Unit 8 · Statistics & ProbabilityS3 · Scatterplots & Models · Telemetry

Scatterplots, Models & Study Design

Interpret slope, residuals, association, random samples, and experiments.

28 min
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7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

residual: Observed value minus predicted value.
correlation: Strength and direction of a linear association.
random assignment: A method that supports cause-and-effect conclusions.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Random sampling supports generalizing to a population. Random assignment supports causation. For a line of best fit, interpret slope with both units and use the model only in a sensible input range.

Key idea

The decision that matters

A model prediction is not the observed value. Their vertical difference, observed minus predicted, is the residual.

Residual

\text{residual}=y-\hat y=\text{observed}-\text{predicted}

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

A model predicts 42 when $x=6$ , but the observed value is 39. The residual is $39-42=-3$ .

Explore in Desmos

Fit a line and inspect a residual

Enter the lists and regression exactly; use a tilde on line 3. Read m and b, evaluate the model at x=4, and subtract that prediction from the observed value 15. Then change the last y-value from 19 to 24 and watch how the fitted slope changes.

The calculator is live: change anything and see what happens.

Fit a line and inspect a residual

Degree mode

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Cue	Best move
Model error	observed minus predicted
Causal claim	look for random assignment

Trap

Claiming causation from correlation

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Compute a residual

Problem

A line of best fit predicts 52.4 centimeters for a plant. The observed height is 49.1 centimeters. Find the residual.

Residual means observed minus predicted.

$49.1-52.4=-3.3$ centimeters.

Which subtraction gives the residual?

The negative sign means the point lies below the model line.

Why is the residual negative?

Worked example

Example 2: Separate sampling from assignment

Problem

Researchers randomly assign 80 volunteers to use either app A or app B, then compare completion times. What conclusion can the design support?

Random assignment balances lurking variables between treatment groups.

A difference in average completion time can support a cause-and-effect claim for these volunteers.

What conclusion does this experimental design support?

Because volunteers were not randomly sampled, extending the result to all people requires caution.

Why does random assignment support that conclusion?

Faded example · you finish it

Example 3: You finish it

Problem

A model predicts 27 when the observed value is 31. Complete residual = observed minus predicted.

Evaluate the model first, then subtract prediction from observation.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.