Fit a line of best fit (linear regression) to two-variable data, interpret the slope and y-intercept in context, and use the line to make predictions (A.DSR, Data and Statistical Reasoning).

What this topic is asking

This Data and Statistical Reasoning (A.DSR) standard asks you to fit a line to two-variable data (a line of best fit, found by linear regression), to interpret its slope and y-intercept in context, and to use it to predict. The Georgia Milestones EOC tests interpretation heavily, because a fitted line is only useful if you can say what its slope and intercept mean for the situation, and it tests prediction by substituting an $x$-value. It connects directly to the linear-function work: a line of best fit is a linear model, written $\hat{y} = a + bx$ (or $\hat{y} = bx + a$).

What a line of best fit is

A line of best fit is the straight line that comes closest to the points in a scatterplot, summarizing a linear association. It is computed by linear regression, which finds the line minimizing the overall distance from the points (technically the sum of squared vertical distances). On the EOC and its calculator tool, you are usually given the line or generate it with technology, then interpret and use it. The fitted value is written $\hat{y}$ ("y-hat") to distinguish a prediction from an actual data value.

Interpreting the slope and intercept

Because a line of best fit is a linear model, its parameters carry the same meanings as any line, stated in context.

• Slope $b$: the predicted change in $y$ for each one-unit increase in $x$, a rate. For $\hat{y} = 60 + 4x$ (hours studied versus score), the slope 4 means each extra hour is associated with about a 4-point higher predicted score.
• y-intercept $a$: the predicted $y$ when $x = 0$. Here, 60 is the predicted score for 0 hours studied.

Making predictions

To predict, substitute an $x$-value into the line.

Interpolation versus extrapolation

Predictions are trustworthy only near the data.

• Interpolation: predicting within the range of the observed $x$-values. Generally reliable.
• Extrapolation: predicting far outside the range. Risky, because the linear trend may not continue.

If the data covered 0 to 8 hours, predicting at 6 hours is interpolation (safe), but predicting at 40 hours is extrapolation (the model likely breaks down).

How the Milestones examines this topic

• Multiple choice. Identify what the slope or y-intercept of a fitted line represents.
• Numeric entry. Predict a $y$-value by substituting an $x$-value into the line.
• Constructed response. Interpret slope and intercept in context and make a prediction, noting reliability.

Why "associated with," not "causes"

The careful wording of a regression interpretation is itself tested, and the reason is important. A line of best fit describes an association in the data, not a cause-and-effect mechanism. Saying the slope means each extra hour "is associated with" a 4-point higher score is accurate; saying it "causes" a 4-point increase claims more than the data support, because other factors (prior knowledge, sleep, motivation) could be driving both. This is the same correlation-is-not-causation caution covered in the next topic, and the EOC enforces it by rewarding hedged language ("predicted," "associated with," "on average") and penalizing causal overstatement. Building the habit of describing what the line predicts, rather than what it makes happen, keeps your interpretations defensible.

Why extrapolation is dangerous

Extrapolation fails because a line of best fit is only known to summarize the trend where there is data. Beyond that range, nothing guarantees the relationship stays linear: a plant's height might grow linearly for a few weeks but then level off, so a line fitted to early data would absurdly predict a giant plant after a year. The model has no information about what happens outside the observed values, so a prediction far beyond the data is an unjustified guess. On the EOC, recognizing that a prediction at an extreme $x$-value is extrapolation, and therefore unreliable, is a common reasoning point, and it is why interpolation (staying within the data) is the safe kind of prediction.

Try this

Q1. For $\hat{y} = 12 + 3x$ (years experience versus salary in thousands), interpret the slope. [1 point]

• Cue. Each additional year of experience is associated with about a $3000 higher predicted salary.

Q2. Data cover $x$ from 1 to 10. Is predicting at $x = 25$ interpolation or extrapolation? [1 point]

• Cue. Extrapolation (far outside the data), so unreliable.