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InstructionsFor 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. The topic I would like to do is comparing average global temperatures and levels of pollution from the past 10 years.A Linear Model Template has been includedTasks for Linear Regression Model (LR)(LR-1) Describe your topic, provide your data, and cite your source. Collect at least 8 data points. 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.

(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

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

Submitted by Example

(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.

𝑟 = −√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 𝑟 = −√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

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