Week 4 — Quiz (auto-graded) · Relationships Between Two Variables

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 3 — scatterplots; correlation r; two-way tables & conditional distributions; association vs. causation.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 4 · Closed to AI.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-04-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file.

Blueprint

No trick questions; distractors target the Week 4 misconceptions named in the lecture outline.

Questions, key, and feedback

Q1 (MC). A used-car marketplace wants to use a car's age to predict its asking price. Which statement assigns the variable roles correctly?
- A. Price is explanatory and age is the response, because price matters most
- B. Age is explanatory and price is the response — age goes on the x-axis ✅
- C. Both are response variables, because both were measured on every car
- D. Age is the response variable because it happens first chronologically
Feedback: The predictor is the explanatory variable (x); the outcome being predicted is the response (y). "Use age to predict price" makes age explanatory. (x explains, y responds.)

Q2 (MC). On that marketplace's scatterplot of 60 cars, the dots run downhill from left to right, stay fairly close to a straight line, and show no stragglers. The best description is —
- A. A weak positive linear association between age and price
- B. A strong curved association with several influential outliers
- C. A fairly strong negative linear association between age and price ✅
- D. No association, because the dots do not fall exactly on one line
Feedback: Downhill = negative; near a line = linear; close to the line = strong. And real data never falls exactly on a line — that's not required for an association.

Q3 (MC). A bakery tests 30 cakes across a range of oven temperatures and scores each cake's texture. The scatterplot is a clear arch — texture improves, peaks, then worsens — and technology reports r = 0.03. What should the bakery conclude?
- A. Temperature and texture are unrelated, since r is almost exactly zero
- B. The data must contain an entry error, because patterns require large r
- C. Texture improves about 3% for each extra degree of oven temperature
- D. There is a strong relationship, but a curved one that r cannot measure ✅
Feedback: r speaks one language: straight lines. An arch's uphill half and downhill half cancel, leaving r near 0 while the relationship stays strong. Plot first, r second.

Q4 (Multiple answer — select all that apply). Which statements about the correlation coefficient r are true?
- A. r is always between −1 and +1 ✅
- B. r = −0.85 indicates a stronger linear association than r = +0.55 ✅
- C. r is measured in the same units as the response variable
- D. A positive r means the scatterplot's pattern runs uphill, left to right ✅
- E. An r very close to 1 proves that one variable causes the other
Feedback: Sign = direction, size = strength — so −0.85 beats +0.55 on strength. r has no units at all, and no value of r, however impressive, earns a causal claim.

Q5 (MC). A dog groomer finds r = 0.72 between dogs' weights (pounds) and grooming times (minutes). If the grooming times are converted from minutes to hours and r is recomputed, the new value will be —
- A. Exactly 0.72, unchanged, because r carries no units at all ✅
- B. 0.72 divided by 60, because every time value shrinks by that factor
- C. Larger than 0.72, because hours make the numbers more compact
- D. Impossible to determine without recomputing from the raw data
Feedback: r is unitless — rescaling a variable (minutes → hours, pounds → kilograms) changes r not at all. Only the story-changing things (outliers, different data) move r.

Q6 (Matching). Match each correlation value to the scatterplot it best describes.
| r value | Correct description |
|---|---|
| r = −0.92 | A tight downhill cloud — a strong negative linear association |
| r = −0.15 | A near-shapeless spray with the barest downhill lean |
| r = +0.55 | A clear but loose uphill trend — a moderate positive association |
| r = +0.98 | Dots hugging an uphill line almost perfectly |
Feedback: Work sign first (uphill or downhill?), then size (how far from 0?). The two negatives differ only in strength — that's the pair to slow down on.

Q7 (MC). A harbor ferry logs 200 crossings' passengers: of the 120 weekday passengers, 30 brought a vehicle; of the 80 weekend passengers, 40 brought a vehicle. What percent of weekend passengers brought a vehicle?
- A. 20%, dividing the 40 weekend vehicles by all 200 passengers
- B. 35%, because 70 of the 200 passengers overall brought vehicles
- C. 50%, dividing the 40 weekend vehicles by the 80 weekend passengers ✅
- D. 57%, dividing the 40 weekend vehicles by all 70 vehicle-bringers
Feedback: "Among weekend passengers" fixes the denominator: that group's total, 80. So 40 ÷ 80 = 50%. Option A is the joint percent; B is the marginal; D conditions on the wrong variable.

Q8 (MC). To decide whether travel mode is associated with day type on that ferry, which comparison settles it?
- A. Compare the total number of weekday passengers to the total number of weekend passengers
- B. Compare the percent bringing vehicles among weekday passengers to the percent among weekend ones ✅
- C. Compare the count in the largest single cell to the table's overall grand total
- D. Check whether the four cell counts of the table are all exactly equal
Feedback: Association means the conditional distributions differ across groups — 25% of weekday vs. 50% of weekend passengers bringing vehicles is exactly such a difference. Margins alone can't show it.

Q9 (True / False). "A strong correlation between two variables is, by itself, proof that changing one variable will change the other."
- True
- False ✅
Feedback: False. A strong r measures the handshake, not a push: the arrow could run either way, or a lurking variable could drive both. Only a randomized experiment supports a causal claim.

Q10 (MC). Across a region's fires, the more firefighters sent to a fire, the greater the property damage. The best explanation is —
- A. A lurking variable — the size of the fire — drives both the firefighter count and the damage ✅
- B. Firefighters cause property damage, so cities should dispatch fewer of them
- C. Property damage causes firefighters, so the correlation runs backward here
- D. The association must be coincidence, because firefighters reduce damage
Feedback: Big fires summon many firefighters AND destroy more property — the classic third-hand. The association is real; the causal reading of it is not.

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item's three true statements are the only true options listed; no positional pattern in the key (B C D · A · C B · A) and no letter exceeds 2 of the 7 single-answer MC items; option lengths within each item are comparable (key/distractor mean-length ratio inside 0.55–1.45, machine-checked); the ferry arithmetic is verified in tools/checks/w04_math.py (40/80 = 50%, 30/120 = 25%, 70/200 = 35%, 40/200 = 20%, 40/70 ≈ 57%); no item asserts a fact outside the Week 4 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces.

Item-bank entries (for variants + the midterm/final)

All ten items are tagged week=4 · objective=3 · topic=relationships-two-variables and deposited in Item Bank: Week 4 — Relationships Between Two Variables with idents w04q1–w04q10. The midterm (Week 9), the final (Week 18), and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w04q1 explanatory-response, w04q2 scatterplot-description, w04q3 r-linear-only, w04q4 r-properties, w04q5 r-unitless, w04q6 r-strength-matching, w04q7 conditional-distribution, w04q8 association-conditionals, w04q9 correlation-causation, w04q10 lurking-variable.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 4 Quiz — Relationships Between Two Variables"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true