Instructions are found in the Linear Project module and examples. this project have different phases. The phase 1 is to post your Linear Model Topic and your Data (SEE EXAMPLE FROM STUDENT ATTACHED IN

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 sho w a downward trend, as stronger competition and trai ning methods result in faster speeds. The trend is som ewhat 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 Time (seconds) Year Summer Olympics: Men's 400 Meter Dash Winning Times Page 2 of 4 (LR-3) Line of Best Fit (Regression Line) y = 0.0431 x + 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 tim e decreases by 0.0431 second a year, and so the win ning time decreases at an average rate of 4(0.0431) = 0.1724 second each 4-ye ar 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 Time (seconds) 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 ne gative so the correlation coefficient r must be negative.

= − 0.6991 = −0.84 Recall that r = 1 corresponds to perfect negative correlation, and s o r = 0.84 indicates moderately strong negative correlati on (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 S ummer 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 throu gh 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 b e nearly 0.4 second less than the existing Olympic record of 43.49 seconds, q uite a feat! Will the regression line's prediction be accurate? In the last two decades, there appears to be more o f 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 dec reased 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 mos t recent winning times, beyond 1968, for Olympic winn ing times in 1972 and beyond (10 winning times), we have the following sca tterplot and regression line.

Page 4 of 4 Using the most recent ten winning times, our regress ion line is y = 0.025 x + 93.834.

When x = 2012, the prediction is y = 0.025(2012) + 93.834 43.5 seconds. This line predicts a winning t ime of 43.5 seconds for 2012 and that would indicate an excellent time close to the existing record of 43.49 seconds, but not dramatical ly 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. T he 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 pr edicts 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 s ee what happens later this summer! UPDATE: When the race was run in Augu st, 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 Time (seconds) Year Summer Olympics: Men's 400 Meter Dash Winning Times