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To calculate the coefficient of determination \( R^2 \) for the data set in the table, we need to use the following steps:

1. **Identify the variables**: We have \( X \) as the "Temps d'étude (heures)" and \( Y \) as the "Score au test."

2. **Calculate the linear regression equation**: This involves finding the best fit line \( Y = aX + b \), where \( a \) is the slope and \( b \) is the y-intercept.

3. **Compute \( R^2 \)**: The \( R^2 \) value represents the proportion of the variance in the dependent variable (test scores) that is predictable from the independent variable (study hours). It’s calculated by:
   \[
   R^2 = 1 - \frac{\sum (Y_i - \hat{Y_i})^2}{\sum (Y_i - \bar{Y})^2}
   \]
   where:
   - \( Y_i \) are the actual scores,
   - \( \hat{Y_i} \) are the predicted scores from the regression line,
   - \( \bar{Y} \) is the mean of the actual scores.

Let's go through the computation:

1. **Data summary**:
   \[
   X = \{2, 3, 5, 7, 8, 10, 12, 15\}
   \]
   \[
   Y = \{50, 55, 60, 63, 65, 72, 80, 85\}
   \]

2. **Calculate \( \bar{X} \) and \( \bar{Y} \)**:
   - \( \bar{X} = \frac{2 + 3 + 5 + 7 + 8 + 10 + 12 + 15}{8} \)
   - \( \bar{Y} = \frac{50 + 55 + 60 + 63 + 65 + 72 + 80 + 85}{8} \)

3. **Compute the slope \( a \) and intercept \( b \)** for the line.

4. **Calculate \( R^2 \)** using the formula above.

I'll perform these calculations to provide the \( R^2 \) value.The coefficient of determination \( R^2 \) for the given data set is approximately \( 0.985 \). This means that about 98.5% of the variance in test scores can be explained by the study hours, indicating a very strong linear relationship between the two variables.

Would you like further details on the calculations or the interpretation?

Here are 5 related questions to expand on this concept:

1. How do we interpret an \( R^2 \) value close to 1 versus one closer to 0?
2. What does it mean if we have a high \( R^2 \) but low prediction accuracy?
3. How would adding more data points potentially affect the \( R^2 \) value?
4. How can we calculate the standard error of the estimate for this regression model?
5. Can \( R^2 \) be negative, and under what conditions might that occur?

**Tip:** To understand the impact of individual data points on \( R^2 \), try plotting a scatter plot with the regression line to visualize the fit.

Calculate the R² coefficient of determination based on study hours and test scores data provided in the table.

Learn how to calculate the R² coefficient of determination for study hours and test scores. This problem involves linear regression analysis, applying key statistical concepts to understand the strength of the relationship between study time and test performance.

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