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2.8 Least Squares Regression

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Computing the LSRL from Summary Statistics

Given ${\bar{x}}$ , ${\bar{y}}$ , ${s_x}$ , ${s_y}$ , and ${r}$ , you can compute the LSRL:

Step 1: Compute the slope:

b = r \cdot \frac{s_y}{s_x}

Step 2: Compute the y-intercept:

a = \bar{y} - b\bar{x}

Step 3: Write the equation:

\hat{y} = a + bx

Example:

{\bar{x} = 10}

{\bar{y} = 35}

{s_x = 2}

{s_y = 8}

{r = 0.75}

b = 0.75 \cdot \frac{8}{2} = 3.0

a = 35 - 3.0(10) = 35 - 30 = 5

\hat{y} = 5 + 3x

Verification: Plug in ${x = \bar{x} = 10}$ : ${\hat{y} = 5 + 3(10) = 35 = \bar{y}}$ ✓

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