REG-3: Solutions — Regression Interpretation and Prediction

Example 1 — Interpreting Slope and Intercept

Full solution with narration is presented directly in the lesson.

Example 2 — Testing (n = 15, r = 0.75)

The full five-step solution is behind the “Show Solution” toggle in the lesson.

Example 3 — Interpolation vs. Extrapolation

Model , fertilizer (g) → yield (kg), observed .

• : → interpolation. kg (reliable).
• : → extrapolation. kg — yield often plateaus at high fertilizer; report only with an explicit caveat.

Example 4 — Find the Error (Age → Reaction Time)

Model , ages ; three errors:

1. Causation language (“aging causes slower reactions”): regression shows association — say “is associated with.”
2. Significance ≠ prediction reliability: means the linear relationship is non-zero, not that any prediction is trustworthy ( within range, but says nothing about extrapolation).
3. Extrapolation as reliable: is outside ; ms is arithmetically right but not a “reliable clinical prediction.”

Problems 1, 2, and 4 now generate a fresh case each time. Their explanation appears after you commit to an answer, so there is no fixed answer key to memorize. Use Try another to practise the same decision with new values.

Problem 3 — Residual Plot Diagnosis

• Random, even scatter around zero supports linearity and equal variance.
• A U-shaped pattern signals non-linearity.
• A fan-shaped spread signals non-constant variance (heteroscedasticity).

Every independent problem is parameterized. Its post-attempt feedback explains the calculation, range decision, significance conclusion, residual diagnosis, or earlier-lesson retrieval point for that specific case.

For a reusable check, ask these questions in order:

1. What does the regression equation predict at the requested x-value?
2. Is that x-value inside the observed range?
3. Does the correlation test support a non-zero linear relationship?
4. What does say about explained variation—and what does it not say about an individual?
5. Do the residuals support a straight-line model with roughly equal spread?

Question 1 — Statistical versus practical significance

A statistically significant relationship may still have weak predictive value. gives the proportion of response variation explained by the model; it is not an individual-accuracy percentage.

Question 2 — Apply

• A slope needs the words predicted and on average; regression alone does not establish causation.
• For a two-tailed correlation test, compare with the critical value. The test statistic keeps the sign of , but the decision uses its absolute value.
• A target inside the observed x-range is interpolation. Even then, check the residual diagnostics.

Question 3 — Error Analysis

Neither nor supplies an individual prediction margin. A prediction interval needs residual spread, leverage at the target x-value, and a critical t-value.

The Boss Fight is a generated data brief. Start with a full recommendation: state the forecast, identify interpolation or extrapolation, assess the correlation result and , then use the residual pattern to qualify your advice. The activity reveals its case-specific reasoning after your attempt and offers a fresh rematch.

The challenge problems also generate fresh cases. Their feedback focuses on three durable ideas:

• An influential point combines leverage (an unusual x-value) with enough pull to change the fitted relationship.
• A larger sample can make the same relationship statistically detectable without making it practically strong.
• describes explained variation across the sample, not the error of one individual prediction.