Week 10 — Quiz (auto-graded) · Sampling Distributions & the Central Limit Theorem

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 5 — sampling distributions of x̄ and p̂; the standard error; the Central Limit Theorem.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 10 · 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-10-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 10 misconceptions named in the lecture outline (wrong ruler, σ vs. σ², wrong tail, n-vs-number-of-samples, the missing √).

Questions, key, and feedback

Q1 (MC). A support center's manager repeatedly imagines drawing random samples of 25 calls and recording each sample's mean hold time. The sampling distribution of the sample mean is best described as —
- A. The 25 individual hold times recorded in one manager's sample
- B. The population of hold times for every call the center receives
- C. The distribution of sample-mean values across all possible samples of 25 calls ✅
- D. The range between the shortest and longest hold time in one sample
Feedback: One entry per SAMPLE — the pile of x̄'s across all possible samples of size 25. Options A and B are the other two of the week's "three distributions" (one sample; the population).

Q2 (MC). The center's true mean hold time is μ = 200 seconds. For random samples of n = 25 calls, the sampling distribution of x̄ has its center at —
- A. 8 seconds, the population mean divided by n
- B. 40 seconds, the population mean divided by √n
- C. A value knowable only by listing every possible sample
- D. 200 seconds — the same as the population mean μ ✅
Feedback: x̄ is unbiased: its sampling distribution is centered exactly at μ. (Dividing by n or √n is what happens to the spread — never the center.)

Q3 (MC). Hold times at the center have a standard deviation of σ = 20 seconds. For random samples of n = 25 calls, the standard error of the sample mean is —
- A. 20 seconds
- B. 4 seconds ✅
- C. 0.8 seconds
- D. 16 seconds
Feedback: SE = σ/√n = 20/√25 = 20/5 = 4 seconds. A uses σ alone (the individuals' ruler); C divides by n instead of √n; D is σ²/n = 400/25 (the σ-vs-σ² slip).

Q4 (MC). To cut that standard error in half, the manager's sample size must become —
- A. Twice as large
- B. Four times as large ✅
- C. Half as large
- D. Larger by exactly 25 calls
Feedback: n sits under a square root, so halving the SE takes √n twice as big — n four times as big (25 → 100). Doubling n only divides the SE by √2 ≈ 1.41.

Q5 (Multiple answer — select all that apply). Which statements about the Central Limit Theorem and sampling distributions are true?
- A. With a large enough sample size, the sampling distribution of x̄ is approximately normal even if the population is strongly skewed ✅
- B. Taking a very large sample changes the shape of the population itself to normal
- C. The sampling distribution of x̄ is centered at the population mean μ ✅
- D. The Central Limit Theorem applies only when the population is already normal
- E. The standard error of x̄ gets smaller as the sample size grows ✅
Feedback: A, C, E are the theorem's content: bell shape for means (any population), center μ, spread σ/√n shrinking with n. B is the classic misread (the population never changes); D is backwards — the CLT matters precisely when the population isn't normal.

Q6 (MC). Hold times: μ = 200 seconds, σ = 20 seconds. For a random sample of n = 25 calls, what is the probability the sample mean exceeds 206 seconds? (The area to the left of z = 1.50 under the standard normal curve is 0.9332.)
- A. 0.0668 ✅
- B. 0.9332
- C. 0.3821
- D. 0.1336
Feedback: Ruler first: SE = 20/√25 = 4. z = (206 − 200)/4 = 1.50; "exceeds" is the right tail: 1 − 0.9332 = 0.0668. B is the wrong tail; C comes from using σ = 20 as the ruler (z = 0.30 — the wrong-ruler error); D doubles the tail for no reason.

Q7 (MC). Two probabilities about the same 206-second threshold: P₁ = the probability that ONE randomly chosen call exceeds 206 seconds; P₂ = the probability that the MEAN of 25 randomly chosen calls exceeds 206 seconds. Which statement is correct?
- A. P₂ is smaller than P₁, because averages vary less than individuals do ✅
- B. P₂ is larger than P₁, because 25 calls give 25 chances to exceed it
- C. P₁ and P₂ are equal, because both use the same mean and threshold
- D. P₁ and P₂ cannot be compared without knowing the sample's median
Feedback: Averages huddle: the mean's ruler is σ/√25 = σ/5, so a threshold 6 seconds above μ is many more SEs away for a mean than for one call — extreme averages are rarer. (This is the two-rulers idea in one question.)

Q8 (True / False). "The Central Limit Theorem guarantees that if you collect one large enough sample, the histogram of the individual values in that sample will be approximately normal, whatever the population's shape."
- True
- False ✅
Feedback: False — one sample's histogram mirrors the population (skew and all) at any n. The CLT's subject is the distribution of sample means across samples. Say it in full: the sampling distribution of the sample mean.

Q9 (Matching). Match each quantity to what it measures.
| Quantity | Correct description |
|---|---|
| σ (the population SD) | The typical distance of one individual value from the population mean |
| σ ⁄ √n | The typical distance of a sample mean from the population mean — the standard error of x̄ |
| μ | The center of both the population and the sampling distribution of x̄ |
| √( p(1 − p) ⁄ n ) | The standard error of the sample proportion p̂ |
Feedback: The two rulers plus their anchors: σ for individuals, σ/√n for sample means, both centered at μ — and p̂ gets its own SE with the √ over the whole fraction.

Q10 (MC). Across all calls to the center, 20% end up escalated to a specialist (p = 0.20). One shift, a random sample of n = 64 calls is audited. What is the probability that 30% or more of the sampled calls were escalated? (The area to the left of z = 2.00 under the standard normal curve is 0.9772.)
- A. 0.9772
- B. 0.0228 ✅
- C. 0.0062
- D. 0.2000
Feedback: SE = √(0.20 × 0.80 ⁄ 64) = √(0.16/64) = 0.4/8 = 0.05 (and np = 12.8, n(1 − p) = 51.2 — the bell applies). z = (0.30 − 0.20)/0.05 = 2.00; "30% or more" is the right tail: 1 − 0.9772 = 0.0228. A is the wrong tail; C follows from mis-setting the SE at 0.04 (an n = 100 slip → z = 2.5); D confuses p itself with a probability about p̂.

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 (C D B B · A A · B) and no letter carries more than three of the seven single-answer MC items; option lengths within each item are comparable (no length giveaway); every numeric claim (SE = 4; quadruple-to-halve; z = 1.50 → 0.0668; the 0.3821 wrong-ruler distractor; SE = 0.05 with np = 12.8 and n(1 − p) = 51.2; z = 2.00 → 0.0228; the 0.0062 distractor) is re-verified in tools/checks/w10_math.py; any table value an item needs is stated in the stem; no item asserts a fact outside the Week 10 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces (the quiz's call-center numbers appear nowhere else in the module).

Item-bank entries (for variants + the final)

All ten items are tagged week=10 · objective=5 · topic=sampling-distributions-clt and deposited in Item Bank: Week 10 — Sampling Distributions & the CLT with idents w10q1–w10q10. The final (Week 18) and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w10q1 sampling-distribution-concept, w10q2 unbiased-center, w10q3 standard-error, w10q4 root-n-economics, w10q5 clt-statements, w10q6 mean-probability, w10q7 two-rulers, w10q8 big-sample-misconception, w10q9 quantity-matching, w10q10 proportion-probability.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 10 Quiz — Sampling Distributions & the Central Limit Theorem"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true