To find the linear regression equation $y = mx + b$ for the given data, we first calculate the slope $m$ and the y-intercept $b$ of the line that best fits the data. Here’s the process:

Given data:

• $x = [1, 2, 3, 4, 5, 6]$
• $y = [90, 110, 129, 152, 168, 175]$

Step-by-Step Calculation:

1. Calculate the mean of $x$ and $y$.
2. Find the slope $m$ using the formula: $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
3. Calculate the y-intercept $b$ using: $b = \bar{y} - m \cdot \bar{x}$

After calculation:

• Slope $m \approx 17.50$
• Intercept $b \approx 71.17$

Final Answer:

The linear regression equation is: $y = 17.50x + 71.17$

Would you like a detailed breakdown of each calculation step?

1. What is the purpose of calculating the mean of $x$ and $y$?
2. How do changes in data points affect the slope and intercept?
3. Why is rounding important in linear regression calculations?
4. How is the sum of squared deviations used in calculating $m$?
5. What are potential sources of error in manual linear regression?

Tip: Linear regression minimizes the sum of the squares of the residuals, which improves prediction accuracy.