Linear Model Project Discussion
Feel free to discuss the Linear Model Project. (Instructions are found in the Linear Project module.)
It is highly recommended that you post your Linear Model Topic and your Data, by the end of Week 4. Give your posting a descriptive title so that we immediately know a bit about your topic. It is to your advantage to post your topic and data as early as possible, to get feedback. Sometimes, students are on the wrong track, so early feedback helps to put you on the right path as soon as possible.
Scatterplots,Linear Regression, and Correlation
When we have a set of data, often we would like to develop a model that fits the data.
First we graph the data points (x, y) to get a scatterplot. Take the data, determine an appropriate scale
on the horizontal axis and the vertical axis, and plot the points, carefully labeling the scale and axes.
Summer Olympics:
Men’s 400 Meter Dash
Winning Times
Year (x)
Time(y)
(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
Burger Fat (x)
(grams)
Calories (y)
Wendy’s Single 20 420
BK Whopper Jr. 24 420
McDonald’s Big Mac 28 530
Wendy’s Big Bacon
Classic 30 580
Hardee’s The Works 30 530
McDonald’s Arch
Deluxe 34 610
BK King Double
Cheeseburger 39 640
Jack in the Box
Jumbo Jack 40 650
BK Big King 43 660
BK King Whopper 46 730
Data from 1997
If the scatterplot shows a relatively linear trend, we try to fit a linear model, to find a line of best fit.
We could pick two arbitrary data points and find the line through them, but that would not necessarily
provide a good linear model representative of all the data points.
A mathematical procedure that finds a line of “best fit” is called linear regression. This procedure is also
called the method of least squares, as it minimizes the sum of the squares of the deviations of the points
from the line. In MATH 107, we use software to find the regression line. (We can use Microsoft Excel, or
Open Office, or a hand-held calculator or an online calculator — more on this in the Technology Tips
topic.)
Linear regression software also typically reports parameters denoted by r or r
2
.
The real number r is called the correlation coefficient and provides a measure of the strength of the
linear relationship.
r is a real number between −1 and 1.
r = 1 indicates perfect positive correlation — the regression line has positive slope and all of the data
points are on the line.
r = −1 indicates perfect negative correlation — the regression line has negative slope and all of the
data points are on the line
The closer |r| is to 1, the stronger the linear correlation. If r = 0, there is no correlation at all. The
following examples provide a sense of what an r value indicates.
Source: The Basic Practice of Statistics, David S. Moore, page 108.
Notice that a positive r value is associated with an increasing trend and a negative r value is associated
with a decreasing trend. The strongest linear models have r values close to 1 or close to −1.
The nonnegative real number r
2
is called the coefficient of determination and is the square of the
correlation coefficient r.
Since 0 ≤ |r| ≤ 1, multiplying through by |r|, we have 0 ≤ |r|
2
≤ |r| and we know that −1 ≤ r ≤ 1.
So, 0 ≤ r
2
≤ 1. The closer r
2
is to 1, the stronger the indication of a linear relationship.
Some software packages (such as Excel) report r
2
, and so to get r, take the square root of r
2
and
determine the sign of r by observing the trend (+ for increasing, − for decreasing).
Page1 of 4
(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
(
se
co
n
d
s)
Year
Summer Olympics: Men’s 400 Meter Dash Winning Times
Page 2 of 4
(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
(
se
co
n
d
s)
Year
Summer Olympics: Men’s 400 Meter Dash Winning Times
Page 3 of 4
(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.
Page 4 of 4
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 r2 = 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
T
im
e
(
se
co
n
d
s)
Year
Summer Olympics: Men’s 400 Meter Dash Winning Times