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 linear model property

A linear model has a constant rate of change. If we check the differences for the given data:
From \(12\) to \(18\) months (\(\Delta x = 6\)), the change in amount is \(2110 - 1500=610\). From \(18\) to \(24\) months (\(\Delta x = 6\)), the change is \(1500 - 870 = 630\). From \(24\) to \(30\) months (\(\Delta x=6\)), the change is \(870-220 = 650\). The differences (\(610\), \(630\), \(650\)) are not constant. So, the amount is not decreasing by the same amount every \(6\) months, which rules out the linear - model - based options (first two options).

Step2: Analyze the exponential model property

An exponential model is of the form \(y = a\cdot b^{x}+c\) (\(0\lt b\lt1\) for decay). A linear model is of the form \(y=mx + b\). If we assume a linear model \(y=mx + b\), using two - point form (say \((x_1,y_1)=(12,2110)\) and \((x_2,y_2)=(18,1500)\)), \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1500 - 2110}{18 - 12}=\frac{- 610}{6}\approx - 101.67\). The equation is \(y-2110=-\frac{610}{6}(x - 12)\). As \(x\) gets large enough (solving \(y = mx + b\lt0\) for \(x\), \(x\gt\frac{-b}{m}\)), the linear model will give a negative amount (since the slope \(m\lt0\)). An exponential model \(y = a\cdot b^{x}+c\) (\(0\lt b\lt1\)) approaches \(c\) (a non - negative value, often \(0\) in practical debt - like situations as \(x\to\infty\)) as \(x\) increases.

Answer:

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