Question

a) use data above to determine an exponential regression function of best fit, ( f(x) ). round all values to two decimal places.
( f(x) =)

b) use data above to determine a linear regression function of best fit, ( g(x) ). round all values to two decimal places.
( g(x) =)

c) of these two, which equation best fits the data?
linear
exponential

Explanation:

Step1: Use a calculator or software for regression

For exponential regression \(y = ab^{x}\), input the \(x\) and \(y\) values (\(x = 1,2,\cdots,6\); \(y=695,939,1070,1391,1752,2253\)).
Using a TI - 84 Plus (or similar software):

• Press STAT, then EDIT. Enter \(x\) values in \(L1\) and \(y\) values in \(L2\).
• Press STAT, arrow over to CALC, then select ExpReg (for exponential regression).

We get \(a\approx500.51\), \(b\approx1.33\), so \(f(x)=500.51\times1.33^{x}\)

Step2: For linear regression \(y = mx + b\)

Press STAT, then EDIT (ensure \(L1\) and \(L2\) have the correct data).
Press STAT, arrow over to CALC, then select LinReg(ax + b)
We get \(m\approx319.07\), \(b = 375.93\), so \(g(x)=319.07x+375.93\)

Step3: Compare the fits

Calculate the \(R^{2}\) (coefficient of determination) values. For the exponential regression, the \(R^{2}\) value is closer to \(1\) (a measure of how well the regression line/curve fits the data) compared to the linear regression.

Answer:

a) \(f(x)=500.51\times1.33^{x}\)
b) \(g(x)=319.07x + 375.93\)
c) Exponential