https://file.aiquickdraw.com/mathbot/file/1729101693653yzwnfx73.jpg

To solve this, we need to:

1. Perform a linear regression to find the equation of the line of best fit in the form \( y = mx + b \), where:
   - \( m \) is the slope,
   - \( b \) is the y-intercept.
   
2. Use the equation to predict the test grade for a homework grade of 44.

### Step 1: Find the slope \( m \)

The formula for the slope \( m \) is:

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

Where:
- \( n \) is the number of data points (in this case, 8),
- \( \sum xy \) is the sum of the products of each \( x \) and \( y \),
- \( \sum x \) is the sum of all homework grades,
- \( \sum y \) is the sum of all test grades,
- \( \sum x^2 \) is the sum of the squares of each homework grade.

### Step 2: Calculate the intercept \( b \)

The formula for \( b \) (y-intercept) is:

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

### Step 3: Use the equation to predict the test grade for a homework grade of 44.

I'll now calculate this.The equation of the line of best fit is approximately:

\[
y = 0.8x + 11.9
\]

Using this equation, the predicted test grade for a homework grade of 44 is approximately:

\[
46.3
\]

Would you like more details on the calculations, or do you have any other questions?

Here are 5 questions for further exploration:
1. How do changes in the homework grade affect the predicted test grade?
2. What does the slope of 0.8 indicate about the relationship between homework and test grades?
3. How would we calculate the residual for each data point using the line of best fit?
4. Can we calculate the correlation coefficient to assess the strength of the relationship?
5. How would you interpret the y-intercept in this context?

**Tip**: When interpreting the slope of a linear regression, remember it represents the average change in the dependent variable (y) for every 1-unit increase in the independent variable (x).

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 44.

Explore how to find the linear regression equation to determine the correlation between test scores and homework grades. This solution includes step-by-step calculations, including slope and intercept formulas, suitable for students in Grades 10-12.

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