# example of a linear model

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(Sample) Curve-Fitting Project – Linear Model: Men’s 400 Meter Dash Submitted by Suzanne Sands

(LR-1) Purpose: To analyze the winning times for the Olympic Men’s 400 Meter Dash using a linear model

Data: The winning times were retrieved from http://www.databaseolympics.com/sport/sportevent.htm?sp=ATH&enum=130

The winning times were gathered for the most recent 16 Summer Olympics, post-WWII. (More data was available, back to 1896.)

DATA:

Summer Olympics:

Men’s 400 Meter Dash

Winning Times

Year

Time

(seconds)

1948 46.20

1952 45.90

1956 46.70

1960 44.90

1964 45.10

1968 43.80

1972 44.66

1976 44.26

1980 44.60

1984 44.27

1988 43.87

1992 43.50

1996 43.49

2000 43.84

2004 44.00

2008 43.75

(LR-2) SCATTERPLOT:

As one would expect, the winning times generally show a downward trend, as stronger competition and training

methods result in faster speeds. The trend is somewhat linear.

43.00

43.50

44.00

44.50

45.00

45.50

46.00

46.50

47.00

1944 1952 1960 1968 1976 1984 1992 2000 2008

T im

e (

s e

c o

n d

s )

Year

Summer Olympics: Men’s 400 Meter Dash Winning Times

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(LR-3)

Line of Best Fit (Regression Line)

y = −0.0431x + 129.84 where x = Year and y = Winning Time (in seconds)

(LR-4) The slope is −0.0431 and is negative since the winning times are generally decreasing.

The slope indicates that in general, the winning time decreases by 0.0431 second a year, and so the winning time decreases at an

average rate of 4(0.0431) = 0.1724 second each 4-year Olympic interval.

y = -0.0431x + 129.84

R² = 0.6991

43.00

43.50

44.00

44.50

45.00

45.50

46.00

46.50

47.00

1944 1952 1960 1968 1976 1984 1992 2000 2008

T im

e (

s e

c o

n d

s )

Year

Summer Olympics: Men’s 400 Meter Dash Winning Times

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(LR-5) Values of r 2 and r:

r 2 = 0.6991

We know that the slope of the regression line is negative so the correlation coefficient r must be negative.

� = −√0.6991 = −0.84

Recall that r = −1 corresponds to perfect negative correlation, and so r = −0.84 indicates moderately strong negative correlation

(relatively close to -1 but not very strong).

(LR-6) Prediction: For the 2012 Summer Olympics, substitute x = 2012 to get y = −0.0431(2012) + 129.84 ≈ 43.1 seconds.

The regression line predicts a winning time of 43.1 seconds for the Men’s 400 Meter Dash in the 2012 Summer Olympics in London.

(LR-7) Narrative:

The data consisted of the winning times for the men’s 400m event in the Summer Olympics, for 1948 through 2008. The data exhibit

a moderately strong downward linear trend, looking overall at the 60 year period.

The regression line predicts a winning time of 43.1 seconds for the 2012 Summer Olympics, which would be nearly 0.4 second less

than the existing Olympic record of 43.49 seconds, quite a feat!

Will the regression line’s prediction be accurate? In the last two decades, there appears to be more of a cyclical (up and down)

trend. Could winning times continue to drop at the same average rate? Extensive searches for talented potential athletes and

improved full-time training methods can lead to decreased winning times, but ultimately, there will be a physical limit for humans.

Note that there were some unusual data points of 46.7 seconds in 1956 and 43.80 in 1968, which are far above and far below the

regression line.

If we restrict ourselves to looking just at the most recent winning times, beyond 1968, for Olympic winning times in 1972 and

beyond (10 winning times), we have the following scatterplot and regression line.

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Using the most recent ten winning times, our regression line is y = −0.025x + 93.834.

When x = 2012, the prediction is y = −0.025(2012) + 93.834 ≈ 43.5 seconds. This line predicts a winning time of 43.5 seconds for 2012 and

that would indicate an excellent time close to the existing record of 43.49 seconds, but not dramatically below it.

Note too that for r 2 = 0.5351 and for the negatively sloping line, the correlation coefficient is � = −√0.5351 = −0.73, not as strong as when

we considered the time period going back to 1948. The most recent set of 10 winning times do not visually exhibit as strong a linear trend as the

set of 16 winning times dating back to 1948.

CONCLUSION:

I have examined two linear models, using different subsets of the Olympic winning times for the men’s 400 meter dash and both have

moderately strong negative correlation coefficients. One model uses data extending back to 1948 and predicts a winning time of 43.1 seconds

for the 2012 Olympics, and the other model uses data from the most recent 10 Olympic games and predicts 43.5 seconds. My guess is that 43.5

will be closer to the actual winning time. We will see what happens later this summer!

UPDATE: When the race was run in August, 2012, the winning time was 43.94 seconds.

y = -0.025x + 93.834

R² = 0.5351

43.40

43.60

43.80

44.00

44.20

44.40

44.60

44.80

1968 1976 1984 1992 2000 2008

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