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The following is the data obtained in the study of the number of absences and the final grade of the seven students in the mathematics class.
Construct a scatter diagram for these data. 

Calculate the correlation of coefficient (***r***). Interpret the results. 

Find the regression equation. 

Plot a regression line on the scatter diagram of part (a). 

Predict the final grade with 10 number of absences. 


### Analysis:

1. **Scatter Diagram**: The scatter diagram of absences vs. final grades shows a negative correlation, where an increase in the number of absences tends to result in lower final grades.

2. **Correlation Coefficient (\(r\))**: The calculated correlation coefficient is approximately \(-0.944\), indicating a strong negative linear relationship between the number of absences and the final grade. This means that as the number of absences increases, the final grade tends to decrease significantly.

3. **Regression Equation**: The equation for the line of best fit (regression line) is given by:
   \[
   y = -3.62x + 102.49
   \]
   Here, \(y\) represents the final grade, and \(x\) represents the number of absences. The negative slope confirms that the grades decrease as absences increase.

4. **Regression Line Plot**: The regression line plotted on the scatter diagram visually represents the downward trend of final grades with increasing absences.

5. **Prediction for 10 Absences**: Using the regression equation, the predicted final grade for a student with 10 absences is approximately \(66.27\).

Would you like more details or have any further questions?

### Related Questions:
1. How can we interpret the slope of the regression equation in this context?
2. What is the coefficient of determination (\(r^2\)), and what does it indicate about the data?
3. Can we infer causation from the correlation coefficient in this case? Why or why not?
4. How would adding more students to this study potentially affect the correlation coefficient?
5. What other factors might influence a student's final grade besides the number of absences?

### Tip:
When interpreting a correlation coefficient, remember that correlation does not imply causation. Even with a strong correlation, it’s important to consider other potential factors that may influence the relationship between variables.

The following is the data obtained in the study of the number of absences and the final grade of the seven students in the mathematics class. Construct a scatter diagram for these data. Calculate the correlation of coefficient (r). Interpret the results. Find the regression equation. Plot a regression line on the scatter diagram of part (a). Predict the final grade with 10 number of absences.

Explore how absences impact final grades in a high school mathematics class using correlation and regression analysis. This problem includes constructing a scatter plot, calculating the correlation coefficient, interpreting results, and predicting grades based on absences.

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