Taipei European SchoolMath Portfolio | VINCENT CHEN| Gold Medal Heights Aim: To consider the winning height for the men’s high jump in the Olympic games Years| 1932| 1936| 1948| 1952| 1956| 1960| 1964| 1968| 1972| 1976| 1980| Height (cm)| 197| 203| 198| 204| 212| 216| 218| 224| 223| 225| 236| Height (cm) Height (cm) As shown from the table above, showing the height achieved by the gold medalists at various Olympic games, the Olympic games were not held in 1940 and 1944 due to World War II. Year (1=1932, 2 = 1936 and so on) Year (1=1932, 2 = 1936 and so on)Using autograph, the graph above is a scatter graph showing the high jump results from the table. The plot suggests that the high jump heights start off with a steep positive slope then coming to a decreasing negative slope, however without the 1940 and 1944 high jump competitions, it may not be certain. Then finally, it starts increasing again with a fairly steep positive slope. However, it would not be realistic if the function has an infinitely increasing range, such as quadratic, exponential and linear because of the limitations that humans have due to natural forces like gravity.Therefore, narrowing down the options that may fit this graph to natural logarithm and logistics Since the statistics given starts from year 1896, in order to make sure that calculations can be as simplified as possible, I have decided to rearrange the table with the assumption that 1896 is 0. Years| 36| 40| 52| 56| 60| 64| 68| 72| 76| 80| 84| Height (cm)| 197| 203| 198| 204| 212| 216| 218| 224| 223| 225| 236| This table shows the rearranged data of winning height for the men’s high jump in the Olympic games High Jump Height vs.
Years after 1896High Jump Height vs. Years after 1896 Height (cm) Height (cm) Years after 1896 Years after 1896 With the natural logarithm equation y=a+b (lnx), the variables “a” and “b” are the unknown parameters that affects the shape of the curve. “a” affects the shift along the y-axis and “b” affects the vertical stretch of the line. “y” is the height and “x” is the year in centimeters. In order to solve for the unknown, in this case “a” and “b”, two sets of given data points are substituted in the equation. (36,197) (84,236) y=33. 1+46(lnx) Height (cm) Height (cm)High Jump Height vs.
Years after 1896 High Jump Height vs. Years after 1896 Years after 1896 The graph above shows that the natural logarithmic function that I worked out doesn’t really fit the graph. y=33. 1+46(lnx) Therefore, by finding the average difference in height compared with the plots can make this equation more accurate. (Average difference = 7. 63cm) y=33.
1+46(lnx) By subtracting the variable “a” by 7. 63, it should give a more precise function that fits the graph. y=25. 5+46(lnx) High Jump Height vs. Years after 1896 High Jump Height vs.Years after 1896 Height (cm) Height (cm) Years after 1896 Years after 1896 Blue line = y=33. 1+46lnx Purple line = y=25. 5+46(lnx) From the new function y=25.
5+46(lnx), it is clear that it fits the data plots more accurately. By using technology, the TI-84 calculator, it is able to calculate a more precise equation for natural logarithm. y=44+41lnx High Jump Height vs. Years after 1896 High Jump Height vs. Years after 1896 Height (cm) Height (cm) Years after 1896 Years after 1896 Blue line = y=33. 1+46lnx Purple line = y=25.
5+46(lnx) Green line = y=44+41lnxBy using the natural logarithm function, it gives a more realistic prediction to the future since it shows a more constant positive slope that isn’t that steep, suggesting the fact that as human beings and the limitations from natural forces such as gravity, it is unlikely for athletes to continue jumping higher and higher in the future. There is a limit to how high athletes can jump, therefore the natural logarithm function demonstrates this logic most appropriately, as shown from the long-run graph below, it remains at a constant, only slowly increasing, at around the height of 240cm.High Jump Height vs. Years after 1896 High Jump Height vs. Years after 1896 Height (cm) Height (cm) Years after 1896 Years after 1896 By using the TI-84 calculator, it is possible to predict the high jump heights for year 1940 and 1944. When x = 44, y = 200. 943, therefore meaning that in year 1940, the predicted height from the natural logarithmic function is at 200.
943cm. When x = 48, y = 204. 547, therefore meaning that in year 1944, the predicted height from the natural logarithmic function is at 204. 547cm. Another equation that may be used is the logistic regression.
By using the TI-84 calculator and autograph, it calculates the logistic regression function. y=39. 6/(1+20872. 7183-0. 1139x))+197 High Jump Height vs.
Years after 1896 High Jump Height vs. Years after 1896 Height (cm) Height (cm) Years after 1896 Years after 1896 Green line = natural logarithmic function Purple line = logistic function From using the logistic regressive function, it gives another logical perspective to the task, as the function starts off with a positive slope that gradually becomes steeper and then returns back to how it started off.This gives a more logical prediction to the past since the logistic regression doesn’t start from point (0,0), meaning that in 1896, people could barely jump, which wouldn’t be realistic. The logistic regression predicts that in 1896, the gold medalist’s jump height is 198cm, which is far more realistic than the natural logarithm function. As shown on the graph above, comparing the two functions, the logistic regression function seems to be fitting the data more, since it has more of a positive slope like the data, having a steeper curve, whereas the logarithmic function is steady, which doesn’t entirely match the data.Furthermore, the natural logarithm steadily increases at a slow rate, which suggests that over time athletes will continue jumping higher and higher, which may exceed the limit of which a human being can jump thus meaning this is unrealistic. The logistic regression function however, does not.
It shows that the high jump height will remain constant at around 238cm. This also has its limitations, since fluctuation may occur in the future, it may not always be 238cm, and it could be higher or lower. Therefore, both models can only be used as predictions to an extent.
The winning height in 1984 can be predicted through both the logistic regression function and the natural logarithmic function. Logistic function suggests that the winning height in 1984, when x = 0, is 198cm. Whereas, the logarithmic regression function cannot predict the height in 1984 due to the fact that the function never reaches x = 0, it comes so close to 0 but just not quite. However, if a prediction was necessary, then it can only be estimated that the height = 0cm.
Estimation for 1940 and 1944 by the logistic function: 1940 = 199. 7cm 1944 = 201. cm Using both the logarithmic regression and logistic regression function, it allows an estimation for the high jump height in 2016. Logarithmic regression = 242. 5cm (x=120) Logistic regression = 236. 8cm (x=120) Table showing the winning heights for all the other Olympic games since 1896, with the assumption that 1896 = 0 Year| 0| 8| 12| 16| 24| 32| 88| 92| 96| 100| 104| 108| 112| Height (cm)| 190| 180| 191| 193| 193| 194| 235| 238| 234| 239| 235| 236| 236| High Jump Height vs. Years after 1896 High Jump Height vs.
Years after 1896 Height (cm)Height (cm) Years after 1896 Years after 1896 Green line = logarithmic regression Purple line = logistic regression The graph above shows the other Olympic games since 1896. Clearly, the logistic function fits better with the data than the logarithmic function, thus meaning that the logistic function is better for predictions than the logarithmic. The graph below shows the long run of the predictions for the winning height for the men’s high jump in the Olympic games High Jump Height vs. Years after 1896 High Jump Height vs. Years after 1896 Height (cm)Height (cm) Years after 1896 Years after 1896 Green line = logarithmic regression Purple line = logistic regression Once again, from this graph, it is clear that the logistic regression makes a much better prediction of the men’s high jump in the Olympic games. From the graph, it seems like starting from 1932 (36 years after 1896), athletes have consistently been jumping higher than before.
This is demonstrated by the logistic regression curve, as there is a steep positive slope, which represents the increase in men’s high jump heights. However, fter 1980 (84 years after 1896), athletes seem to have been stuck once again, just like from 1896 – 1928, perhaps due to the limitations of natural forces or possibly there haven’t been breakthroughs in technology that will enable athletes to jump even higher. There seems to be more fluctuation from 1896 – 1928, which the logistic regression curve predicts the poorer out of all the other years after 1928. Overall, the logistic regression curve definitely predicts the men’s high jump in the Olympic games fairly well, illustrating the trend line, from my calculations.