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To solve the problem of finding the linear equation for the given data using linear regression, we'll follow these steps:

1. The given data points are:
   - \( (x_1, y_1) = (1, 633) \)
   - \( (x_2, y_2) = (2, 687) \)
   - \( (x_3, y_3) = (3, 662) \)
   - \( (x_4, y_4) = (4, 649) \)
   - \( (x_5, y_5) = (5, 677) \)
   - \( (x_6, y_6) = (6, 713) \)

2. The formula for a linear equation is:
   \[
   y = mx + b
   \]
   where \(m\) is the slope and \(b\) is the y-intercept. To find the best fit line, we need to calculate the slope (\(m\)) and intercept (\(b\)) using the formulas for linear regression.

   The slope \(m\) is given by:
   \[
   m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}
   \]
   and the y-intercept \(b\) is:
   \[
   b = \frac{\sum y - m \sum x}{n}
   \]
   where \(n\) is the number of data points.

3. Now, let's calculate the required sums:
   - \( \sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21 \)
   - \( \sum y = 633 + 687 + 662 + 649 + 677 + 713 = 4021 \)
   - \( \sum xy = (1 \cdot 633) + (2 \cdot 687) + (3 \cdot 662) + (4 \cdot 649) + (5 \cdot 677) + (6 \cdot 713) = 18712 \)
   - \( \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91 \)

4. Using these values, we can now calculate the slope \(m\):
   \[
   m = \frac{6(18712) - (21)(4021)}{6(91) - (21)^2}
   = \frac{112272 - 84441}{546 - 441}
   = \frac{27831}{105} = 265.06
   \]

5. Now, calculate the y-intercept \(b\):
   \[
   b = \frac{4021 - (265.06 \cdot 21)}{6}
   = \frac{4021 - 5566.26}{6}
   = \frac{-1545.26}{6} = -257.54
   \]

6. Therefore, the equation of the best fit line is approximately:
   \[
   y = 265.06x - 257.54
   \]

Would you like further details on this, or do you have any questions?

Here are 5 related questions to explore further:
1. How does linear regression work for larger datasets?
2. What is the difference between linear regression and multiple regression?
3. How does the correlation coefficient relate to the accuracy of the regression line?
4. Can we apply linear regression to non-linear data?
5. How can we calculate linear regression using a calculator or software?

**Tip**: Always plot your data to visually check how well the line fits before relying solely on calculations!

Use linear regression to find the equation for the linear function that best fits this data. Round parameters to two decimal places.

Solve a linear regression problem with given x-values (1-6) and y-values (633-713). Find the best fit line equation using the least squares method. Suitable for College Algebra students.

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