Predicted, On Average, Inside the Range

You'll learnwhat the hat on a predicted value means, the four elements a slope interpretation is graded on, when the intercept means something and when it is an extrapolation in disguise, how to make a prediction, and why the arithmetic keeps working long after the prediction stops meaning anything.

A regression line turns an association into a prediction rule: score = 58.5 + 3.5 · hours. The two sentences that interpret it are graded on two words each — predicted, and on average — and this page lets the data do the grading: the line says 83 at seven hours, and the seven-hour student scored 86, so "will score 83" is refuted by the very cloud the line came from. Then the other boundary: at 20 hours the equation returns 128.5 points, on an exam whose maximum is 100. The arithmetic is fine. The prediction is worthless. Where those two facts part company is what this topic is about.

8 STEPS · 6 QUICK CHECKS · THE HAT · THE SLOPE · THE INTERCEPT · EXTRAPOLATION · v1

predicted 83, actual 86 · each hour moves the prediction exactly 3.5 · at 20 hours the equation says 128.5 of 100

Before you start

What you're looking at

The eight students' cloud and the least-squares line through it, used the way a line is actually used: to predict.

The question

What exactly does 58.5 + 3.5 · hours promise — for whom, how precisely, and over what range of hours?

Watch for

The gap between the line and the points. Every interpretation error in this topic is a way of talking as if that gap were zero.

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