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All courses Math I · S-ID.7 57 of 59
Interpret slope and intercept of a linear model in the data context.

Interpret the slope of a fitted linear model

Problem
Interpret the slope of S(h)=2.5h+70, where h is study time in hours and S is predicted test score in points.
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Hint

Identify the coefficient of the input and attach output units per input unit.

In y=mx+b, m=Δy/Δx. Interpret its sign and units as a predicted output change for a +1 input-unit change.

Solution walkthrough

01

Identify the slope

\[S(h)=2.5h+70~\text{has}~\text{slope}~2.5\]

The coefficient of study time is the predicted score change per hour.

02

Attach units and direction

\[+2.5~\text{score}~\text{points}~\text{per}~\text{study}~\text{hour}\]

The positive sign means predicted score increases with study time.

03

Use a one-hour difference

\[S(h+1)-S(h)=2.5~\text{points}\]

Increasing study time by one hour raises the model prediction by 2.5 points.

04

State the interpretation

\[\text{The}~\text{predicted}~\text{test}~\text{score}~\text{increases}~\text{by}~2.5~\text{points}~\text{for}~\text{each}~\text{additional}~\text{study}~\text{hour}.\]

This is an association in the fitted model, not a causal guarantee for every student.

+

Another way

  1. Compare the model at h and h+1; the constant cancels and leaves 2.5.

!

Common mistake

Do not confuse slope 2.5 with intercept 70 or claim exact causation.