The Regression Line and Making Predictions
Fit the least-squares line through the data, read its gradient and intercept in the units of the problem, and predict only inside the range the data covers.
What a learner can do afterwards
- Interpret the gradient of a fitted line in the units of the situation
- Predict a value for an x inside the range of the data
- Explain why a prediction well outside that range is not supported
1 · Read
A fitted line reads y equals a plus b x. The intercept a is the predicted y when x hits zero. The gradient b is the predicted change in y for each extra unit of x. Both live in the units of the story, such as marks per hour. The line sums up average drift, and no single pupil must obey it.
A class line for marks on hours studied is y equals 5 plus 2 x. For 6 hours, swap in 6 to get 5 plus 12, which is 17 marks. The data cover x from 1 to 8, and 6 sits inside that range, so the value is supported.
Outside the observed range the line guesses into unknown land. Curves and ceilings may lurk there. Kim predicts at x equals 30 and trusts it fully, but 30 sits far past 8. State the value, then judge it: supported inside, unsupported outside.
Read the gradient in story units, predict inside the range, and distrust guesses far outside it.
More to read
13.3 Linear Equations
A straight line model reads y = a + bx, and each symbol earns its keep. The intercept a is the predicted y when x hits zero, while the gradient b is the predicted change in y for each extra unit of x. Both live in the units of the story: pounds per week, marks per hour. Read a fitted equation by translating each number back into the situation. The line is a summary of drift, not a rule any individual must obey.
13.4 The Regression Equation
The regression line is the straight line that hugs the scatter best by least squares. Its gradient comes from the correlation scaled by the two spreads, and it always passes through the mean of x paired with the mean of y. That anchor point is worth memorising: mean meets mean on every regression line. Fit the line, check the scatter hugs it, and only then predict.
12.5 Prediction
Prediction means plugging an x into the line and reading off y, but trust depends on where that x sits. Inside the observed range the line interpolates between real neighbours, which is fair. Outside it the line extrapolates into unknown land where curves and ceilings may lurk. Exam answers earn full marks by stating the value and then judging it: supported inside, unsupported outside.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Optional connections
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.