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To complete the Linear Model portion of the project, you will need to use technology (or hand-drawing) to create a scatterplot, find the regression line, plot the regression line, and find r and r2.
Below are some options, together with some videos. Each video is limited to 5 minutes or less. It takes a bit of time for the video to initially download. When playing the video, if you want to slow it down to read the text, hit the pause icon. (If you run the mouse over the bottom of the video screen, the video controls will appear.) You may need to adjust the volume.
The basic options are to:
(1) Generate by hand and scan.
(2) Use Microsoft Excel.
Visit
Scatterplot – Start
(VIDEO) to see how to create a scatter plot using Microsoft Excel and format the axes.
Visit
Scatterplot – Regression Line
(VIDEO) to see how to add labels and title to the scatterplot, how to generate and graph the line of best fit (regression) and obtain the value of r2 in Microsoft Excel.
Using Excel to obtain precise values of slope m and y-intercept b of the regression line:
Video
,
Spreadsheet
(3) Use Open Office.
(4) Use a hand-held graphing calculator (See section 2.5 in your textbook for help with Texas Instruments hand-held calculators.)
(5) Use a free online tool
(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:
(LR-2) SCATTERPLOT:
Summer Olympics:
Men’s 400 Meter Dash
Winning Times
Time
(seconds)
46.20
45.90
46.70
44.90
45.10
43.80
44.66
44.26
44.60
44.27
43.87
43.50
43.49
43.84
44.00
43.75
47.00
46.50
46.00
Time (seconds)
Year
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
Summer Olympics: Men’s 400 Meter Dash Winning Times
45.50
45.00
44.50
44.00
43.50
43.00
1944
1952
1960
1968
1976
1984
1992
2000
2008
Year
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.
Page 1 of 4
(LR-3)
Summer Olympics: Men’s 400 Meter Dash Winning Times
47.00
46.50
y = -0.0431x + 129.84
R² = 0.6991
Time (seconds)
46.00
45.50
45.00
44.50
44.00
43.50
43.00
1944
1952
1960
1968
1976
1984
1992
2000
2008
Year
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.
Page 2 of 4
(LR-5) Values of r2 and r:
r2 = 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 3 of 4
Summer Olympics: Men’s 400 Meter Dash Winning Times
44.80
Time (seconds)
44.60
y = -0.025x + 93.834
R² = 0.5351
44.40
44.20
44.00
43.80
43.60
43.40
1968
1976
1984
1992
2000
2008
Year
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.
Page 4 of 4
Curve-fitting Project – Linear Model (due at the end of Week 5)
Instructions
For this assignment, collect data exhibiting a relatively linear trend, find the line of best fit, plot
the data and the line, interpret the slope, and use the linear equation to make a prediction. Also,
find r2 (coefficient of determination) and r (correlation coefficient). Discuss your findings. Your
topic may be that is related to sports, your work, a hobby, or something you find interesting. If
you choose, you may use the suggestions described below.
A Linear Model Example and Technology Tips are provided in separate documents.
Tasks for Linear Regression Model (LR)
(LR-1) Describe your topic, provide your data, and cite your source. Collect at least 8 data
points. Label appropriately. (Highly recommended: Post this information in the Linear
Model Project discussion as well as in your completed project. Include a brief informative
description in the title of your posting. Each student must use different data.)
The idea with the discussion posting is two-fold: (1) To share your interesting project idea with
your classmates, and (2) To give me a chance to give you a brief thumbs-up or thumbs-down
about your proposed topic and data. Sometimes students get off on the wrong foot or
misunderstand the intent of the project, and your posting provides an opportunity for some
feedback. Remark: Students may choose similar topics, but must have different data sets.
For example, several students may be interested in a particular Olympic sport, and that is fine,
but they must collect different data, perhaps from different events or different gender.
(LR-2) Plot the points (x, y) to obtain a scatterplot. Use an appropriate scale on the horizontal
and vertical axes and be sure to label carefully. Visually judge whether the data points exhibit a
relatively linear trend. (If so, proceed. If not, try a different topic or data set.)
(LR-3) Find the line of best fit (regression line) and graph it on the scatterplot. State
the equation of the line.
(LR-4) State the slope of the line of best fit. Carefully interpret the meaning of the slope in a
sentence or two.
(LR-5) Find and state the value of r2, the coefficient of determination, and r, the correlation
coefficient. Discuss your findings in a few sentences. Is r positive or negative? Why? Is a line a
good curve to fit to this data? Why or why not? Is the linear relationship very strong, moderately
strong, weak, or nonexistent?
(LR-6) Choose a value of interest and use the line of best fit to make an estimate or prediction.
Show calculation work.
(LR-7) Write a brief narrative of a paragraph or two. Summarize your findings and be sure to
mention any aspect of the linear model project (topic, data, scatterplot, line, r, or estimate, etc.)
that you found particularly important or interesting.
You may submit all of your project in one document or a combination of documents, which may
consist of word processing documents or spreadsheets or scanned handwritten work, provided it
is clearly labeled where each task can be found. Be sure to include your name. Projects are
graded on the basis of completeness, correctness, ease in locating all of the checklist items, and
strength of the narrative portions.
To complete the Linear Model portion of the project, you will need to use technology
(or hand-drawing) to create a scatterplot, find the regression line, plot the regression
line, and find r and r2.
Below are some options, together with some videos. Each video is limited to 5
minutes or less. It takes a bit of time for the video to initially download. When
playing the video, if you want to slow it down to read the text, hit the pause icon.
(If you run the mouse over the bottom of the video screen, the video controls will
appear.) You may need to adjust the volume.
The basic options are to:
(1) Generate by hand and scan.
(2) Use Microsoft Excel.
Visit Scatterplot – Start (VIDEO) to see how to create a scatter plot using Microsoft
Excel and format the axes.
Visit Scatterplot – Regression Line (VIDEO) to see how to add labels and title to the
scatterplot, how to generate and graph the line of best fit (regression) and obtain the
value of r2 in Microsoft Excel.
Using Excel to obtain precise values of slope m and y-intercept b of the regression
line: Video, Spreadsheet
(3) Use Open Office.
(4) Use a hand-held graphing calculator (See section 2.5 in your textbook for help
with Texas Instruments hand-held calculators.)
(5) Use a free online tool
