Winning_Speeds_for_the_Indianapolis_500_auto-race

=Winning Speeds for the Indianapolis 500 auto-race=


 * Year || Speed (mph) ||
 * 1961 || 139.1 ||
 * 1962 || 140.3 ||
 * 1963 || 143.1 ||
 * 1964 || 147.4 ||
 * 1965 || 151.4 ||
 * 1966 || 144.3 ||
 * 1967 || 151.2 ||
 * 1968 || 152.9 ||
 * 1969 || 156.9 ||
 * 1970 || 155.7 ||
 * 1971 || 157.7 ||
 * 1972 || 163.5 ||
 * 1973 || 159 ||
 * 1974 || 158.6 ||
 * 1975 || 149.2 ||
 * 1976 || 148.7 ||
 * 1977 || 161.3 ||
 * 1978 || 161.4 ||
 * 1979 || 158.9 ||
 * 1980 || 142.9 ||

Part 1: Logarithmic Model
1) Finding the equation

A- By Hand i) Determine the common ratio of the inverse function Choose two points to represent the data (143.1, 1963) & (161.3, 1977) y=ab^x ab^143.1=1963 ab^161.3=1977 OR ab^143.1*b^18.2=1977 So, 1963*b^18.2=1977 b=(1977/1963)^(1/18.2)=1.000391 ii) Finding the constant multiplier of the inverse function Substitute b into ab^x=y with (143.1, 1963) a*1.000391^143.1=1963 a=1963/1.057525 =1856.202532 Final exponential equation: y=1856.202532*1.000391^x iii) Flip x and y to undo the inverse and find the logarithmic equation x=1856.202532*1.000391^y (x/1856.202532)=1.000391^y log (x/1845.202532) / log 1.000391= y (log x - log 1845.202532)/ 1.6977595 E-4= y (log x - 3.268578)/ 1.6977594 E-4= y log x /1.6977594 E-4 - 3.268578/1.6977594 E-4 = y The final logarithmic equation, in the form f(x)= A + B log X is... -19252.30395 + (1/1.6977594 E-4) log x = y

B- Exponential Regression i) Equation of the inverse function: y=1898* 1.000246^x ii) Flip x and y ot undo the inverse and find the logarithmic equation x=1898* 1.000246^y (x/1898)=1.000246^y log (x/1898) / log 1.000246= y (log x - 3.278296)/ 1.068233 E-4= y log x /1.068233 E-4 - 3.278296/1.068233 E-4 = y The final logarithmic equation, in the form f(x)= A + B log X is... -30688.96018 + (1/1.068233 E-4) log x = y

2) Graph of Inverse Function

3) Doubling Time for the Function 1963=1856.202532*1.000391^143.1 x=1856.202532*1.000391^286.2 x= 2075.94211 2075.94211-1856.202532= 210.739578 It takes 210.739578 years to double the speed.

4) Ln Regression Equation of Logarithmic Data: y= -11910.7247+1590.210753 lnx

Residual Plots i) Calculated Inverse Function

ii) Inverse of Exponential Regression Function

iii) Ln Regression Inverse Function

5) We think that the natural logarithm regression function is a better fit, since it gives us a better overall trend of the data and where it is heading towards rather than simply connecting 2 points, as seen in the calculated version, which has too steep of a slope that we can't really see the function curving downwards. The trend shown in the inverse equation of the exponential regression is more based on the years according to the speeds that occur than the speeds according to when they occur. This is because the year and speed axis were flipped around when finding the equation. That is why it only models the top half of the data, but not the bottom half. The natural logarithm regression has a smaller residual sum (excluding outliers), which can also tell us that it is a better fit.

6) Based on the generalized equation, we can say that the winning speed was about 161.13 mph in 1981, 20 years since the first recording. It was about 169.14 mph in 1991, and 177.11 mph in 2001. We can estimate the winning times for the future years. The winning speed for the Indianapolis 500 auto-race would be about 184.24 mph in 2010, 192.14 mph in 2020, 199.99 mph in 2030, 207.8 mph in 2040, and 215.58 mph in 2050.

7) Exponential Data Scatter Plot & Equations

Part 2
Linearlizing the Data

Part 3
The best graph that linearlized the data was the log-log graph. Apart from the three outliars in the right botton corner in the above graph, all other points were on an invisible linear line. Just graphing paper made the most random shape, and semi-log graphs formed a somewhat linear graph but not quite as accurate as the log-log graph. Because of technical difficulties, our group could not post up the graphs but the equation of the log-log scatterplot is: (L3,L4) where L3 is equal to logX, and L4 is equal to logY. Window: 3.29, 3.3, 0.01, 2.1, 2.25, 0.05, 1