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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%