Let's go through each part of the problem from the images you've uploaded and solve it.

Part A: Slope of the Regression Line

The given regression equation is:

$\hat{y} = 4.5x + 69.7$

• Slope is the coefficient of $x$, which is 4.5.
• This means for each additional hour of study, the student's exam grade is predicted to increase by 4.5 points on average.

So, the correct interpretation from the options is:

• If a student studies an extra hour, his/her exam grade is predicted to increase by 4.5 points, on average.

Part B: Y-intercept of the Regression Line

The y-intercept of the regression line is the constant term, which is 69.7. This represents the predicted exam grade when $x = 0$ (i.e., no study hours).

However, the model's data is based on study hours between 0 and 6 hours, so interpreting the y-intercept makes sense here.

The correct interpretation of the y-intercept is:

• If a student does not study, his/her exam grade is predicted to be 69.7.

Part C: Predict the exam grade of a student who studies for 4 hours

To predict the grade for 4 hours of study, substitute $x = 4$ into the regression equation:

$\hat{y} = 4.5(4) + 69.7 = 18 + 69.7 = 87.7$

The predicted grade is 87.7.

Part D: Predict the exam grade of a student who studies for 9 hours

Since the range of study hours in the data is 0 to 6, predicting for 9 hours falls outside this range and may not be appropriate. Therefore, the correct response is:

• It is not appropriate to predict the grade of a student who studies 9 hours because 9 is outside the scope of the model.

Part E: Predict the exam grade of a student who studies for 4.5 hours

Now, substitute $x = 4.5$ into the regression equation:

$\hat{y} = 4.5(4.5) + 69.7 = 20.25 + 69.7 = 89.95$

The predicted grade is 89.95, which rounds to 90.

Would you like further details or have any questions? Here are some related questions for practice:

1. What does the slope represent in any regression equation?
2. Why might it not be appropriate to use a regression model to make predictions outside the data range?
3. How do you interpret the y-intercept in a real-world context?
4. If the slope was negative, what would that indicate about the relationship between study hours and grades?
5. How can we improve a regression model to include predictions for a wider range of hours?

Tip: Always check the range of data when using a regression model to ensure accurate predictions!