Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Verify least-squares normal equations in a small regression.
- Justify the conclusion "Residuals=(−1/6,1/3,−1/6); Σr=Σxr=0" using the stated assumptions.
Before you start
Linear regression and dot products.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For points (0,1),(1,2),(2,2), fit y=a+bx.
Why this math matters
Verify least-squares normal equations in a small regression. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The model includes an intercept.
- Each observation has equal weight.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Check a fitted line through orthogonal residuals
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For points (0,1),(1,2),(2,2), fit y=a+bx.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
x̄=1, ȳ=5/3; b=Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²=1/2
Centering separates slope estimation from the intercept.
Work through the mathematics
a=ȳ−bx̄=7/6
The fitted line passes through the sample centroid.
Check the conclusion
Residuals=(−1/6,1/3,−1/6); Σr=Σxr=0
Orthogonality to the intercept and predictor columns certifies the least-squares fit.
The result
Residuals=(−1/6,1/3,−1/6); Σr=Σxr=0
Orthogonality to the intercept and predictor columns certifies the least-squares fit.
Common mistakes to catch
- A fitted line need not pass through every observation.
- Extrapolation accuracy is not guaranteed by a small training residual.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the fitted value at x=3?
Show a hint
Substitute into a+bx.
Reveal answer and explanation
8/3
7/6+3/2=16/6.
Practice 2
Does Σr=0 prove a model is scientifically correct?
Show a hint
It follows mechanically from the fitted intercept.
Reveal answer and explanation
No
This algebraic property does not validate model assumptions or causal interpretation.
Take the idea with you
Distinguish a numerical least-squares check from evidence that a linear model is appropriate.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Verify stationarity through balanced transitions
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