A scientific study on fishing gives the following data table.
 
Fishing LinesFish Caught

4

13

5

15

7

25

11

29

12

26
 
Using technology, it was determined that the total sum of squares (SST) was 203.20 and the sum of squares due to error (SSE) was 41.62. Calculate R2 and determine its meaning. Round your answer to four decimal places.

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.

Learn how to calculate the coefficient of determination (R^2) in regression analysis using the sum of squares method. Given SST = 203.20 and SSE = 41.62, find out the meaning of R^2 and interpret its value. Ideal for students and professionals interested in statistical analysis.

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