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=**Cost at Four- Year Private Colleges**=

Data:

 * Academic Year || Total Annual Cost in $US ||
 * 1975 || 4205 ||
 * 1976 || 4460 ||
 * 1977 || 4680 ||
 * 1978 || 4960 ||
 * 1979 || 5510 ||
 * 1980 || 6060 ||
 * 1981 || 6845 ||
 * 1982 || 7600 ||
 * 1983 || 8435 ||
 * 1984 || 9000 ||
 * 1985 || 9659 ||

Equation by Hand
1) 4205 Ratio = 1.06064 2) 4460 Ratio = 1.04932 3) 4680 Ratio = 1.05982 4) 4960 Ratio = 1.110887 5) 5510 Ratio =1.099818 6) 6060 Ratio = 1.1295379 7) 6845 Ratio = 1.1103 8) 7600 Ratio = 1.109868 9) 8435 Ratio = 1.066983 10) 9000 Common Ratio = 1.09 U(0): 4205 / 1.09 = 3857.798165 Equation by hand: Y = 3857.798165 * 1.09^x

Exponential Regression by Calculator:
Y = 3653.0261 * 1.093220139^x R^2 = 0.988839 R = 0.994403



Initial Analysis of the Two Models
We think that the ExpReg function is better. The two functions basically have the same shape, but the hand-making function is a little bit above some points. The problem is that the function is hard to be adjusted by hand, and calculation on the TI-84 is more accurate. Therefore, we think that the ExpReg function is better, but they have the same nature and shape.

Prediction for doubling the total annual cost in $US in 1985 (11)
2 x U(11) = 3653.03 * 1.09322^x 19318 / 3653.03 = 1.09322^x x = (10^5.2882128) / (10^1.09322) x = 15667.25096

Doubling the starting amount
2U(0) = U(0) * 1.09322^x 2 = 1.09322^x x = (10^2) / (1.09322^2) x = 83.6729 years

Final Analysis of the Two Models
The ExpReg function is obviously better because it is accurate and it is the best-fit line for these cases. The function obtained by division by hand is not precise and the common rate has a range since it takes too much time to consider all the cases.

Part II - Linearizing the Data


(x, log y) linearizes the graph. This is because our original data is an exponential function. Therefore, (x, log y) which linearized an exponential function also linearizes the graph.

(x, log y) LinReg: y = 0.038707862x + 3.562652775 r^2 = 0.98884 r = 0.9944038

Transformation: Log y = 0.038707862x + 3.5652775 Y = 10^(0.038707862x) * 3675.170569

Part III - A Graph
Normal Graph:

Semi-log Graph

Graph observations: Normal graph paper: Appears to be an exponential shape, but there is a little curve down in the last few cases. Semi-log graph paper: An almost straight line. Good enough to see why it is linearized, thus, the original data can be expressed exponentially yet it shows a curve.