To solve this problem, we need to compute the linear regression equation for the data. The general form of the linear regression equation is:

$y = mx + b$

Where:

• $y$ is the dependent variable (test grade),
• $x$ is the independent variable (homework grade),
• $m$ is the slope of the regression line, and
• $b$ is the y-intercept of the regression line.

Step 1: Compute the necessary statistics.

We will need to calculate the following:

• The means of the $x$ (homework grades) and $y$ (test grades),
• The slope $m$, which is calculated using the formula:

$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

Where $n$ is the number of data points.

• The intercept $b$, which is calculated using the formula:

$b = \bar{y} - m\bar{x}$

Where $\bar{x}$ and $\bar{y}$ are the means of the $x$ and $y$ data, respectively.

Step 2: Plug the data into the formulas.

Given dataTo solve this problem, we need to compute the linear regression equation for the data. The general form of the linear regression equation is:

$y = mx + b$

Where:

• $y$ is the dependent variable (test grade),
• $x$ is the independent variable (homework grade),
• $m$ is the slope of the regression line, and
• $b$ is the y-intercept of the regression line.

Step 1: Compute the necessary statistics.

We will need to calculate the following:

• The means of the $x$ (homework grades) and $y$ (test grades),
• The slope $m$, which is calculated using the formula:

$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

Where $n$ is the number of data points.

• The intercept $b$, which is calculated using the formula:

$b = \bar{y} - m\bar{x}$

Where $\bar{x}$ and $\bar{y}$ are the means of the $x$ and $y$ data, respectively.

Step 2: Plug the data into the formulas.

Given

\hline \text{Homework Grade (x)} & \text{Test Grade (y)} \\ \hline 58 & 57 \\ 62 & 60 \\ 84 & 79 \\ 70 & 69 \\ 80 & 67 \\ 82 & 81 \\ 83 & 78 \\ \hline \end{array}$$ Let me calculate these values for you now.The linear regression equation representing the relationship between homework grade $$x$$ and test grade $$y$$ is: $$y = 0.8x + 10.2$$ ### Step 3: Estimate the homework grade for a student with a test grade of 31. We rearrange the equation to solve for $$x$$ (homework grade): $$x = \frac{y - b}{m}$$ Substitute $$y = 31$$, $$m = 0.8$$, and $$b = 10.2$$ to estimate the homework grade. Let me calculate that.The estimated homework grade for a student with a test grade of 31 is approximately **26**. Would you like more details or have any other questions? Here are 5 related questions to consider: 1. How would you interpret the slope of the regression line in this context? 2. How does a higher or lower homework grade influence the test grade, according to the equation? 3. What is the meaning of the y-intercept in this scenario? 4. How could this regression model be used to predict future student performance? 5. What other factors could affect the correlation between homework and test scores? **Tip:** Always check the residuals (the differences between actual and predicted values) to ensure the model's accuracy in real-world predictions.