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Grade 10 · Inspect regression residuals · 37 of 750

Inspect regression residuals · practice 4

For inputs [1, 2, 3, 4, 5] and corresponding outputs [7.75, 6.5, 9, 11.5, 10.25], find the least-squares slope, mean output, and sum of squared residuals. Follow the calculation, then test the quantities in the animated example.

Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.

01 · Make a prediction

Your practice question

For inputs [1, 2, 3, 4, 5] and corresponding outputs [7.75, 6.5, 9, 11.5, 10.25], find the least-squares slope, mean output, and sum of squared residuals.

Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.

02 · Explore the model

See the mathematical relationship move.

Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.

Watch the relationship

Paused
Inspect regression residuals · practice 4. Points revealed: 1 of 5. Full-data least-squares slope: 1. Full-data mean output: 9. Correlation: 0.8Points, least-squares line, and residuals-8-4048-40040xRead numeric axes; drawing scales differ.
Data are constructed, not observations. The residual pattern is fixed and c≥0. Axes use x∈[−8,8], y∈[−40,40]. Full-data correlation is undefined when every output is equal; no causal or population inference is made.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Points revealed
1 of 5
Full-data least-squares slope
1
Full-data mean output
9
Correlation
0.8

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

The mathematical idea

A scatter plot separates a trend from the deviations around it. These five constructed points have residuals whose sum and linear trend both vanish, so y=ax+b is their exact least-squares line. Correlation measures linear association and cannot, by itself, establish cause. Starting quantities: Trend slope = 1; Trend intercept = 6; Residual scale = 0.75.

yᵢ = axᵢ+b+c·eᵢ; e=(1,−2,0,2,−1), x=(1,2,3,4,5)

03 · Reflect and transfer

Explain what changes and why.

Why does the same fitted slope allow different amounts of scatter around the fitted line?

This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.