Interpreting a regression model: slope, r², extrapolation

Warm-up

Three quick questions from earlier lessons.

Discover

$\widehat{\text{price}} = 2 + 0.5\,\text{weight}$ for bags of rice (dollars, kg). What does 0.5 mean?

Each extra kg adds 0.5 dollars to the predicted price
Every bag costs exactly 0.5 dollars per kg

Key idea

Interpret in context, and know the limits.

Slope: the predicted change in

y

for each one-unit increase in

x

. Intercept: the predicted

y

when

x = 0

(meaningful only if

x = 0

makes sense).

r^2

: the fraction of the variation in

y

explained by the linear model. Predicting outside the range of the data is extrapolation and is unreliable.

Worked example

$r = 0.9$ for study hours and scores.

r^2 = 0.81

Square r.

81% of the variation in scores is explained by the linear model on hours

Interpretation.

19% is not explained

The rest is other factors and chance.

Your turn, guided

Your turn: $r = -0.7$ . Find $r^2$ .

0.49

(−0.7)².

Practice ★

A regression has $r = 0.2$ . What fraction of the variation in $y$ is explained by the linear model ( $r^2$ )?
A regression has $r = 0.8$ . What fraction of the variation in $y$ is explained by the linear model ( $r^2$ )?

For trees, $\widehat{\text{height}} = 3 + 8\,\text{age}$ (feet, years). What does $8$ mean?
1. For each extra year of age, the predicted height increases by 8 feet
2. Every tree grows exactly 8 feet each year
3. A tree of age 0 is 8 feet tall

Watch out!

Practice ★★

A model for children aged 2 to 6 predicts height from age. Is it reasonable to use it for a 40-year-old?
1. No: 40 is far outside the ages used to fit the model (extrapolation)
2. Yes, if r is close to 1
3. Yes: a line can be extended forever

Exit ticket

Three questions to check you have got it.

A regression has $r = 0.2$ . What fraction of the variation in $y$ is explained by the linear model ( $r^2$ )?
A regression has $r = 0.8$ . What fraction of the variation in $y$ is explained by the linear model ( $r^2$ )?
A regression has $r = 0.4$ . What fraction of the variation in $y$ is explained by the linear model ( $r^2$ )?

Summary card

Interpreting regression

Slope: predicted change per unit of $x$ .
$r^2$ : share of variation explained.
No extrapolation.

Watch out: $r^2$ , not $r$ , is the explained fraction.

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