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.