https://file.aiquickdraw.com/mathbot/file/1729886228175nlkamio2.jpg

The task is to determine which linear model (function) best represents the relationship between the number of absences and the grades of Mr. Washington's students. The table on the right provides data for absences and grades, and the options for possible linear models are listed below:

1. \( G(x) = -2.8x + 93 \)
2. \( G(x) = -3.5x + 97 \)
3. \( G(x) = -3.7x + 98 \)
4. \( G(x) = -4.2x + 101 \)

This requires finding the equation of the best-fit line (linear regression) for the given data. We will calculate the slope and intercept of the linear model using the method of least squares.

I'll guide you through the process:

1. **Calculate the slope (m)** using the formula:
   \[
   m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
   \]
   where:
   - \(x\) is the number of absences,
   - \(y\) is the grade,
   - \(n\) is the number of data points.

2. **Calculate the intercept (b)** using the formula:
   \[
   b = \frac{\sum{y} - m\sum{x}}{n}
   \]

Let me calculate these values for you and compare them with the options provided.The linear model calculated based on the given data is approximately:

\[
G(x) = -3.70x + 97.95
\]

This closely matches the option:

\[
\boxed{G(x) = -3.7x + 98}
\]

Thus, the third option is the best linear model for predicting grades based on the number of absences.

Would you like more details or have any other questions? 

Here are some related questions:
1. How does a linear regression work?
2. What other methods can be used to analyze this type of data?
3. Can we make predictions using this model for students with higher absences?
4. How can the accuracy of this model be measured?
5. How would the equation change if outliers (like extreme absences) were removed?

**Tip:** Always check the residuals (the difference between actual and predicted values) to assess how well the model fits the data.

Which of the following functions represents the linear model that will make the best predictions of Mr. Washington’s students’ class grades based on how many classes they have missed?

Determine the best linear model for predicting Mr. Washington’s students' grades based on absences using linear regression. Analyze the relationship between the number of absences and grades with the least squares method. Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation