equired Readings
Helen Petrakis
52-year-old heterosexual female of Greek descent
Lives in a four bedroom house with her husband John, age 60, and three adult children, son Alec, age 27, and daughters, Dmitra, age 23, and Athina, age 18
Works full time in the billing department of a hospital
Serves as the caregiver for her husband's mother, Magda Petrakis, a widow, age 81, who lives in an apartment about 30 minutes from Helen's home
Began visiting a social worker because she is feeling “blue” and feeling overwhelmed by her family responsibilities
John Petrakis
60-year-old male married to Helen, age 52, both of Greek descent
Lives in a four bedroom house with his wife, Helen, and three adult children, son Alec, age 27, and daughters, Dmitra, age 23, and Athina, age 18
Works full time managing a grocery store
Does not expect his children to contribute to the finances or upkeep of the home
His wife Helen is the caregiver for his elderly mother, age 81, who lives about 30 minutes away and who has some physical limitations and early dementia symptoms
Magda Petrakis
81-year-old widow and mother of John, age 60
Lives in an apartment 30 minutes from her son and daughter-in-law's home
Daughter-in-law Helen, age 52, takes the responsibility for her care and visits Magda several times a week
Recently, Helen has allowed her son, Alec, to move in with Magda, and Alec has begun stealing from his grandmother to support a drug habit
Alec Petrakis
27-year-old male of Greek descent
Currently unemployed
Lives with his parents, John, age 60, and Helen, 52, and sisters, Dmitra, age 23, and Athina, age 18
Recently moved in with his elderly grandmother promising to help take care of her, but instead, has begun stealing from her to support a drug habit
Dmitra Petrakis
23-year-old female of Greek descent
Works as a sales consultant for a local department store
Lives with her parents, John, age 60, and Helen, 52, and with a brother, Alex, age 27, and a sister, Athina, age 18
Has a grandmother, Magda Petrakis, age 81, who lives about 30 minutes from the family home
Athina Petrakis
18-year-old female of Greek descent
Works part time as a hostess at a family friend's restaurant
Is an honors student at a local college
Lives with her parents, John, age 60, and Helen, 52, and with a brother, Alex, age 27, and a sister, Dmitra, age 23
Has a grandmother, Magda Petrakis, age 81, who lives about 30 minutes from the family home
Page 1 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 (
s e
c o
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 (
s e
c o
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 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
T im
e (
s e
c o
n d
s )
Year
Summer Olympics: Men's 400 Meter Dash Winning Times
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 a free online tool
Use the free Desmos calculator: See DesmosLinearRegressionGuide.pdf to view how to generate a scatterplot and carry out linear regression.
The result of the free tool might not be as nice looking as the Microsoft Excel version, but it is free, accurate and easy to use.
(3) 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
(4) Use Open Office.
(5) Use a hand-held graphing calculator (See section 2.5 in your textbook for help with Texas Instruments hand-held calculators.)
The Linear Project Example uses Microsoft Excel.
Get help from top-rated tutors in any subject.
Efficiently complete your homework and academic assignments by getting help from the experts at homeworkarchive.com