To calculate the coefficient of determination, $R^2$, we use the formula:

$R^2 = \frac{SSR}{SST}$

where:

• $SSR$ is the Sum of Squares Regression,
• $SST$ is the Total Sum of Squares,
• $SSE$ is the Sum of Squares due to Error.

First, we find $SSR$ using the relationship:

$SST = SSR + SSE$

Given:

• $SST = 203.20$
• $SSE = 41.62$

So,

$SSR = SST - SSE = 203.20 - 41.62 = 161.58$

Now, calculate $R^2$:

$R^2 = \frac{161.58}{203.20}$

$R^2 \approx 0.7952$

Interpretation of $R^2$:

The $R^2$ value of approximately 0.7952 means that about 79.52% of the variability in the number of fish caught can be explained by the number of fishing lines used. The remaining 20.48% of the variability is due to factors not explained by the model.

Would you like more details, or have any questions about this? Here are 5 related questions:

1. How does adding more independent variables to a model typically affect the $R^2$ value?
2. What are some common ways to improve the $R^2$ value in a regression model?
3. What does an $R^2$ value close to 1 indicate about the model?
4. How can overfitting impact the interpretation of $R^2$?
5. What are the assumptions behind using $R^2$ in linear regression?

Tip: Always check the assumptions of the regression model (like linearity, independence, etc.) to ensure the $R^2$ value is meaningful.