Question

she calculates two regression models. which is true? the linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months. the linear model better represents the situation because according to the exponential model, the repayment amount will never be 0. the exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months. the exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.

Explanation:

Step1: Analyze the decrease amount for linear model

For the linear model, if it were a linear decrease (constant amount per period), from \(12\) to \(18\) months (\(\Delta x = 6\)), the decrease is \(2110 - 1500=610\). From \(18\) to \(24\) months (\(\Delta x = 6\)), the decrease is \(1500 - 870 = 630\). From \(24\) to \(30\) months (\(\Delta x=6\)), the decrease is \(870 - 220=650\). The decrease amount is not constant, so the linear - model - because - of - constant - decrease - per - 6 - months is wrong.

Step2: Analyze the nature of linear and exponential models

A linear model has the form \(y=mx + b\). As \(x\) (months) increases indefinitely, if \(m<0\) (since the amount is decreasing), \(y\) (amount owed) will eventually be negative (\(y=mx + b<0\) when \(x>-\frac{b}{m}\)). An exponential model has the form \(y = a\cdot r^{x}+k\) (\(0k\) (assuming \(a>0\) and \(k\geq0\)) and approaches \(k\) (not necessarily \(0\)) as \(x\to\infty\).

Answer:

The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.