Week 7 — Quiz (auto-graded) · Binomial & Normal Models

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objectives tested: Objective 4 (binomial random variables — settings, probabilities, mean & SD) · Objective 5 (the normal approximation to the binomial).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 7.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-07-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file. All arithmetic is pre-computed and every single-answer item has exactly one correct option.

Blueprint

No trick questions; distractors target the Week 7 misconceptions named in the lecture outline. All "choose" values and probabilities are pre-computed.

Questions, key, and feedback

Q1 (MC). Which of the following is NOT one of the four conditions for a binomial setting?
- A. Each trial has exactly two outcomes (success / failure)
- B. The number of trials n is fixed in advance
- C. The trials are independent
- D. The probability of success p changes from trial to trial ✅
- Feedback: A binomial requires the same p on every trial; a changing p breaks the setting. The four conditions are B-I-N-S: Binary, Independent, fixed N, Same p.

Q2 (Multiple answer — select all that apply). Which of the following are binomial settings?
- A. Flip a fair coin 10 times and count the heads ✅
- B. Roll a die repeatedly until you get a 6, and count the rolls
- C. Ask 4 randomly chosen students "Do you have a campus job?" and count the "yes" answers ✅
- D. Deal 5 cards from one deck (no reshuffling) and count the aces
- E. Inspect 50 items where each is defective with probability 0.02, and count the defectives ✅
- Feedback: A, C, E meet all four conditions. B has no fixed n ("until a 6"); D is without replacement, so trials aren't independent and p changes. Both fail the setting.

Q3 (MC). A fair coin is flipped 4 times. Let X = the number of heads, so X ~ B(4, 0.5). Using P(X = k) = C(4, k)·(0.5)ᵏ·(0.5)⁴⁻ᵏ with C(4, 2) = 6, what is P(X = 2)?
- A. 0.0625
- B. 0.25
- C. 0.375 ✅
- D. 0.5
- Feedback: P(X = 2) = 6 × (0.5)² × (0.5)² = 6 × 0.0625 = 0.375 (= 6/16 = 3/8).

Q4 (MC). A basketball player makes 50% of her free throws and shoots 3, so X ~ B(3, 0.5). What is the probability she makes all 3 (P(X = 3))? (C(3, 3) = 1.)
- A. 0.375
- B. 0.125 ✅
- C. 0.5
- D. 0.25
- Feedback: P(X = 3) = 1 × (0.5)³ = 0.125 (= 1/8). (Distractor A = P(exactly 2), a common "which count?" slip.)

Q5 (MC). Each of 2 manufactured parts is defective with probability 0.3, independently. Let X = the number defective, X ~ B(2, 0.3). What is P(X = 0) (neither is defective)? (C(2, 0) = 1.)
- A. 0.09
- B. 0.42
- C. 0.49 ✅
- D. 0.7
- Feedback: P(X = 0) = 1 × (0.3)⁰ × (0.7)² = (0.7)² = 0.49. (Distractor D = 0.7 is just (1−p), forgetting the square.)

Q6 (MC). For a binomial X ~ B(64, 0.5), what is the mean number of successes? (Mean = np.)
- A. 16
- B. 32 ✅
- C. 64
- D. 8
- Feedback: Mean = np = 64 × 0.5 = 32.

Q7 (MC). For X ~ B(100, 0.5), the variance is np(1−p) = 100 × 0.5 × 0.5 = 25. What is the standard deviation?
- A. 25
- B. 5 ✅
- C. 50
- D. 10
- Feedback: SD = √(variance) = √25 = 5. ("Expect about 50 heads, give or take about 5." Distractor A = the variance itself, not its square root.)

Q8 (MC). A quiz-app sends 50 reminder emails, each opened with probability 0.2, independently. Let X = the number opened, X ~ B(50, 0.2). What is the expected number opened? (Mean = np.)
- A. 10 ✅
- B. 40
- C. 25
- D. 5
- Feedback: Mean = np = 50 × 0.2 = 10. (Distractor B = 50 × 0.8, using the failure probability by mistake.)

Q9 (True / False). "The standard deviation of a binomial random variable is √(np)."
- True
- False ✅
- Feedback: False. The SD is √(np(1−p)) — the (1−p) factor is essential. √(np) drops it and overstates the spread.

Q10 (MC). The normal approximation to the binomial may be used only when np ≥ 10 AND n(1−p) ≥ 10. For which of the following is the normal approximation NOT appropriate?
- A. n = 100, p = 0.5
- B. n = 50, p = 0.2
- C. n = 200, p = 0.1
- D. n = 30, p = 0.05 ✅
- Feedback: For D, np = 30 × 0.05 = 1.5, far below 10 → fails the rule; use the exact binomial. (A: 50 & 50 ✓; B: 10 & 40 ✓; C: 20 & 180 ✓.)

Answer key (quick reference)

Quality gate (self-checked): each single-answer item (Q1, Q3–Q8, Q10) has exactly one correct option; the multiple-answer item (Q2) lists all three binomial scenarios (A, C, E) and both non-binomial traps (B, D) as distractors; Q9 is a clean True/False. Arithmetic re-verified: B(4,0.5) P(2)=0.375; B(3,0.5) P(3)=0.125; B(2,0.3) P(0)=0.49; mean B(64,0.5)=32; SD B(100,0.5)=5; mean B(50,0.2)=10; normal-approx fails only for B(30,0.05) among the four options. No item asserts a fact outside the Week 7 course definitions.

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

All ten items are tagged course=MATH11 · week=7 · objective=4,5 · topic=binomial-and-normal-models and deposited in Item Bank: Week 7 — Binomial & Normal Models. The midterm (Week 8) and the per-term variant updates draw fresh items from this bank. (Tags: q1 binomial-conditions, q2 binomial-identify, q3 binomial-prob, q4 binomial-prob, q5 binomial-prob, q6 binomial-mean, q7 binomial-sd, q8 binomial-mean, q9 sd-misconception, q10 normal-approximation.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 7 Quiz — Binomial & Normal Models"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start (Sun Oct 18)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"