Question
Which regression model best fits the data set?
(2,13),(4,8),(5,7.5),(7,9),(8,12)
(1 point)
Responses
linear
linear
quadratic
quadratic
exponential growth
exponential growth
exponential decay
(2,13),(4,8),(5,7.5),(7,9),(8,12)
(1 point)
Responses
linear
linear
quadratic
quadratic
exponential growth
exponential growth
exponential decay
Answers
GPT-4o mini
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:
1. **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.
2. **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.
3. **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.
1. **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.
2. **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.
3. **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.