Week 9 — Quiz (auto-graded) · The Normal Distribution

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objective tested: Objective 5 — use normal distributions to reason about variability (empirical rule; z-scores; normal area/percentile; assessing normality).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 9.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-09-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file.

Embedded z-table (cumulative area to the LEFT of z) — supplied to students on any item that needs it; every item is engineered to land on these:

z	−2.0	−1.5	−1.0	−0.5	0	+0.5	+1.0	+1.5	+2.0	+2.5
area to LEFT	.0228	.0668	.1587	.3085	.5000	.6915	.8413	.9332	.9772	.9938

Blueprint

No trick questions; distractors target the Week 9 misconceptions named in the lecture outline. All arithmetic is pre-computed and lands exactly on the embedded table.

Questions, key, and feedback

Q1 (MC). In any normal distribution, about what percentage of the data falls within 1 standard deviation of the mean?
- A. 34%
- B. 68% ✅
- C. 95%
- D. 99.7%
Feedback: The empirical rule, in order: about 68% within 1 SD, 95% within 2, 99.7% within 3. (34% is one half of the 68% band — a common slip.)

Q2 (MC). Exam scores are normal with mean 70 and standard deviation 10. About what percentage of students scored between 50 and 90?
- A. 68%
- B. 95% ✅
- C. 99.7%
- D. 50%
Feedback: 50 and 90 are exactly 2 SDs below and above the mean (70 ∓ 2·10), so this is the within-2-SD band → 95%.

Q3 (Multiple answer — select all that apply). Which of the following statements about a normal distribution are true?
- A. The curve is symmetric about its mean ✅
- B. The total area under the curve equals 1 ✅
- C. A z-score can never be negative
- D. About 95% of the data lie within 2 standard deviations of the mean ✅
- E. The empirical rule applies to any data set, whatever its shape
Feedback: A, B, and D are core facts of the normal curve. A z-score is negative whenever a value is below the mean (C is false), and the empirical rule applies only to roughly normal data (E is false).

Q4 (MC). A value has a z-score of −1.5. What does the negative sign tell you?
- A. A calculation error was made
- B. The value is below the mean ✅
- C. The value itself is a negative number
- D. The data are not normally distributed
Feedback: A z-score's sign is only a direction: negative means the value is below the mean (here, 1.5 SDs below). It is not an error and doesn't mean the value is negative.

Q5 (MC). SAT-style scores are normal with mean 1050 and standard deviation 200. A student scores 1450. What is the student's z-score?
- A. +1.0
- B. +2.0 ✅
- C. +2.5
- D. −2.0
Feedback: z = (1450 − 1050) ÷ 200 = 400 ÷ 200 = +2.0 — two standard deviations above the mean.

Q6 (MC). Exam scores are normal with mean 70 and standard deviation 10. A score of 90 has a z-score of +2.0. Using the embedded table (area to the left of +2.0 = .9772), the score of 90 is at about the —
- A. 23rd percentile
- B. 84th percentile
- C. 98th percentile ✅
- D. 50th percentile
Feedback: The area to the LEFT of a z-score is its percentile: .9772 → about the 98th percentile (precisely 97.72nd). About 98% of students scored at or below 90.

Q7 (MC). Adult heights (in a population) are normal with mean 64 in and standard deviation 2.5 in. A height of 66.5 in has a z-score of +1.0. Using the embedded table (area to the left of +1.0 = .8413), about what fraction of people are taller than 66.5 in?
- A. About 15.87% ✅
- B. About 84.13%
- C. About 50%
- D. About 68%
Feedback: "Taller than" is the area to the RIGHT = 1 − (area to the left) = 1 − .8413 = .1587 ≈ 15.87%. (84.13% is the area to the left — the fraction who are shorter.)

Q8 (Matching). Match each z-score to its cumulative area to the left (from the embedded table).
| z-score | Correct area to the left |
|---|---|
| z = −1.0 | .1587 |
| z = 0 | .5000 |
| z = +1.0 | .8413 |
| z = +2.0 | .9772 |
Feedback: Read each straight off the table. Note the symmetry: the area left of −1.0 (.1587) and the area right of +1.0 are the same value.

Q9 (True / False). "The 68–95–99.7 rule can be applied to any data set, even one that is strongly skewed."
- True
- False ✅
Feedback: False. The empirical rule is a fact about the normal (symmetric, bell-shaped) curve. On skewed data (incomes, home prices), it can be flat wrong — check that the data are roughly normal first.

Q10 (MC). On a biology test (mean 75, SD 5) Maria scores 87.5, giving z = +2.5. On a chemistry test (mean 80, SD 8) Jon scores 88, giving z = +1.0. Relative to their own classes, whose score is more unusual (more impressive)?
- A. Maria — her z-score (+2.5) is larger ✅
- B. Jon — his raw score (88) is higher
- C. They are equally unusual
- D. You cannot compare scores from two different tests
Feedback: Across different distributions you compare z-scores, not raw numbers. Maria's +2.5 is further into the tail than Jon's +1.0, so her score is more unusual — even though Jon's raw 88 is higher. (B is the "bigger raw number wins" trap; the z-score is the great equalizer.)

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item (Q3) lists all three true statements (A, B, D) and two false ones (C, E). All computation is pre-computed and verified against the embedded table — Q2 (50–90 = ±2 SD → 95%), Q5 ((1450−1050)/200 = +2.0), Q6 (left of +2.0 = .9772 → 98th pctile), Q7 (1 − .8413 = .1587), Q8 (table reads). No item asserts a fact outside the Week 9 course definitions.

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

All ten items are tagged course=MATH11 · week=9 · objective=5 · topic=normal-distribution and deposited in Item Bank: Week 9 — The Normal Distribution. The final (Week 16) and the per-term variant updates draw fresh items from this bank. (Tags: q1 empirical-rule-1sd, q2 empirical-rule-band, q3 normal-curve-properties, q4 zscore-sign, q5 zscore-compute, q6 area-left-percentile, q7 area-right, q8 zscore-area-match, q9 normality-misconception, q10 compare-zscores.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 9 Quiz — The Normal Distribution"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start
published       = true
shuffle_answers = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"