https://file.aiquickdraw.com/mathbot/file/1727113718268ebefg9vu.jpg

https://file.aiquickdraw.com/mathbot/file/1727113718269in9qpvxj.jpg

https://file.aiquickdraw.com/mathbot/file/1727113718269umiyfsgq.jpg

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!

A study is conducted to predict the grade of a student, y, on a statistics exam based on his/her hours of study, x. The least-squares prediction equation is given as ŷ = 4.5x + 69.7.

Explore a detailed solution to a regression analysis problem where the predicted exam grade of a student is based on study hours using the linear regression equation ŷ = 4.5x + 69.7. Learn about the slope, y-intercept, and how to interpret results. Suitable for Grades 10-12.

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