L-REG-3: Solutions — Regression Interpretation and Prediction

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Example 1 — What This Line Claims

For , where is the daily high temperature (°C, observed from 4 to 29) and is the number of bike-share rentals:

• Slope: For each additional degree of daily high temperature, the predicted number of rentals increases by 18.6, on average.
• Intercept: at . Zero degrees lies outside the observed 4–29 °C, so 412 is an extrapolation, not a supported baseline.
• At 20 °C: interpolation. rentals.
• At 38 °C: extrapolation. , with no data that hot behind it.

Example 2 — Testing

A random sample of students gives between weekly practice hours and quiz score; the pairs are independent and the residual plot shows no curve, no fan and no point pulling the line.

1. and , with .
2. The licence holds: independent pairs and a residual plot with no pattern.
3. .
4. Compute the test statistic:

5. The t-table gives at . Since , reject (two-sided ). There is evidence of a positive linear relationship between practice hours and quiz scores.

Also, : the line explains about 38% of the variation in quiz scores.

Example 3 — Settling the Quartet at

• Set I: defend . No pattern in the residuals, and data on both sides of .
• Set II: do not. The residuals form an arch; the data bend. The observed value at is 9.14.
• Set III: not as it stands. One outlier at tilts an exact line; without it, gives 7.46 at .
• Set IV: do not. Ten points at and one at ; the slope comes from the single far point, and nothing lies near .

Example 4 — Find the Errors

The analysis contains four errors:

1. Causation: an observational regression shows association, not that experience causes better hazard perception.
2. Significance as trust in every prediction: is evidence that , not that every prediction is reliable.
3. Extrapolation: years is far outside 1–25 years, and the result, 103.0, exceeds the maximum score of 100.
4. Deleting an outlier to raise : a real, correctly recorded observation stays in; report the fit with and without it.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — What the Line Claims (representative)

For , with the daily high temperature (°C, observed from 9 to 31) and a garden’s water use (litres): “For each additional degree Celsius of daily high temperature, the predicted water use increases by 73.3 litres, on average. The intercept is an extrapolation: a daily high of 0 °C is outside the observed range.” When the observed range starts at 0, the intercept is a supported baseline instead.

Problem 2 — A Prediction, Step by Step (representative)

For (trunk diameter in cm, observed from 10 to 60; height in m) at : extrapolation (104 is past 60), m, and the honest claim is that no data lie there, so the trend is an assumption.

Problem 3 — Read the Residual Plot

Name what the drawn plot shows: an even band (no pattern), a bend (a curve), a band that widens or narrows steadily (a fan), or one residual far from all the rest (one point).

Problem 4 — Leverage or Influence?

Far out in and on the trend: high leverage, little influence. Far out in and off the trend: likely influential. Off the trend near the middle of : an outlier with low leverage. Confirm influence by refitting without the point.

Problem 5 — Test (representative)

, : , , . Since , fail to reject : not enough evidence of a linear relationship.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — A Forecast Request (representative)

For (practice quizzes, observed from 2 to 14; exam score) at : points, and is just past the edge of the data, so this is extrapolation.

Problem 2 — Test, Then Size (representative)

, : , , reject ; , about 37% of the variation explained.

Contrasting cases. Study A (25 students, ) explains a real share of the variation. Study B (6,200 users, ) is significant mainly because is huge: it explains about 80 times less.

Problem 3 — Find the Error

The error can sit in any step. The reviewed slips: leaving out the intercept, calling a target just past the edge “close enough”, reporting an extrapolation as reliable, reading a small p-value as accurate predictions, reading as a hit rate or as an error margin, seeing a pattern in noise, calling a fan or a curve harmless, taking leverage from , deleting an outlier to raise , and claiming least squares makes every point equally influential.

Problem 4 — What Kind of Unusual Point?

Use two facts per card: how far is from , and how much the slope changes. Near the middle with a big residual: outlier, little influence. Far out and on the trend: high leverage, no influence. Far out and off the trend: influential.

Problem 5 — Which Point Is Holding the Line?

The store at (40, 70). Without it the slope rises from about 1.09 to about 2.03. The store with the largest residual, (13, 68), barely matters (slope 1.11 without it); the store with the smallest , (4, 31), changes it only to 0.97.

Problem 6 — From Summaries to a Decision (representative)

, , , , , : , , and the test gives , so reject .

Mixed Review — L-REG-1 and L-REG-2

L-REG-1: square for , keep the sign for the direction, and refuse the causal claim. L-REG-2: , , , and ; a positive residual means the case is above the line.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Question 1 — Feynman Test

A strong answer says that and measure how tightly points hug the best straight line, not whether a straight line is right; uses the quartet (same , one curve, one outlier, one single-point slope); names the residual plot as the check; and separates “probably not zero” (significance) from “big enough to matter” ().

Question 2 — What the Line Claims

The slope is negative, so the predicted value decreases by per unit of , on average; zero lies outside the observed range, so the intercept is an extrapolation; the forecast is inside the data or outside it depending on the target shown.

Question 3 — What Should the Team Do?

No pattern: use the line inside the data. A curve: fit a curved model first. A fan: expect bigger misses where the band is wide. One point: check that observation before relying on the line.

Question 4 — Which Runner Is Holding the Line?

The runner at (1, 26.5). Without that runner the slope steepens from about −0.38 to about −0.76. The runner with the biggest residual, (8, 23.0), leaves it at about −0.37.

Question 5 — The Test Statistic (representative)

, : , below : not significant.

Question 6 — Significant, but How Big? (representative)

, : the line explains about 77% of the variation (), and with so few pairs it is significant because the relationship itself is strong. With thousands of pairs and , it would be significant mainly because the sample is very large.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Path A — The Forecast Desk

Compute at the target, then decide: far outside the data, or any pattern in the residual plot, means the forecast does not go out; just past the edge with a clean plot means it goes out with a warning; inside with a clean plot means use it. The memo states the target’s position, what the plot shows, and that decision.

Path B — The Quartet Audit

All four branches share , and the line by construction. Diagnose each plot: an even scatter, a curve the line cuts across, one point tilting an exact line, or one far point setting the slope. Only the even-scatter branch can support a forecast inside its data; if the target is beyond every branch’s data, none can.

A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so you know where to look back.

Problem 1 — How Much Is One Point Holding Up the Slope? (representative)

Subtract the point from each sum, then and with the new . In the representative case, , , so without the point against with it: the far point is influential.

Problem 2 — Same Correlation, Four Sample Sizes (representative)

With : , , and for , , and , and throughout. Larger gives more evidence that the correlation is not zero; it does not make the relationship stronger.

Problem 3 — Delete the Outlier? (representative)

A point near the middle of barely moves the slope (here from about 1.435 to 1.468) but drags down (0.931 without it, 0.556 with it). It is an outlier with low leverage; if it is real, keep it and report both fits.