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To solve this problem, we need to find the least squares regression line, which has the equation:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the regression line.
- \( b \) is the y-intercept.

### Steps for calculating the regression line:

1. **Calculate the means of \( x \) and \( y \):**
   The given data consists of the percentage of carbon (\( x \)) and the weight supported in kilograms (\( y \)).

   We calculate:
   \[
   \overline{x} = \frac{1.04 + 1.12 + 1.13 + 1.25 + 1.35 + 1.36 + 1.51 + 1.82}{8}
   \]
   \[
   \overline{y} = \frac{1103 + 1029 + 1041 + 1158 + 1278 + 1127 + 1156 + 1375}{8}
   \]

2. **Calculate the slope \( m \):**
   The formula for the slope \( m \) is:
   \[
   m = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}}
   \]

3. **Calculate the intercept \( b \):**
   Once we have the slope, we calculate the intercept using:
   \[
   b = \overline{y} - m \cdot \overline{x}
   \]

4. **Substitute the slope and intercept into the equation.**

Let me calculate these values for you.The equation of the least squares regression line is:

\[
y = 388.219x + 644.956
\]

This equation represents the relationship between the percentage of carbon in the car frame and the weight it can support.

Do you need further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of the slope in a regression line?
2. How do changes in the percentage of carbon affect the supported weight according to the regression line?
3. What is the purpose of finding the regression line in real-world applications?
4. How does rounding affect the accuracy of the regression line equation?
5. How would you predict the weight supported for a carbon percentage not listed in the table?

**Tip:** Always check the correlation coefficient to understand the strength of the relationship between two variables.

Find the equation for the least squares regression line of the data described below. A French car company wants to cut costs by using cheaper low-carbon steel in its car frames. To determine how varying the carbon content will affect strength, company engineers manufactured several car frames. For each car frame, the engineers noted the percentage of carbon, x, as well as the weight it could support (in kilograms), y.

Learn how to calculate the least squares regression line for data involving carbon content and weight supported by car frames. This problem demonstrates the application of linear regression using the least squares method, suitable for Grades 10-12.

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