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The problem asks for the linear regression equation for the given set of data, and to use this equation to predict a test grade for a homework grade of 39.

We will follow these steps:
1. Use the points \((x, y)\), where \(x\) is the homework grade and \(y\) is the test grade, to calculate the linear regression equation \(y = mx + b\).
2. Use the regression equation to project the test grade for \(x = 39\).

The data points provided are:

\[
\begin{array}{|c|c|}
\hline
\text{Homework Grade (x)} & \text{Test Grade (y)} \\
\hline
75 & 64 \\
82 & 80 \\
89 & 93 \\
90 & 84 \\
78 & 66 \\
60 & 61 \\
76 & 74 \\
50 & 55 \\
\hline
\end{array}
\]

I will calculate the regression equation and find the predicted test grade for \(x = 39\).The linear regression equation for the given data is:

\[
y = 0.83x + 9.87
\]

Using this equation, the projected test grade for a homework grade of 39 is approximately:

\[
\text{Projected Test Grade} = 42
\]

Would you like more details on how the linear regression was calculated? Here are some related questions for further understanding:

1. How do you calculate the slope in linear regression?
2. What is the role of the intercept in a regression equation?
3. How can we measure the accuracy of a linear regression model?
4. What does the \( r \)-value in linear regression indicate?
5. How would outliers affect the regression equation?

**Tip:** When performing linear regression, always check the correlation between variables to assess how well they relate to each other.

A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 39.

This page provides a detailed solution to a linear regression problem involving the correlation between homework grades and test grades. It includes the calculation of the regression equation and the projection of a test score for a given homework grade of 39. The data points are provided for students in Grades 9-12.

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