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To find the regression equation for this data, we will calculate a linear regression model in the form:

\[
y = mx + b
\]

where:
- \( y \) represents the final grade,
- \( x \) represents the number of absences,
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

### Steps to solve:
1. **Calculate the slope (m)**:
   \[
   m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
   \]

2. **Calculate the intercept (b)**:
   \[
   b = \frac{\sum y - m \sum x}{n}
   \]

where:
- \( n \) is the number of data points,
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \) value,
- \( \sum x \) and \( \sum y \) are the sums of the \( x \) and \( y \) values, respectively,
- \( \sum x^2 \) is the sum of the squares of the \( x \) values.

Given the data from the table:

| Student | Absences (\( x \)) | Grade (\( y \)) |
|---------|---------------------|-----------------|
| A       | 6                  | 82              |
| B       | 2                  | 86              |
| C       | 15                 | 43              |
| D       | 9                  | 74              |
| E       | 12                 | 58              |
| F       | 5                  | 90              |
| G       | 8                  | 78              |

Let me calculate the necessary values and provide the regression equation.The regression equation for the given data is:

\[
y = -3.62x + 102.49
\]

where:
- \( y \) is the predicted final grade,
- \( x \) is the number of absences.

This equation suggests that, on average, each additional absence reduces the final grade by approximately 3.62 points.

Would you like further details on how this was calculated, or do you have any questions?

Here are some related questions to expand your understanding:
1. How would you interpret the y-intercept in this context?
2. What does a negative slope indicate in this situation?
3. How reliable might this regression line be for predicting grades if absences were much higher than observed here?
4. How would you calculate the correlation coefficient to measure the strength of this relationship?
5. What would the predicted grade be for a student with zero absences?

**Tip:** In real-world data, make sure to check the residuals to assess the accuracy of a regression model.

Learn how to find the linear regression equation for the relationship between the number of absences and final grades of students. This problem includes calculating the slope and intercept using the least squares method, and interpreting the resulting regression line. Suitable for high school students in Grades 10-12.

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