Math Problem Statement

What is the R²-value of the linear, quadratic, or exponential function that best fits the data?

Solution

To determine the R2R^2-value of the best fitting function (linear, quadratic, or exponential) for the given data, I will first extract the data points:

Data:

\hline x & y \\ \hline 8 & 1.72 \\ 24 & 11.99 \\ 11 & 2.91 \\ 19 & 7.70 \\ 17 & 6.12 \\ 9 & 2.13 \\ 14 & 4.46 \\ 23 & 11.07 \\ 22 & 10.15 \\ \hline \end{array}$$ Using these points, I will now compute the $$R^2$$-value for each type of regression model. Let me perform the calculations and return the results.The $$R^2$$-values for the different regression models are: - **Linear regression**: $$R^2 = 0.9834$$ - **Quadratic (2nd-degree polynomial) regression**: $$R^2 = 0.9998$$ - **Exponential regression**: $$R^2 = 0.9830$$ The quadratic model provides the best fit with an $$R^2$$-value of 0.9998. Would you like more details on the calculations or explanations of these models? Here are five related questions: 1. What does an $$R^2$$-value represent in the context of regression analysis? 2. How would a cubic (3rd-degree polynomial) model perform on this dataset? 3. How do residuals help in determining the best fit for a regression model? 4. Can we compare the models using other metrics like RMSE or MAE? 5. What are the implications of using a quadratic model in real-world data fitting? **Tip**: A higher $$R^2$$ value indicates a better fit, but beware of overfitting, especially when using higher-degree polynomials.

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Math Problem Analysis

Mathematical Concepts

Regression Analysis
Linear Regression
Quadratic Regression
Exponential Regression

Formulas

R² = 1 - (SS_res / SS_tot)
Linear Regression: y = mx + b
Quadratic Regression: y = ax² + bx + c
Exponential Regression: y = ab^x

Theorems

Least Squares Method

Suitable Grade Level

Undergraduate

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