To determine which regression model best fits the data set \((2,13),(4,8),(5,7.5),(7,9),(8,12)\), we can analyze the pattern of the data points:
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Linear: A linear regression model fits data that shows a consistent rate of change. This data has varying differences between \(y\) values as \(x\) increases, which suggests that it may not perfectly align with a straight line.
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Quadratic: A quadratic regression model can accommodate curves and changes in direction. If the data appears to have a parabolic trend, a quadratic model may be more appropriate.
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Exponential Growth/Decay: Exponential models typically apply to situations where rates of change increase (growth) or decrease (decay) rapidly. The provided data doesn’t show this kind of rapid change.
To conduct a preliminary manual check:
- The differences in \(y\) as \(x\) increases are not consistent enough for a linear fit.
- A quadratic fit might be plausible, considering it could curve upward or have a vertex that fits the changing slope of the data points.
The best approach to confirm this would be to calculate the coefficients for each regression model (linear, quadratic, and possibly a logarithmic fit for exponential) and check their R-squared values to see which model minimizes the error.
Given a qualitative analysis and without performing numerical calculations, it leans towards:
Quadratic
However, for a definitive answer, performing regression analysis using statistical software or tools would provide precise results, including residual plots and R-squared values for evaluation.