson |
11.5 |
|
1956 |
Morrow |
10.5 |
Cuthbert |
11.5 |
|
1960 |
Har |
10.2 |
Rudolph |
11.0 |
|
1964 |
Hayes |
10.0 |
Tyus |
11.4 |
|
1968 |
Hines |
9.95 |
Tyus |
11.0 |
|
1972 |
Borsov |
10.14 |
Stecher |
11.07 |
|
1976 |
Crawford |
10.06 |
Richter |
11.08 |
|
1980 |
Wells |
10.25 |
Kondratyeva |
11.06 |
|
1984 |
Lewis |
9.99 |
Ashford |
10.97 |
|
1988 |
Lewis |
9.92 |
Joyner |
10.54 |
|
1992 |
Christie |
9.96 |
Devers |
10.82 |
|
1996 |
Bailey |
9.84 |
Devers |
10.94 |
|
2000 |
Greene |
9.87 |
Jones |
10.75 |
a. Use EXCEL's trendline feature to find the best straight lines (one for Men and one for Women) through the data, where
is the straight line for the best time (T) as a function of the Olympic year (Y) with EXCEL determining the slope (m) and intercept (b). Write the equations for the best linear models and show (on a single graph) the graphs of the data and linear model for both Men and Women. Be sure to label which lines correspond to the data for the Men and Women.
b. Use the model to determine the predicted year when the best time is 10.0 sec for Men and 11.0 sec for Women, then compare your prediction to the actual data.
c. Use the model to predict the time for the 2000 and 2004 Olympics for both Men and Women in this event. Give the percent error between the actual and predicted value in 2000.
d. According to the model, which Olympics will first see Women outrunning the Men? Give a short discussion on the validity of this prediction and why you think it is true or false. What fundamental premise do you consider to be critical? Can you formulate another model that might be more valid?