Week 10 — Quiz (auto-graded) · Sampling Distributions

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objective tested: Objective 5 — use normal and sampling distributions to reason about variability (sampling variability; the standard error σ/√n; the Central Limit Theorem; the sampling distribution of x̄).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 10.

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.

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, and every standard error is pre-computed from friendly numbers:

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 10 misconceptions named in the lecture outline. All arithmetic is pre-computed; every standard error uses a perfect-square n and every z lands exactly on the embedded table.

Questions, key, and feedback

Q1 (MC). A researcher takes a random sample of 40 students and computes their average study time. A second researcher takes a different random sample of 40 students and gets a different average. The main reason the two averages differ is —
- A. One researcher made a calculation error
- B. Sampling variability — a sample mean changes from one sample to the next ✅
- C. The population's true mean changed between the two samples
- D. There is no true average study time
Feedback: A sample mean x̄ is a moving target: a fresh sample gives a fresh average, with no one making a mistake and nothing about the population changing. That sample-to-sample wobble is sampling variability.

Q2 (MC). If you took thousands of random samples of the same size and made a histogram of all their sample means x̄, that sampling distribution of x̄ would be centered at —
- A. Zero
- B. The population mean μ ✅
- C. The sample size n
- D. The population standard deviation σ
Feedback: The averages aim at the truth: the mean of the sampling distribution of x̄ is μ. (Its spread is the standard error σ/√n — a different quantity.)

Q3 (Multiple answer — select all that apply). Which of the following statements about the sampling distribution of the sample mean x̄ are true?
- A. Its mean (center) is the population mean μ ✅
- B. Its standard deviation is the standard error, σ/√n ✅
- C. Its spread is the same as the population's spread, σ
- D. For large n, it is approximately normal even if the population is not (the Central Limit Theorem) ✅
- E. A bigger sample size n makes the population's standard deviation σ smaller
Feedback: A, B, and D are the core facts. The spread of x̄ is σ/√n, which is smaller than σ (so C is false), and n shrinks the standard error, never σ itself (so E is false).

Q4 (MC). A population has standard deviation σ = 20. For random samples of size n = 100, the standard error of the sample mean (σ/√n) is —
- A. 20
- B. 2 ✅
- C. 0.2
- D. 200
Feedback: σ/√n = 20 ÷ √100 = 20 ÷ 10 = 2. The averages of 100 spread far less than individuals (whose spread is 20). (Using 20 is the "σ not σ/√n" trap; 0.2 divides by n instead of √n.)

Q5 (MC). Keeping σ fixed, what happens to the standard error σ/√n as the sample size n increases?
- A. It gets smaller (the sample mean becomes more precise) ✅
- B. It gets larger
- C. It stays exactly the same
- D. It becomes equal to σ
Feedback: Bigger n means a bigger √n in the denominator, so σ/√n shrinks — larger samples give a tighter, more precise average. (To halve the standard error you must quadruple n.)

Q6 (MC). A population of purchase amounts is strongly right-skewed (not bell-shaped), with μ = $4.00 and σ = $2.00. For random samples of n = 100, the shape of the sampling distribution of x̄ is —
- A. Strongly right-skewed, just like the population
- B. Approximately normal (bell-shaped), by the Central Limit Theorem ✅
- C. Uniform (flat)
- D. Impossible to determine
Feedback: The Central Limit Theorem: for large n the distribution of the averages is approximately normal regardless of the population's shape. The skew belongs to the individuals, not to x̄. (A is the "CLT doesn't change the shape" trap.)

Q7 (MC). Purchase amounts have μ = $4.00 and σ = $2.00; for n = 100 the standard error is σ/√n = 2 ÷ 10 = $0.20. Using z = (x̄ − μ) ÷ (σ/√n) and the embedded table, the probability that the sample mean is less than $3.60 is about —
- A. .0228 ✅
- B. .9772
- C. .1587
- D. .42
Feedback: z = (3.60 − 4.00) ÷ 0.20 = −2.0; the area to the LEFT of −2.0 is .0228 (about 2.28%). Note you divide by the standard error 0.20, not by σ = 2. (.9772 is the right-tail/“above” value; .42 comes from wrongly using σ = 2 to get z = −0.2.)

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 = −2.0 | .0228 |
| z = 0 | .5000 |
| z = +1.0 | .8413 |
| z = +2.0 | .9772 |
Feedback: Read each straight off the table. The same three moves as Week 9 work on a sample mean — you just standardize with the standard error σ/√n first.

Q9 (True / False). "The Central Limit Theorem says that if the sample size is large, the individual data values in the population become normally distributed."
- True
- False ✅
Feedback: False. The CLT is about the sampling distribution of x̄ (the averages) — those become approximately normal for large n. The raw population keeps its own shape (a skewed population stays skewed).

Q10 (MC). You want the probability that the average of a sample of n = 25 values falls in some range. When you standardize the sample mean x̄, you should divide by —
- A. σ (the population standard deviation)
- B. σ/√n (the standard error) ✅
- C. n
- D. √n by itself
Feedback: For an average, the right spread is the standard error σ/√n, so z = (x̄ − μ) ÷ (σ/√n). (Dividing by σ is the week's signature mistake — that's the spread of a single value, not of an average.)

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 — Q4 (20/√100 = 20/10 = 2), Q7 (SE = 2/√100 = 0.20; z = (3.60−4)/0.20 = −2; left area = .0228), Q8 (table reads). No item asserts a fact outside the Week 10 course definitions.

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

All ten items are tagged course=MATH11 · week=10 · objective=5 · topic=sampling-distributions and deposited in Item Bank: Week 10 — Sampling Distributions. The final (Week 16) and the per-term variant updates draw fresh items from this bank. (Tags: q1 sampling-variability, q2 mean-of-xbar, q3 sampling-distribution-properties, q4 standard-error-compute, q5 standard-error-vs-n, q6 clt-shape, q7 clt-probability, q8 zscore-area-match, q9 clt-misconception, q10 sigma-vs-standard-error.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 10 Quiz — Sampling Distributions"
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"