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express the model using function notation

To express the model that relates the number of hours a student plays video games per week, \( h \), to the cumulative grade-point average, \( G \), in function notation, we need to determine the linear model (or any other model) that fits the data points.

The data provided is as follows:

| Hours of Video Games per Week, \( h \) | Grade-point Average, \( G \) |
|----------------------------------------|------------------------------|
| 2                                      | 3.79                         |
| 2                                      | 3.45                         |
| 4                                      | 3.14                         |
| 5                                      | 3.22                         |
| 5                                      | 3.59                         |
| 7                                      | 3.18                         |
| 10                                     | 2.21                         |
| 10                                     | 2.94                         |
| 12                                     | 1.93                         |
| 14                                     | 2.91                         |

To express the model in function notation, we need to find a linear regression equation of the form:

\[
G(h) = a h + b
\]

Where:
- \( G(h) \) is the grade-point average (dependent variable).
- \( h \) is the number of hours spent playing video games per week (independent variable).
- \( a \) is the slope of the line (indicating how much the grade-point average changes with each additional hour of video games).
- \( b \) is the y-intercept (the predicted grade-point average when no hours are spent playing video games).

Let's compute the values for \( a \) and \( b \) using the least squares method. I will calculate this next.The linear model that relates the number of hours a student plays video games per week (\( h \)) to their grade-point average (\( G \)) is:

\[
G(h) = -0.106h + 3.787
\]

This means that for each additional hour spent playing video games, the grade-point average decreases by approximately 0.106 points. The \( r \)-value of -0.764 indicates a moderate negative correlation between the hours of video games played and the GPA.

Do you have any questions or would you like to explore this further?

Here are 5 related questions:
1. How does the \( r \)-value impact the interpretation of this model?
2. What is the significance of the intercept in this model?
3. Can we use this model to predict GPA for students playing more than 14 hours of video games per week?
4. What are the limitations of using a linear model for this data?
5. How could you improve the accuracy of this model?

**Tip**: Always check the \( r \)-value to understand how well the linear model fits the data. A value closer to -1 or 1 indicates a stronger linear relationship.

A professor wanted to find a linear model that relates the number of hours a student plays video games each week to the cumulative grade-point average (G) of the student. The table shows the data for 10 students: Hours of Video Games per Week (h) and Grade-point Average (G). Complete parts (a) through (f).

Explore how to find a linear model relating the number of hours spent playing video games to the cumulative GPA of students. Learn how to use linear regression and interpret the model with real data points, including calculating the regression equation and r-value.

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