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Math ia #2

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  • 1. SIriben Somboon Olympic Games are held every 4 years. The heights for men’s high jump in the OlympicGames are collected from 1896 to 2008. The aim if this task is to consider the winning height for themen’s high jump in the Olympic Games.Table 1: A data showing the heights achieved by for men’s high jumped by the gold medalists atOlympic Games from 1932 to 1980.Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980Number 1 2 5 6 7 8 9 10 11 12 13Height 197 203 198 204 212 216 218 224 223 225 236(cm)The heights are collected from year 1932 – 1980 for the first set of data. Since the Olympic Gamesare held every 4 years, so every 4 years, the number is increasing by 1. Let year 1932 be number 1.Example is from year 1932 to year 1936, the number is 1 for year 1932 and 2 for year 1936. OlympicGames were not held in 1940 and 1944 because there were wars during that time (WW2). So thenumber skipped from 2 to 5 from year 1936 to 1948.Figure 1 : A diagram showing the height for men’s high jump in Olympic Games from 1932 to 1980(number from 1 to 13). X-axis is the number and y-axis is the height in cm that the gold medalistsachieved in each Olympic Games.
  • 2. According to the figure 1, I observe that the numbers are increasing between each Olympic Games(except from year 1936 to 1948). The amount that the heights for men’s high jump in the OlympicGames that are increasing each year is quite constant, so I use a linear line to find the equation ofheights for men’s high jump for Olympic Games. With many points, I have to choose 2 points thatcan represent my graph if I want to find the linear equation for this data. I wouldn’t get an accurateresult for this data, so I decided to use a median-median line.I use a median-median line to find the equation of the line. The equation for median-median line isy = ax + b. Where “a” = the slope of the median from the data; “b” = y-intercept for the data; “x” =the number (the year); and “y” = height that the gold medalist achieved in that Olympic Game. Thenumbers are divided into 3 groups; in this case 4/3/4 because there are 11 numbers and 11 is notdivisible by 3. The first and the last group must have the same size. Then I find the median for eachgroup (both the number and the height).Table 2: Finding median for first group.Number 1 2 5 6Height (cm) 197 203 198 204Median for number = = 3.5Median for height = = 200.5Table 3: Finding median for middle group.Number 7 8 9Height (cm) 212 216 218Median for middle group is (8,216).Table 4: Finding median for last group.Number 10 11 12 13Height (cm) 224 223 225 236Median for number = = 11.5Median for height = = 224.5The median for the first group is 3.5 for number and 200.5 for height (3.5, 200.5). The median forthe middle group is (8, 216). The median for the last group is (11.5, 224.5). To find “a” for themedian-median line equation, I find the slope of the median I got from last group and the first group.To get “b” for the median-median line equation, I need to find the y-intercept of the line.
  • 3. To find “a” = = =3 “a” = To find “b” = = = 190.67 “b” = 190.67The median-median line equation is y = 3x + 190.67.Then I graph the line by using my equation that Ifound.Figure 2: A diagram showing the height for men’s high jump in Olympic Games from 1932 to 1980.The slope of the line Is 3 and the y-intercept is 190.7. This shows that every 4 year (every OlympicGames) the height for men’s high jump increases by 3 cm and at the year 1932, the height thatshould be achieved by the gold medalist is 190.7 cm.
  • 4. Figure 3: A diagram showing the height for men’s high jump in Olympic Games from 1932 to 1980.The black line is the line of best fit (linear line). The gray line is from the equation that I got by usingmedian-median line equation. The slope of the black line is 3.02 and the y-intercept for the black lineis 191.1 cm. This shows that every 4 year (every Olympic Games) the height for men’s high jumpincreases by 3.02 cm and at the year 1932, the height that should be achieved by gold medalist is191.1 cm. The slope of the line Is 3 and the y-intercept is 190.7 cm. This shows that every 4 year(every Olympic Games) the height for men’s high jump increases by 3 cm and at the year 1932, theheight that should be achieved by the gold medalist is 190.7 cm.Then to find the differences between the two lines, I use the percent error from my median-medianequation and the line of best fit when I used technology to find the line for this set of data. Mymedian-median line and the line of best fit that the computer found has the percent error of 0.67%.The limitation for my model is as time passes, the percent error for my model and the computermodel would be bigger because the slope and y-intercept is higher for the computer model than mymodel.To find the percent error: x 100 = x 100 = 0.67%
  • 5. Another function that models this data is quadratic equation. I use quadratic equation because theheight for men’s high jump in the Olympic Games is increasing for most of the Olympic Games.Figure 4: A diagram showing the height for men’s high jump in Olympic Games from 1932 to 1980.The gray line is the equation median-median line. The black line is a quadratic function that modelsthe data. The slope of the line Is 3 and the y-intercept is 190.7. This shows that every 4 year (everyOlympic Games) the height for men’s high jump increases by 3 cm and at the year 1932, the heightthat should be achieved by the gold medalist is 190.7 cm. The “a” value for the black line is 0.1807,“b” is 0.5004 and “c” is 197.3. This shows that according to the black curve, at year 1932, the heightthat should be achieved by the gold medalists is 197.3 cm.The differences between the median-median line and the quadratic curve are the y-intercept isdifferent and the quadratic line is closer to the points than the median-median line. The y-interceptfor the median-median line is 191.1 and the y-intercept for the quadratic line is 197.3.The quadraticline is closer to the points than the median-median line because the median-median line has the R2of 0.883 and the quadratic line has the R2 of 0.924. This shows that the quadratic line is closer to thepoints than the median-median line. The quadratic line also has a limit from 1 13 because ifthere is no limit for the quadratic line, in a few decades, the height that should be achieved by goldmedalists at the Olympic Games would be too high.
  • 6. If the game is held in 1940 and 1944, I use both the median-median line and the quadratic line toestimate the winning heights that would have been if the games are held.Table 5: A table showing the heights estimation for men’s high jump if the game is held in 1940 and1944. The estimation heights for men’s high jump in year 1940 and 1944 are calculated by usingmedian-median line and quadratic line.Year Number Height (cm) Median-median line Quadratic line Height % Height difference Y = 3.02x + 191.1 Y = 0.1807x2 + 0.5004x + difference (compare to 197.3 Quadratic line)1940 3 200.16 200.43 0.27 0.13%1944 4 203.10 202.19 0.91 0.45%The estimated winning heights if the Olympic Game in 1940 is 200.4 cm and 202.2 cm in year 1994.The quadratic line is being used instead of median-median line because the quadratic line is closer tothe points in the data than median-median line. Also when I plugged 6 for “x” (year 1956) in bothequations, for the median-median line, I got 209.2 cm; for quadratic line, I got 206.8 cm. This showsthat the quadratic line give more similar number to the data than the median-median line.Table 6: A table showing the heights estimation for men’s high jump if the game is held in 1984 and2016. The estimation heights for men’s high jump in year 1940 and 1944 are calculated by usingmedian-median line and quadratic line.Year Number Height (cm) Median-median line Quadratic line Height % Height difference Y = 3.02x + 191.1 Y = 0.1807x2 + 0.5004x + difference (compare to 197.3 Quadratic line)1984 14 233.38 239.72 6.34 2.72%2016 20 251.5 279.59 28.09 11.17%By using the median-median line, in 1984, the predicted height achieved by gold medalists in theOlympic Games is 233.4 cm. In 2016, the predicted height achieved by gold medalists in the OlympicGames is 251.5 cm. By using the quadratic line, in 1984, the predicted height achieved by goldmedalists in the Olympic Games is 239.7 cm. In 2016, the predicted height achieved by goldmedalists in the Olympic Games is 279.6 cm. When I used the quadratic line to estimate the heightthat gold medalists can achieve in 2016, the answer comes out to be 279.6 cm which is quite high forme. In 32 years (8 Olympic Games), the height that gold medalists can achieve from 1984 to 2016increases by 18.1 cm by using median-median line. The height that gold medalists can achieve from1984 to 2016 increase by 39.9 cm by using quadratic line. There is also a higher percentage heightdifference between the median-median line and the quadratic line in year 2016 than in year 1984.This means that one line (the quadratic line) is increasing a lot more than another line (median-median line).
  • 7. Table 7: A data showing the heights achieved by for men’s high jump by the gold medalists atOlympic Games from other Olympic Games since 1896.Year 1896 1904 1908 1912 1920 1928 1984 1988 1992 1996 2000 2004 2008Number -9 -7 -6 -5 -3 -1 14 15 16 17 18 19 20Height 190 180 191 193 193 194 235 238 234 239 235 236 236(cm)Table 8: A table showing the estimated heights achieved for men’s high jump by gold medalists inthe Olympic Games since 1896 and the collected data for men’s high jumped by gold medalists inOlympic Games. Height difference from the % difference Year Height (cm) collected data from the collected data Number Median- Median- Median- Collected median Quadratic median Quadratic median Quadratic data Model Model Model Model Model Model 1896 -9 190 163.92 207.43 26.08 -17.43 15.91% 8.40% 1904 -7 180 169.96 202.65 10.04 -22.65 5.91% 11.18% 1908 -6 191 172.98 200.80 18.02 -9.80 10.42% 4.88% 1912 -5 193 176 199.32 17 -6.32 9.66% 3.17% 1920 -3 193 182.04 197.43 10.96 -4.43 6.02% 2.24% 1928 -1 194 188.08 196.98 5.92 -2.98 3.15% 1.51% 1984 14 235 233.38 239.72 1.62 -4.72 0.69% 1.97% 1988 15 238 236.4 245.46 1.6 -7.46 0.69% 3.04% 1992 16 234 239.42 251.57 -5.42 -17.57 2.26% 6.98% 1996 17 239 242.44 258.03 -3.44 -19.03 1.42% 7.37% 2000 18 235 245.46 264.85 -10.46 -29.85 4.26% 11.27% 2004 19 236 248.48 272.04 -12.48 -36.04 5.02% 13.25% 2008 20 236 251.5 279.59 -15.5 -43.59 6.16% 15.59%Median-median function and quadratic function are being used to estimate the heights. The heightsare then compared to the collected data to find the percentage difference estimated data is fromthe collected data.According to table 8, my models (both median-median line and quadratic line) do not fit well whenadditional data are being added because the percentage difference the estimated data are from thecollected data is quite big. The height achieved for men’s high jump level off at around 236, it doesnot keep on going up, so both of the equations do not fit the additional data because both of theequations keep on going up.
  • 8. Figure 5: A diagram showing the height for men’s high jump in Olympic Games since 1896. This lineis a logarithm line where “a” = 54.19 and “b” = 720.I used a logarithm curve because first the height achieved by the gold medalists in the OlympicGames for men’s high jump kept on increasing, but since year 1980, there is almost no change in theheights achieved by gold medalists in the Olympic Games. The overall trend from 1896 to 2008 is theheights achieved by the gold medalists in the Olympic Games increase until year 1980. Thesefluctuations happen because technologies improved, so people have more resources to train with(so that they could jump higher). There is almost no change in the height achieved by the goldmedalists in the Olympic Games for men’s high jump because technologies have increased theresources for high jumpers to train to a certain level.I would modify my graphs (from median-median and quadratic curve) to logarithm because theheight achieved by gold medalists in the Olympic Games did not increase since 1980, so I cannot useequation that would keep on going up. It is also impossible for human to keep on increasing theheight achieved by the gold medalists in the Olympic Games for men’s high jump.

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