To determine which regression model better represents the given data on the amount owed over time, we'll analyze the nature of the decrease in the amounts using both a linear and an exponential approach.
Consider the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Months (x)} & \text{Amount Owed (y)} \\
\hline
12 & 2110 \\
\hline
18 & 1500 \\
\hline
24 & 870 \\
\hline
30 & 220 \\
\hline
\end{array}
\][/tex]
1. Evaluating Linear Model:
- Linear Decrease: To ensure a linear model fits, the differences between successive amounts should be approximately constant.
[tex]\[
\begin{aligned}
&\text{Difference between 12 and 18 months: } 2110 - 1500 = 610 \\
&\text{Difference between 18 and 24 months: } 1500 - 870 = 630 \\
&\text{Difference between 24 and 30 months: } 870 - 220 = 650
\end{aligned}
\][/tex]
- These differences (610, 630, 650) are relatively close, indicating a nearly linear decrease.
2. Evaluating Exponential Model:
- Exponential Decay: When data decreases exponentially, the ratio of successive amounts should be approximately constant.
[tex]\[
\begin{aligned}
&\text{Ratio between 12 and 18 months: } \frac{2110}{1500} \approx 1.41 \\
&\text{Ratio between 18 and 24 months: } \frac{1500}{870} \approx 1.72 \\
&\text{Ratio between 24 and 30 months: } \frac{870}{220} \approx 3.95
\end{aligned}
\][/tex]
- These ratios (1.41, 1.72, 3.95) vary significantly, suggesting the decrease does not follow a straightforward exponential decay.
3. Effects of the Models:
- Linear Model Implications:
- If the decrease remains constant, the amount owed will eventually drop to zero and might even become negative, which is not feasible in real-world scenarios, as debt can't be negative.
- Exponential Model Implications:
- Exponential decay implies that the amount will decrease at a decelerating rate, approaching zero but never actually reaching zero.
Combining the insights from these analyses, we conclude that the exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative, which doesn't make practical sense. Therefore, the statement that best describes the situation is:
The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
