Week 6 — Quiz (auto-graded) · Random Variables

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objective tested: Objective 4 — work with random variables: discrete vs. continuous; valid probability distributions; expected value; variance & standard deviation; interpretation.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 6.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-06-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 6 misconceptions named in the lecture outline. All arithmetic is pre-computed and double-checked.

Questions, key, and feedback

Q1 (MC). Which of the following is a discrete random variable?
- A. The exact height of a student
- B. The time it takes to run a mile
- C. The number of defective bulbs in a box of 12 ✅
- D. The weight of a suitcase
Feedback: A count of bulbs (0, 1, 2, …) is separate and listable → discrete. Height, time, and weight are measured on a scale → continuous. (Test: count or measure?)

Q2 (MC). Which of the following is a continuous random variable?
- A. The number of siblings a student has
- B. The number of cars a dealership sells in a day
- C. The exact amount of coffee (in ounces) a machine dispenses ✅
- D. The number of heads in 10 coin flips
Feedback: The exact amount dispensed can be any value in a range — it's measured → continuous. The other three are counts → discrete.

Q3 (Multiple answer — select all that apply). Which statements about a valid discrete probability distribution are true?
- A. Every probability is between 0 and 1 (inclusive) ✅
- B. The probabilities add up to exactly 1 ✅
- C. The x-values (the outcomes) must add up to 1
- D. A probability of exactly 0 is allowed for an outcome ✅
- E. A probability may be greater than 1 if the outcome is very rare
Feedback: A, B, and D are correct. It's the probabilities that sum to 1 (not the x-values, so C is false), and no probability may exceed 1 (E is false). A probability of 0 is legal — that outcome simply never happens.

Q4 (MC). A discrete random variable has three possible values. Which table below is a valid probability distribution?
- A. Probabilities 0.3, 0.3, 0.3
- B. Probabilities 0.5, 0.2, 0.3 ✅
- C. Probabilities 0.6, 0.6, −0.2
- D. Probabilities 0.4, 0.4, 0.4
Feedback: B's probabilities sum to exactly 1 (0.5 + 0.2 + 0.3 = 1.00) and each is in [0, 1]. A sums to 0.9 and D sums to 1.2 (both fail the "add to 1" rule); C contains a negative probability and a value above the rest's legal range. (Always check both gates.)

Q5 (MC). A discrete random variable X takes the value 1 with probability 0.2, 2 with probability 0.5, and 3 with probability 0.3. What is the expected value E(X)?
- A. 2.0
- B. 2.1 ✅
- C. 3.0
- D. 6
Feedback: E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1. (A = the middle value 2; D = the sum of the x-values 1+2+3 — neither is the probability-weighted average.)

Q6 (Matching). Match each term to its description.
| Term | Correct description |
|---|---|
| Random variable | A rule that attaches a number to the outcome of a random process |
| Discrete | Values are separate and countable — you can list them |
| Continuous | Values fill an entire interval — you measure them |
| Expected value | The long-run probability-weighted average, Σ [ x · P(x) ] |
Feedback: A random variable turns a chance outcome into a number; discrete = count, continuous = measure; the expected value is the weighted average of the outcomes.

Q7 (MC). A discrete random variable X takes the value −1 with probability 0.5, 0 with probability 0.3, and 2 with probability 0.2. What is E(X)?
- A. −0.1 ✅
- B. 0.1
- C. 0.3
- D. 1
Feedback: E(X) = (−1)(0.5) + (0)(0.3) + (2)(0.2) = −0.5 + 0 + 0.4 = −0.1. (A negative-valued outcome contributes a negative term; expected values can be negative.)

Q8 (True / False). "The expected value of a random variable must be one of the values the variable can actually take."
- True
- False ✅
Feedback: False. E(X) is a long-run weighted average, not a possible outcome. A fair die has E(X) = 3.5, which can never be rolled.

Q9 (MC). A discrete random variable X takes the value 0 with probability 0.2, 1 with probability 0.6, and 2 with probability 0.2. Its mean is E(X) = 1.0 and E(X²) = 1.4. What is the variance of X?
- A. 0.4 ✅
- B. 1.4
- C. 1.0
- D. 0.63
Feedback: Variance = E(X²) − [E(X)]² = 1.4 − (1.0)² = 1.4 − 1.0 = 0.4. (B = E(X²) before subtracting the square of the mean — the classic slip; D ≈ √0.4 is the standard deviation; C is the mean.)

Q10 (MC). A random variable for a daily tip total has E(X) = $50 and standard deviation σ = $8. Which plain-language interpretation is best?
- A. Every day's tip total is exactly $50
- B. Tip totals average about $50 per day, typically landing within roughly $8 of that ✅
- C. The most common tip total is $8
- D. $50 is the largest possible tip total
Feedback: E(X) is the long-run average (about $50), and σ describes the typical swing around it (about $8). It is not a guarantee (A), not the mode (C), and not a maximum (D).

Answer key (quick reference)

Quality gate (self-checked): every single-answer item has exactly one correct option; the multiple-answer item (Q3) lists the three true statements (A, B, D); all arithmetic was pre-computed and verified — Q4 valid table sums to 1.00 (0.5+0.2+0.3), Q5 E(X) = 0.2+1.0+0.9 = 2.1, Q7 E(X) = −0.5+0+0.4 = −0.1, Q9 variance = 1.4 − 1.0 = 0.4 (with E(X)=1.0 from 0.6+0.4 and E(X²)=1.4 from 0.6+0.8). No item asserts a fact outside the Week 6 course definitions.

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

All ten items are tagged course=MATH11 · week=6 · objective=4 · topic=random-variables and deposited in Item Bank: Week 6 — Random Variables. The midterm (Week 8) and the per-term variant updates draw fresh items from this bank. (Tags: q1 discrete-identify, q2 continuous-identify, q3 valid-distribution-rules, q4 valid-distribution-check, q5 expected-value-compute, q6 rv-term-matching, q7 expected-value-negative, q8 expected-value-misconception, q9 variance-compute, q10 interpret-mean-sd.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 6 Quiz — Random Variables"
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"