Week 14 — Quiz (auto-graded) · Tests for Means & Proportions

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objective tested: Objective 7 — conduct and interpret hypothesis tests for means and proportions (choose the test · compute t or z · interpret a p-value decision · one- vs. two-sample).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 14 (Sun Dec 6).

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-14-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 14 misconceptions named in the lecture outline (wrong test, dropped √n, p̂-for-p₀, percent-not-decimal, one-vs-two-sample, significant≠big). Every test statistic is engineered to a clean value and every p-value is supplied in the stem.

Questions, key, and feedback

Q1 (MC). A coffee shop wants to test whether the average time to serve a drink is different from the target of 90 seconds. They time one sample of orders. Which test is appropriate?
- A. One-sample t-test for a mean ✅
- B. One-proportion z-test
- C. Two-sample comparison of means
- D. No test applies
Feedback: The data are a mean (an average time) compared to one fixed number (90), so it's a one-sample t-test. (B is for proportions; C needs two groups compared to each other.)

Q2 (MC). A streaming service claims that more than 40% of its users finish a series they start. Analysts survey one sample of users and record the share who finished. Which test fits?
- A. One-sample t-test for a mean
- B. One-proportion z-test ✅
- C. Two-sample comparison of means
- D. A correlation test
Feedback: The data are a proportion / share (the % who finished) compared to one fixed value (0.40), so it's a one-proportion z-test. (A is for averages; C needs two groups.)

Q3 (MC). A medical study compares the mean blood-pressure drop in a drug group versus a placebo group to see whether the two means differ. Which test is appropriate?
- A. One-sample t-test for a mean
- B. One-proportion z-test
- C. Two-sample comparison of means ✅
- D. A one-proportion z-test for the placebo group only
Feedback: Two groups are compared to each other (drug vs. placebo), so it's a two-sample comparison of means — not one group against a fixed number. (Count the groups: two datasets → two-sample.)

Q4 (MC). A one-sample t-test has x̄ = 52, μ₀ = 50, s = 5, n = 25. The standard error is s/√n = 5/√25 = 5/5 = 1. What is the test statistic t = (x̄ − μ₀)/(s/√n)?
- A. t = 0.4
- B. t = 2.0 ✅
- C. t = 10
- D. t = 0.5
Feedback: SE = 5/√25 = 1, so t = (52 − 50)/1 = 2.0. (A drops the √n and divides by s = 5; the denominator is the standard error s/√n, not s.)

Q5 (MC). A one-proportion z-test has p̂ = 0.40, p₀ = 0.50, n = 100. The standard error is √(p₀(1−p₀)/n) = √(0.50·0.50/100) = √0.0025 = 0.05. What is z = (p̂ − p₀)/SE?
- A. z = −0.10
- B. z = −2.0 ✅
- C. z = −20
- D. z = 2.0
Feedback: z = (0.40 − 0.50)/0.05 = −0.10/0.05 = −2.0 (negative because p̂ is below p₀). (The standard error uses p₀ = 0.50, and the values go in as decimals.)

Q6 (MC). A one-sample t-test of H₀: μ = 60 vs. Hₐ: μ > 60 (a method that may raise mean study time) sets α = 0.05 and technology reports p = 0.028. Which is the best conclusion in context?
- A. We proved the method works
- B. At the 0.05 level, there is significant evidence that the mean study time is above 60 minutes ✅
- C. There is a 2.8% chance the null hypothesis is true
- D. The effect is large and important
Feedback: Since 0.028 ≤ 0.05 we reject H₀; a good conclusion restates the real-world claim at the stated level. (C misreads the p-value; D confuses "significant" with "large"; A overclaims — a test never "proves.")

Q7 (True / False). "For a one-proportion z-test, the standard error in the denominator is built from p₀ (the null value), and for a one-sample t-test the denominator is s/√n, not s."
- True ✅
- False
Feedback: True. A proportion test uses p₀ in √(p₀(1−p₀)/n) because everything is computed assuming H₀ is true, and the t-test denominator is the standard error s/√n — keep the √n.

Q8 (Multiple answer — select all that apply). Which of the following are true about a one-sample t-test for a mean?
- A. The hypotheses are about the population mean μ, not the sample mean x̄ ✅
- B. The denominator of t is just s (you do not divide by √n)
- C. The test statistic is t = (x̄ − μ₀)/(s/√n) ✅
- D. The degrees of freedom are n − 1 ✅
- E. A larger t always means the effect is large and important
Feedback: The hypotheses concern μ (A), the statistic divides the gap by the standard error s/√n (C), and df = n − 1 (D). (B drops the √n; E confuses a big t with a big effect — t measures surprise relative to wobble, not size.)

Q9 (MC). A two-sample test compares the mean test scores of two teaching methods. H₀: the two means are equal; Hₐ: they differ. With α = 0.05, technology reports p = 0.30. What is the best conclusion?
- A. There is significant evidence the two methods differ
- B. There is not enough evidence that the two methods' mean scores differ ✅
- C. The two methods are proven to be identical
- D. The first method is definitely better
Feedback: 0.30 > 0.05 → fail to reject H₀, so there isn't enough evidence of a difference. (That is not proof the methods are identical — "absence of evidence is not evidence of absence.")

Q10 (Matching). Match each scenario to the correct test.
| Scenario | Correct test |
|---|---|
| Is the mean wait time different from 15 minutes? (one sample) | One-sample t-test for a mean |
| Do more than 30% of users click an ad? (one sample, a rate) | One-proportion z-test |
| Does the mean recovery time differ between a drug group and a placebo group? | Two-sample comparison of means |
| Has the campus pass rate changed from the historical 70%? (one sample, a rate) | One-proportion z-test |
Feedback: Mean vs. a number → one-sample t; a share/rate vs. a number → one-proportion z; two groups' means compared → two-sample. (Two scenarios are proportion tests — both are rates measured against a fixed value.)

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item (Q8) lists the three correct statements (A, C, D) and excludes the two distractors (B dropped-√n, E significant≠big); the matching item pairs four scenarios with the correct tests (drawing from a 3-test menu, with two correct proportion answers as designed). Every computed statistic is engineered to a clean value and the arithmetic is shown in the stem — Q4: 5/√25 = 1 → t = (52−50)/1 = 2.0; Q5: √(0.5·0.5/100) = 0.05 → z = (0.40−0.50)/0.05 = −2.0. Every p-value (0.028, 0.30) is supplied — no tail areas to look up, nothing to mis-key. No item asserts a fact outside the Week 14 course definitions.

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

All ten items are tagged course=MATH11 · week=14 · objective=7 · topic=tests-means-proportions and deposited in Item Bank: Week 14 — Tests for Means & Proportions. The final (Week 16) and the per-term variant updates draw fresh items from this bank. (Tags: q1 choose-test-mean, q2 choose-test-proportion, q3 one-vs-two-sample, q4 compute-t, q5 compute-z, q6 conclude-in-context, q7 standard-error-rule, q8 t-test-setup, q9 interpret-two-sample, q10 match-scenario-to-test.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 14 Quiz — Tests for Means & Proportions"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 5        # Sun Dec 6 (module starts Tue Dec 1)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"