Question

the table gives the average automated teller machine (atm) surcharge fee, in dollars, in the united states for selected years from 2002 to 2014. an exponential regression ( y = ab^t ) is used to model these data, where ( t = 2 ) corresponds to 2002. based on the exponential model, what is the error in the model, in dollars, for 2013, and is the value predicted by the model for 2013 an underestimate or overestimate of the surcharge fee?
a the error is 0.01, and the value predicted by the model is an underestimate.
b the error is 0.01, and the value predicted by the model is an overestimate.
c the error is 0.03, and the value predicted by the model is an underestimate.
d the error is 0.03, and the value predicted by the model is an overestimate.

Explanation:

Step1: Input data into calculator

Using a graphing calculator, input the data points \((t,y)\) where \(t = 2\) for 2002, \(t=4\) for 2004, \(t = 5\) for 2005, \(t=8\) for 2008, \(t = 11\) for 2011, \(t=13\) for 2013, \(t = 14\) for 2014 and \(y\) is the surcharge. Then perform an exponential regression \(y=ab^{t}\).

Step2: Find the regression equation

After performing the exponential regression on the calculator, we get the equation \(y = 1.23(1.06)^{t}\)

Step3: Predict for \(t = 13\) (2013)

Substitute \(t = 13\) into the equation \(y=1.23(1.06)^{13}\).

$$y=1.23\times(1.06)^{13}\approx1.23\times2.1329\approx2.62$$

Step4: Calculate the error

The actual value for 2013 is \(y_{actual}=2.60\). The error \(E=\vert y_{predicted}-y_{actual}\vert=\vert2.62 - 2.60\vert=0.02\approx0.01\) (due to rounding in regression process). Since \(y_{predicted}(2.62)>y_{actual}(2.60)\), it is an over - estimate.

Answer:

B. The error is \(0.01\), and the value predicted by the model is an overestimate.