To solve for $r^2$ (the coefficient of determination) and interpret it, we follow these steps:

Step 1: Compute $r^2$

The coefficient of determination $r^2$ is calculated using the following formula:

$r^2 = \left( \frac{\text{Explained Variation}}{\text{Total Variation}} \right)$

Alternatively, $r^2$ can be directly calculated using the correlation coefficient $r$:

$r^2 = r^2$

Here, $r$ is obtained from the regression equation or computed using the formula:

[ r = \frac{n \sum (x y) - \sum x \sum y}{\sqrt{\left[n \sum x^2 - (\sum x)^2\right] \cdot \left[n \sum y^2 - (\sum y)^2\right]}} ]

Given:

• Regression equation: $\hat{y} = -0.542x + 212.802$
• $x$ values: $209, 212, 219, 223, 259, 265, 276, 198, 207, 203, 216, 206, 257, 243$
• $y$ values: $108, 104, 100, 98, 88, 83, 49, 102, 105, 99, 95, 86, 67, 66$

We will compute $r^2$ numerically using the data.

Step 2: Interpretation of $r^2$

• $r^2$ represents the proportion of the variance in the dependent variable ($y$) explained by the independent variable ($x$).
• An $r^2$ close to 1 indicates a strong relationship, while an $r^2$ close to 0 indicates a weak relationship.

Let me calculate $r^2$ for the given data. numpy as np import pandas as pd

Data

goals_allowed = np.array([209, 212, 219, 223, 259, 265, 276, 198, 207, 203, 216, 206, 257, 243]) points = np.array([108, 104, 100, 98, 88, 83, 49, 102, 105, 99, 95, 86, 67, 66])

Regression coefficients from the equation y = -0.542x + 212.802

slope = -0.542 intercept = 212.802

Calculate predicted y values

predicted_points = slope * goals_allowed + intercept

Calculate residuals and total variance

residuals = points - predicted_points total_variance = np.var(points, ddof=1) explained_variance = total_variance - np.var(residuals, ddof=1)

Coefficient of determination r^2

r_squared = explained_variance / total_variance r_squaredThe coefficient of determination, $r^2$, is approximately 0.660.

Interpretation:

This means that about 66.0% of the variance in the number of points earned ($y$) is explained by the number of goals allowed ($x$). The remaining 34.0% is due to other factors not captured by this linear model or random variation.

Would you like me to proceed with calculating the standard error of the estimate ($s_e$)?

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Related Questions:

1. What does an $r^2$ value close to 0 indicate about the relationship between $x$ and $y$?
2. How would removing outliers affect the coefficient of determination in this data set?
3. Can $r^2$ alone determine whether the regression model is a good fit for the data?
4. What is the difference between $r^2$ and adjusted $r^2$?
5. How might other factors (besides goals allowed) influence the total points earned?

Tip:

Always check residual plots to evaluate whether the assumptions of the regression model are satisfied.