Grade 10 · Inspect regression residuals · 719 of 750
Inspect regression residuals · practice 78
For inputs [1, 2, 3, 4, 5] and corresponding outputs [5.75, -0.5, 4.5, 9.5, 3.25], 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 [5.75, -0.5, 4.5, 9.5, 3.25], 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.
A starting point
The input mean is 3 and the sum of squared input deviations is 10.
Work through the reasoning
Step 1
Translate the givens
Mean output = (5.75 + -0.5 + 4.5 + 9.5 + 3.25)/5 = 4.5; mean input = 3.
Step 2
Calculate with the model
The cross-deviation sum is 5 and the squared-input-deviation sum is 10, so slope = 5/10 = 0.5; intercept = 4.5 − 3 × (0.5) = 3.
Step 3
Check and interpret
Residuals from y = 0.5x + (3) are [2.25, -4.5, 0, 4.5, -2.25]. Their squares sum to 50.625. The sum of residuals and their input-weighted sum are both zero.
The answer
Slope = 0.5; mean output = 4.5; residual sum of squares = 50.625.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
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
PausedHD animation studio
From experiment to screen.
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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 = 0.5; Trend intercept = 3; Residual scale = 2.25.
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.