The least-squares regression line

Warm-up

Three quick questions from earlier lessons.

Discover

If $r = 1$ , every point is on a line. A student 1 SD above the mean in $x$ is then 1 SD above the mean in $y$ . If $r = 0.5$ , how many SDs above the mean in $y$ would you predict?

0.5 SD
1 SD

Key idea

The line that minimizes the sum of squared residuals.

Least-squares regression line

\hat{y} = a + bx

with slope

b = r\,\dfrac{s_y}{s_x}

and intercept

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

; it always passes through

(\bar{x}, \bar{y})

.

Worked example

$r = 0.6$ , $\bar{x} = 10$ , $s_x = 2$ , $\bar{y} = 50$ , $s_y = 8$ .

Step 1The data.
Step 2The point of means (x̄, ȳ) = (10, 50).
Step 3The line ŷ = 26 + 2.4x passes through it.
Step 4Residuals: no other line makes the sum of their squares smaller.

b = 2.4

0.6 × 8/2.

a = 26

50 − 2.4 × 10.

\hat{y} = 26 + 2.4 \times 12 = 54.8

At x = 12.

Your turn, guided

Your turn: $r = 0.5$ , $s_x = 4$ , $s_y = 6$ . The slope.

b = 0.75

0.5 × 6/4.

Practice ★

$r = 0.6$ , $s_x = 15$ and $s_y = 16$ . Find the slope of the least-squares regression line of $y$ on $x$ .
$r = 0.5$ , $s_x = 11$ and $s_y = 17$ . Find the slope of the least-squares regression line of $y$ on $x$ .

The regression line is $\hat{y} = 27 + 9x$ . Predict $y$ for $x = 7$ .

Watch out!

Practice ★★

The least-squares line has slope $2$ , and $\bar{x} = 37$ , $\bar{y} = 52$ . Find its intercept.
The least-squares line has slope $5$ , and $\bar{x} = 28$ , $\bar{y} = 53$ . Find its intercept.

Boss problem. $r = 0.8$ , $\bar{x} = 15$ , $s_x = 4$ , $\bar{y} = 42$ , $s_y = 8$ .

Exit ticket

Three questions to check you have got it.

$r = 0.6$ , $s_x = 15$ and $s_y = 16$ . Find the slope of the least-squares regression line of $y$ on $x$ .
$r = 0.5$ , $s_x = 11$ and $s_y = 17$ . Find the slope of the least-squares regression line of $y$ on $x$ .
$r = 0.9$ , $s_x = 19$ and $s_y = 12$ . Find the slope of the least-squares regression line of $y$ on $x$ .

Summary card

Least-squares line

$b = r\,s_y/s_x$ , $a = \bar{y} - b\bar{x}$ , through $(\bar{x}, \bar{y})$ .

Watch out: $s_y$ on top.

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