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The question in your image asks to find the linear regression equation between homework grades \( x \) and test grades \( y \) and then use that equation to predict the test grade when the homework grade is 80.

To do this, we will:
1. Find the linear regression equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. Use the least-squares method to determine \( m \) and \( b \), and round the coefficients to the nearest tenth.
3. Plug in \( x = 80 \) into the equation to find the corresponding test grade.

### Step 1: Organize the Data
From the table:
\[
\begin{array}{|c|c|}
\hline
\text{Homework Grade} (x) & \text{Test Grade} (y) \\
\hline
71 & 63 \\
72 & 75 \\
62 & 58 \\
80 & 77 \\
85 & 71 \\
74 & 65 \\
90 & 95 \\
83 & 68 \\
62 & 57 \\
\hline
\end{array}
\]

### Step 2: Calculate the Linear Regression Coefficients
To get \( m \) and \( b \), we use the formulas for the slope and intercept:
\[
m = \frac{n \sum (xy) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}
\]
\[
b = \frac{\sum y - m \sum x}{n}
\]
Where \( n \) is the number of data points.

I'll now calculate these values.The linear regression equation that represents the relationship between homework grade \( x \) and test grade \( y \) is:

\[
y = 0.96x - 2.8
\]

Now, for a homework grade of 80, the projected test grade can be calculated as:

\[
y = 0.96(80) - 2.8 = 74.3
\]

Rounding to the nearest integer, the projected test grade is approximately **74**.

Would you like further details or have any questions about the process?

Here are 5 related questions you might find useful:
1. What does the slope in this context represent?
2. How does the correlation between homework and test grades affect predictions?
3. Can linear regression always provide accurate predictions for all data types?
4. How would you calculate this manually step by step?
5. How does rounding coefficients impact the accuracy of predictions?

**Tip:** When interpreting linear regression, the slope tells you how much the dependent variable (test grade) changes with each unit increase in the independent variable (homework grade).

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 tenth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.

Learn how to calculate the linear regression equation for a set of homework and test grades, and use it to predict a test grade for a student with a homework grade of 80. This problem involves key algebraic and statistical concepts suitable for students in Grades 9-12.

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