Given that \( n = 10 \), we can now calculate the slope \( m \) and the intercept \( b \) for the least squares line.

### Step 1: Calculate the Slope \( m \)

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

Substituting the given values:

\[
m = \frac{10(1810.095) - (235)(77.564)}{10(5606) - (235)^2}
\]

\[
m = \frac{18100.95 - 18224.54}{56060 - 55225}
\]

\[
m = \frac{-123.59}{835}
\]

\[
m \approx -0.148
\]

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

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

Substituting the values:

\[
b = \frac{77.564 - (-0.148)(235)}{10}
\]

\[
b = \frac{77.564 + 34.78}{10}
\]

\[
b = \frac{112.344}{10}
\]

\[
b \approx 11.234
\]

### Final Equation of the Least Squares Line

The equation for the least squares line is:

\[
y = -0.148x + 11.234
\]

This equation represents the line that best fits the given data points using the least squares method.

Would you like further details or have any questions?

---

1. How does the negative slope affect the interpretation of the least squares line?
2. What are some real-world scenarios where the least squares method is applied?
3. How can the accuracy of this regression line be evaluated?
4. What is the significance of the y-intercept in the context of your data?
5. How would the equation change if the data points were different?

**Tip:** The sign of the slope (\( m \)) gives insight into the relationship between the variables; a negative slope indicates an inverse relationship.

Learn how to calculate the slope and intercept using the least squares method for n = 10 data points. This problem involves statistical concepts, including linear regression and the least squares method. Suitable for advanced high school students and beyond.

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