Week 8 — Quiz (auto-graded) · The Normal Distribution

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 5 (normal-distribution portion) — density curves, the empirical rule, z-scores, forward & inverse normal calculations, assessing normality.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 8 · 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-08-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file. Any z-table areas an item needs are stated in the item — no outside table required.

Blueprint

No trick questions; distractors target the Week 8 misconceptions named in the lecture outline (wrong tail, rule-without-the-bell, z-sign misreads, numeric-therefore-normal).

Questions, key, and feedback

Q1 (MC). A density curve is drawn for a large set of measurements. Which statement must be true?
- A. The total area under the curve equals the number of measurements collected
- B. The area under the curve between two values equals the proportion of the data in that interval ✅
- C. The curve's height at a value equals the proportion of the data exactly at that value
- D. The curve must be bell-shaped and symmetric to qualify as a density curve
Feedback: Density curves carry proportion as area: total area 1, and area over an interval = fraction of the data there. Height is never proportion, and plenty of density curves (like a uniform rectangle) aren't bells.

Q2 (MC). A brand of AA batteries has lifetimes that are approximately normal with mean 100 hours and SD 8 hours. According to the empirical rule, about 68% of these batteries last between —
- A. 84 and 116 hours
- B. 96 and 104 hours
- C. 92 and 108 hours ✅
- D. 76 and 124 hours
Feedback: 68% pairs with 1 SD: 100 ± 8 → 92 to 108 hours. (84–116 is the 2-SD band — that one holds about 95%.)

Q3 (MC). A tire model's tread life is approximately normal with mean 60,000 miles and SD 4,000 miles. One tire wore out at 55,000 miles. Its z-score is —
- A. z = −1.25 — the tire wore out 1.25 standard deviations below the mean ✅
- B. z = 1.25 — the tire wore out 1.25 standard deviations above the mean
- C. z = −5,000 — the tire wore out 5,000 standard deviations below the mean
- D. z = −0.80 — the tire wore out 0.80 standard deviations below the mean
Feedback: z = (55,000 − 60,000) ÷ 4,000 = −5,000 ÷ 4,000 = −1.25. Subtract first, then divide by the SD; keep the sign (below the mean). (−5,000 forgot to divide; −0.80 divided the SD by the distance.)

Q4 (MC). Maya is 69 inches tall; women's heights in her region are approximately N(64, 2.5). Her brother Theo is 74.5 inches tall; men's heights are approximately N(70, 3). Who is relatively taller within their own distribution?
- A. Theo, because 74.5 inches is a greater height than 69 inches
- B. Theo, because his z-score of 1.5 is greater than zero
- C. Neither — z-scores cannot compare values from two different distributions
- D. Maya, because her z-score of 2.0 is greater than Theo's z-score of 1.5 ✅
Feedback: Maya: (69 − 64)/2.5 = 2.0. Theo: (74.5 − 70)/3 = 1.5. Standardizing puts both heights on the shared SD ruler — and comparing across different distributions is exactly what z-scores are for.

Q5 (MC). A bottling machine fills bottles with volumes approximately N(500 mL, 4 mL). Using the course z-table (the area to the left of z = 1.5 is 0.9332), what proportion of bottles hold less than 506 mL?
- A. 0.0668
- B. 0.9332 ✅
- C. 0.8413
- D. 0.1587
Feedback: z = (506 − 500)/4 = 1.5, and "less than" is the left-tail area — read it straight off: 0.9332. (0.0668 is the wrong tail — the area above.)

Q6 (Multiple answer — select all that apply). Which statements are true of every normal distribution?
- A. It is symmetric about its mean ✅
- B. The total area beneath the curve equals 1 ✅
- C. About 68% of its values lie within 1 SD of the mean ✅
- D. It is skewed toward its larger values
- E. Every one of its values lies within 3 SDs of the mean
Feedback: Symmetry, total area 1, and the empirical rule hold for every normal curve. Skew contradicts symmetry — and 99.7% within 3 SDs is not 100%; the tails go on forever.

Q7 (MC). A tire manufacturer's tread life is approximately N(60,000 miles, 4,000 miles). It wants to advertise a warranty mileage low enough that only about 2.28% of tires wear out before it. The warranty mileage should be —
- A. 68,000 miles — two standard deviations above the mean
- B. 56,000 miles — one standard deviation below the mean
- C. 58,632 miles — 2.28 percent of the way down from the mean
- D. 52,000 miles — two standard deviations below the mean ✅
Feedback: "Only 2.28% wear out before it" means left-tail area 0.0228 → z = −2 → x = 60,000 − 2(4,000) = 52,000 miles. Inverse problems walk from percent to z to value — never "percent of the mean."

Q8 (True / False). "For any dataset — whatever its shape — about 95% of the values fall within 2 standard deviations of the mean."
- True
- False ✅
Feedback: False. The 68–95–99.7 figures are properties of the normal curve, not of data in general — strongly skewed data can miss them badly. IF bell-shaped, THEN 68–95–99.7.

Q9 (MC). A technician reviews 200 recorded fill volumes from an aging bottling machine. The histogram is strongly right-skewed, with several extreme high values. The best conclusion is —
- A. A normal model fits poorly here, so z-based percentage claims would be unreliable ✅
- B. A normal model applies anyway, because 200 measurements is a large sample
- C. A normal model applies automatically, because volume is numeric and continuous
- D. The empirical rule still guarantees that 95% of the volumes lie within 2 SDs
Feedback: Shape is an empirical fact: heavy skew breaks the model and every percentage built on it. Sample size never straightens skew (more skewed data just draws the skew more sharply), and "numeric" is not "normal."

Q10 (Matching). Match each region of a normal distribution to the approximate proportion of values it contains.
| Region | Correct proportion |
|---|---|
| Within 1 SD of the mean | About 68% |
| Within 2 SDs of the mean | About 95% |
| Within 3 SDs of the mean | About 99.7% |
| Below the mean | Exactly 50% (symmetry) |
Feedback: The empirical rule's three bands — 68, 95, 99.7 — pair with 1, 2, and 3 SDs; and symmetry puts half the curve below the mean, always.

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 A D B · D · A) and no letter carries more than two of the seven single-answer MC items; option lengths within each item are comparable (no length giveaway); every numeric claim (92–108; z = −1.25; z = 2.0 vs 1.5; 0.9332; 52,000; the 68/95/99.7/50 map) is re-verified in tools/checks/w08_math.py; any table value an item needs is stated in the stem; no item asserts a fact outside the Week 8 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces (different contexts and numbers throughout).

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

All ten items are tagged week=8 · objective=5 · topic=normal-distribution and deposited in Item Bank: Week 8 — The Normal Distribution with idents w08q1–w08q10. 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: w08q1 density-area, w08q2 empirical-rule, w08q3 z-score, w08q4 z-comparison, w08q5 forward-normal, w08q6 normal-properties, w08q7 inverse-normal, w08q8 rule-requires-normal, w08q9 assessing-normality, w08q10 empirical-rule-map.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 8 Quiz — The Normal Distribution"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true