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

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 4 — random variables: distributions, expected value, SD, transformations, and the density idea.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 6 · 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-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 (plain-average E(X), "expected = most likely," σ vs. σ², shift-changes-spread, height-as-probability).

Questions, key, and feedback

Q1 (MC). In a tabletop board game, which of the following is a discrete random variable?
- A. The exact number of minutes a full game lasts
- B. The number of spaces a player's token moves on one turn ✅
- C. The exact weight of the game box, in kilograms
- D. The exact temperature of the room where the game is played
Feedback: Spaces moved are separate, countable values you could list (1, 2, 3, …) — discrete. Times, weights, and temperatures are measurements that fill an interval — continuous. Discrete you count, continuous you measure.

Q2 (MC). A store's records for its tablet protection plans show X = the number of claims filed per plan: P(0) = 0.55, P(1) = 0.25, P(3) = 0.05 — and the probability for X = 2 got smudged. What must P(X = 2) be?
- A. 0.15 ✅
- B. 0.85
- C. 0.25
- D. 0.05
Feedback: A legitimate distribution totals exactly 1, so P(2) = 1 − (0.55 + 0.25 + 0.05) = 0.15. (0.85 is the sum of the given three — the step you subtract from 1, not the answer.)

Q3 (MC). Using the completed table from Q2 — P(0) = 0.55, P(1) = 0.25, P(2) = 0.15, P(3) = 0.05 — the expected number of claims per plan, E(X), is:
- A. 1.50
- B. 0.45
- C. 0.70 ✅
- D. 1.00
Feedback: E(X) = 0(0.55) + 1(0.25) + 2(0.15) + 3(0.05) = 0.25 + 0.30 + 0.15 = 0.70. (1.50 is the unweighted average of 0–3 — the probabilities are not optional; 0.45 is P(X ≥ 1); 1.00 is the sum of the probabilities.)

Q4 (MC). A food truck's number of weekly catering bookings X has expected value E(X) = 0.9. Which statement reads that number correctly?
- A. The truck is guaranteed 0.9 bookings every single week
- B. The most likely number of bookings in any week is 0.9
- C. The truck gets exactly 1 booking per week, since 0.9 rounds up
- D. Over many weeks, bookings average out to about 0.9 per week ✅
Feedback: E(X) is a long-run average — not a guarantee, not the most likely value (0.9 bookings can't even happen in a single week), and not something to round. Expected value is what you'd average, not what you'd expect.

Q5 (Multiple answer — select all that apply). Which of the following must be true of every legitimate probability distribution of a discrete random variable X?
- A. Each listed probability is between 0 and 1 ✅
- B. The listed probabilities add up to exactly 1 ✅
- C. The possible values of X are numbers, not category labels ✅
- D. Every possible value of X has the same probability
- E. The expected value E(X) equals one of the listed values of X
Feedback: The two legitimacy rules are (A) and (B), and random variables are by definition numeric (C). Values need not be equally likely (D), and E(X) is a weighted average that often lands on an impossible value (E).

Q6 (Matching). Match each term to its meaning.
| Term | Correct meaning |
|---|---|
| Random variable | A numeric outcome of a chance process — a number chance hasn't decided yet |
| Probability distribution | The complete list of a variable's possible values with the probability of each |
| Expected value E(X) | The long-run average value over many, many repetitions |
| Standard deviation of X | The typical distance between an outcome and the long-run average |
Feedback: The distribution is the variable's whole personality; E(X) and SD(X) are its center and spread — the Week 3 pair, reborn for chance processes.

Q7 (MC). For that same food truck, the bookings variable has variance 0.49. The standard deviation of X is:
- A. 0.49
- B. 0.70 ✅
- C. 0.2401
- D. 0.245
Feedback: SD = √variance = √0.49 = 0.70 bookings. (0.49 is the variance restated — the classic σ vs. σ² slip; 0.2401 squares it again; 0.245 halves it. Variance is scratch work; the SD is the answer you report.)

Q8 (True / False). "Adding the same constant to every value of a random variable increases its standard deviation by that constant."
- True
- False ✅
Feedback: False. Adding a constant slides every outcome equally — the center moves, the gaps don't. Only a multiplier rescales the SD. Adding shifts the center; multiplying stretches both.

Q9 (MC). In a board game, one draw from the move deck advances your token X spaces, where X has mean 2 and SD 0.8. Your score from the draw is Y = 5X + 10 points. What are the mean and SD of Y?
- A. Mean 20 points and SD 4 points ✅
- B. Mean 20 points and SD 14 points
- C. Mean 10 points and SD 4 points
- D. Mean 20 points and SD 0.8 points
Feedback: E(Y) = 5(2) + 10 = 20; SD(Y) = 5(0.8) = 4 — the +10 shifts the mean but never touches the spread. (B added the 10 to the SD; C forgot the shift entirely; D forgot the multiplier rescales spread.)

Q10 (MC). X is a continuous random variable with density curve f(x). Which statement is true?
- A. P(X = a) equals the height of the density curve at a
- B. The total area under the density curve depends on the variable
- C. The probability that X equals any single exact value is 0 ✅
- D. P(X < a) is always strictly smaller than P(X ≤ a)
Feedback: A single point has width 0 — no width, no area, no probability. Height is not probability (A), total area is always exactly 1 (B), and because the boundary point carries nothing, P(X < a) = P(X ≤ a) for continuous X (D).

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item's three correct statements are the only must-be-true options listed; every computed key re-verified in the week's arithmetic script (P(2) = 0.15; E = 0.70; Var 0.49 → SD 0.70 — and 0.49 really is the variance of the 0.30/0.50/0.20 bookings table whose mean is 0.9; 5(2)+10 = 20 and 5(0.8) = 4); no positional pattern in the MC key (B A C D B A C) and no letter carries more than a third of the single-answer items; no length giveaway (options within each item are comparable lengths); no item asserts a fact outside the Week 6 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces (quiz-only surfaces: one board game, one tablet protection plan, one catering food truck).

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

All ten items are tagged week=6 · objective=4 · topic=random-variables and deposited in Item Bank: Week 6 — Random Variables with idents w06q1–w06q10. 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: w06q1 discrete-vs-continuous, w06q2 legitimacy-missing-probability, w06q3 expected-value-computation, w06q4 expected-value-interpretation, w06q5 distribution-rules, w06q6 vocabulary, w06q7 sd-vs-variance, w06q8 shift-invariance, w06q9 linear-transformation, w06q10 continuous-density.)

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        # end of the module's week
published        = true
shuffle_answers  = true