LEARN · EXPLAIN · REVISE

An observation at x=2 has y=2t+5. Find its residual under ŷ=3+tx.

Read the idea, work independently, then explain what changed.

高三選擇性必修第三册(A版).pdf · 8.2 · PDF 110 / printed page 105

TOPIC 01

Linear regression and applications

Fit a least-squares line and distinguish prediction, residual and extrapolation.

Fit a least-squares line and distinguish prediction, residual and extrapolation.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

t=9

Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the paired-data summaries, preserving which observations belong together.

Hint 2

Use this intermediate relation.

e=y-\hat y

Worked solution

The requested value is 2.

Checks and common pitfalls: A positive residual means the observation is above the fitted line.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Fit a least-squares line and distinguish prediction, residual and extrapolation.
Which condition is essential in linear regression and applications?
Must the fitted line pass through every point?

Defining relation: Fit a least-squares line and distinguish prediction, residual and extrapolation.
$\hat y=a+bx;\quad b=S_{xy}/S_{xx};\quad a=\bar y-b\bar x$
Conditions: Sxx>0; extrapolation beyond observed x needs separate justification and predictions are not certain outcomes.