Math Problem Statement
express the model using function notation
Solution
To express the model that relates the number of hours a student plays video games per week, , to the cumulative grade-point average, , 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, | Grade-point Average, |
---|---|
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:
Where:
- is the grade-point average (dependent variable).
- is the number of hours spent playing video games per week (independent variable).
- is the slope of the line (indicating how much the grade-point average changes with each additional hour of video games).
- is the y-intercept (the predicted grade-point average when no hours are spent playing video games).
Let's compute the values for and 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 () to their grade-point average () is:
This means that for each additional hour spent playing video games, the grade-point average decreases by approximately 0.106 points. The -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:
- How does the -value impact the interpretation of this model?
- What is the significance of the intercept in this model?
- Can we use this model to predict GPA for students playing more than 14 hours of video games per week?
- What are the limitations of using a linear model for this data?
- How could you improve the accuracy of this model?
Tip: Always check the -value to understand how well the linear model fits the data. A value closer to -1 or 1 indicates a stronger linear relationship.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Algebra
Statistics
Formulas
Linear equation G(h) = a * h + b
Least squares method for regression
Theorems
Least Squares Regression Theorem
Suitable Grade Level
College-level Statistics or Algebra
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