To determine which type of function best models the relationship between the temperature and pressure of the gas, we can evaluate how well linear and quadratic functions fit the given data and compare their fit using the R^2 value. The R^2 value, known as the coefficient of determination, measures how well the regression predictions approximate the real data points, with a value of 1 indicating a perfect fit.
Here is the detailed analysis:
1. Linear Regression:
- We fit a linear function to the temperature and pressure data points.
- The linear function that fits the data has a slope of 0.25 and an intercept of 80.
- The R^2 value for the linear regression is 1.0, indicating a perfect fit to the data.
2. Quadratic Regression:
- We also fit a quadratic function to the data points.
- The quadratic function that fits the data has coefficients [tex]\([-8.4820794 \times 10^{-17}, 0.25, 80 \]\)[/tex].
- The R^2 value for the quadratic regression is also 1.0, indicating a perfect fit to the data.
Given that both linear and quadratic models achieve an R^2 value of 1.0, meaning they both fit the data perfectly, it's reasonable to conclude that:
- Since linear regression provides a perfect fit with a simpler form, it is the best choice for modeling the data.
Therefore, the type of function that best models the relationship between the temperature and pressure is linear.
