How is the intercept $b_0$ calculated once the slope $b_1$ is known?

Conditions

• The slope $b_1$ has already been computed.
• The sample means $\bar{x}$ and $\bar{y}$ are known.
• The regression line is assumed to pass through $(\bar{x}, \bar{y})$.

Reasoning, step by step

1. Write the general regression equation $\hat{y} = b_0 + b_1x$.
2. Substitute the mean values $x = \bar{x}$ and $\hat{y} = \bar{y}$ into the equation.
3. Substitute the computed slope $b_1$ into the equation.
4. Perform the multiplication to simplify the term $b_1\bar{x}$.
5. Subtract the product from $\bar{y}$ to isolate $b_0$.

Example

The video writes $4 = b_0 + .6(3)$, simplifies it to $4 = b_0 + 1.8$, and subtracts 1.8 from both sides to get $2.2 = b_0$.

Common misconceptions

• Substituting an arbitrary data point instead of the mean point.
• Forgetting to multiply the slope by the x-value before isolating the intercept.
• Assuming the intercept is simply the mean of the y-values.

Watch the explanation

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