To determine which regression model best fits the given data set \((2, 13), (4, 8), (5, 7.5), (7, 9), (8, 12)\), we should look at the characteristics of the points.
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Linear Regression: A linear model has the general form \(y = mx + b\) and aims to fit a straight line to the data. Given the points, there seems to be a decreasing trend, but it’s not perfectly linear since the changes are not constant.
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Quadratic Regression: This model has the form \(y = ax^2 + bx + c\). Given the nature of the points, a quadratic model might capture any curvature in the data if the points do not lie on a straight line.
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Exponential Growth or Decay: These models have the forms \(y = ae^{bx}\) (growth) or \(y = ae^{-bx}\) (decay). Looking at the y-values, they do not consistently increase or decrease exponentially.
Observing the points:
- The values start high (13) at \(x=2\), and then decrease to a minimum (7.5) at \(x=5\), followed by a slight increase (12) at \(x=8\).
- The y-values decrease from 13 to 7.5 and then increase again, which suggests variability.
Conclusion:
Given this information, a quadratic regression is likely the best fit as it can depict the rise and fall of y-values more effectively than a linear or exponential model.
Thus, the best fitting model is quadratic.
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