Week 15 — Quiz (auto-graded) · Testing Proportions & Two-Sample Inference

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 7 — the one-proportion z-test, two-proportion and two-sample-t procedures (interpret-the-output), and choosing the right procedure.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 15 · 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-15-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file. Any table areas or output values an item needs are stated in the item — no outside table required.

Blueprint

No trick questions; distractors target the Week 15 misconceptions named in the lecture outline (hats in hypotheses, p̂-built SEs, accept-the-null, p-value misreads, paired-as-independent).

Questions, key, and feedback

Q1 (MC). A company's service standard says 80% of live-chat customers should rate their chat "satisfied." For one support team, a manager pulls a random sample of 100 recent chats and finds 74 rated satisfied. To test whether this team's true satisfaction rate differs from the standard, the hypotheses are —
- A. H₀: p̂ = 0.80 and Hₐ: p̂ ≠ 0.80
- B. H₀: p = 0.80 and Hₐ: p ≠ 0.80 ✅
- C. H₀: p = 0.74 and Hₐ: p ≠ 0.74
- D. H₀: μ = 0.80 and Hₐ: μ ≠ 0.80
Feedback: Hypotheses are always about the population proportion p — never about p̂ (the evidence), never about the sample's 0.74 (hypotheses are written before the data speak), and not about μ (this is a percentage, not a mean).

Q2 (MC). For that same test (p₀ = 0.80, n = 100, p̂ = 0.74), the standard error under H₀ is √(0.80 × 0.20 ⁄ 100) = 0.04. The test statistic is —
- A. z = −1.50 — the sample rate sits 1.5 standard errors below the claimed rate ✅
- B. z = +1.50 — the sample rate sits 1.5 standard errors above the claimed rate
- C. z = −0.06 — the plain gap between the sample rate and the claimed rate
- D. z = −1.37 — the gap divided by an SE built from the sample proportion
Feedback: z = (0.74 − 0.80) ⁄ 0.04 = −0.06 ⁄ 0.04 = −1.50. Subtract first, divide by the SE, and keep the sign. (−0.06 forgot to divide; −1.37 built the SE from p̂ = 0.74 — tests use p₀; +1.50 dropped the direction.)

Q3 (MC). Technology reports the two-sided p-value for that test: p = 0.134. At α = 0.05, the correct conclusion is —
- A. Reject H₀ — the team's true satisfaction rate differs from 0.80
- B. Accept H₀ — the sample proves the team's true rate is exactly 0.80
- C. There is a 13.4% probability that the null hypothesis is true
- D. Fail to reject H₀ — no convincing evidence the team's rate differs from 0.80 ✅
Feedback: 0.134 > 0.05, so the evidence doesn't clear the bar — and that is all it means. "Fail to reject" is never "accept" (B), and a p-value is the probability of data this extreme assuming H₀, never the probability H₀ is true (C).

Q4 (MC). A plant nursery compares two seed-starting treatments, no fertilizer in either: 150 seeds soaked overnight — 96 germinated (p̂ = 0.64); 150 seeds planted dry — 81 germinated (p̂ = 0.54). Technology reports: pooled p̂ = 0.59 · z = 1.76 · two-sided p = 0.078. At α = 0.05, the correct conclusion is —
- A. Reject H₀ — the data prove soaking raises the true germination rate
- B. Accept H₀ — the data prove the two treatments share one germination rate
- C. Fail to reject H₀ — not convincing evidence the treatments' true rates differ ✅
- D. No conclusion is possible because the two samples have exactly equal sizes
Feedback: 0.078 > 0.05 → fail to reject. The 10-point sample gap is within chance's usual wobble for these sizes. That is not proof the treatments tie (B — the accept trap); equal sample sizes are perfectly fine (D).

Q5 (MC). An online craft-supply store runs a true A/B test, randomly assigning visitors: redesigned product page — n = 50 completed orders, x̄ = 9.5 items, s = 2.0; current page — n = 50, x̄ = 10.5 items, s = 2.0. Technology reports: t = −2.5 · df = 98 · two-sided p = 0.014. At α = 0.05, the correct conclusion is —
- A. Reject H₀ — convincing evidence the true mean items per order is lower on the redesign ✅
- B. Fail to reject H₀ — a negative t statistic can never lead to rejecting a null hypothesis
- C. Reject H₀ — proof that every single order on the redesign contains exactly one item fewer
- D. Fail to reject H₀ — the p-value 0.014 is smaller than the significance level 0.05
Feedback: 0.014 < 0.05 → reject; the sign of t just points the direction (fewer items on average on the redesign). A negative t rejects as readily as a positive one (B); tests speak about means, not about every individual order (C); and a smaller p-value is more evidence against H₀, not less (D).

Q6 (Matching). Match each procedure to the setting it's built for.
| Procedure | Correct setting |
|---|---|
| One-proportion z-test | One sample's success count is compared with a claimed percentage |
| Two-proportion z-test | Success counts from two independent samples are compared with each other |
| Two-sample t-test | Means of a measured amount from two independent samples are compared |
| Paired t-test | The same individuals are measured twice and their differences are analyzed |
Feedback: Two questions pick the tool: counting successes vs. measuring amounts, then one sample / two independent samples / the same individuals twice.

Q7 (Multiple answer — select all that apply). Which of the following are required checks before running a one-proportion z-test?
- A. The sample was randomly selected ✅
- B. np₀ ≥ 10 and n(1 − p₀) ≥ 10, computed with the hypothesized proportion ✅
- C. The sample is less than 10% of the population when sampling without replacement ✅
- D. The population of individual responses follows a normal distribution
- E. The sample proportion p̂ must come out equal to the hypothesized proportion p₀
Feedback: Random selection, large expected counts under p₀, and the 10% condition are the entry requirements. Individual yes/no responses are never "normally distributed" (D) — it's p̂'s sampling distribution that goes normal — and p̂ matching p₀ would leave nothing to test (E).

Q8 (True / False). "With a very large sample, a difference that is too small to matter in practice can still produce a very small p-value."
- True ✅
- False
Feedback: True. p-values measure surprise under H₀, not importance — giant samples detect microscopic differences. That's why results are reported as a difference in real units plus a p-value: statistical significance and practical significance are separate questions.

Q9 (MC). Team East resolved 33 of 150 tickets within one hour; Team West resolved 27 of 100. For the two-proportion z-test comparing the teams, the pooled proportion is —
- A. 0.245 — the simple average of the two sample proportions
- B. 0.24 — the combined successes divided by the combined sample sizes ✅
- C. 0.22 — the smaller of the two teams' sample proportions
- D. 0.27 — the larger of the two teams' sample proportions
Feedback: p̂_pool = (33 + 27) ⁄ (150 + 100) = 60 ⁄ 250 = 0.24. Pooling combines counts, so the larger sample rightly gets more say — the simple average (0.245) would weight a 100-ticket team equally with a 150-ticket team.

Q10 (MC). Twelve delivery couriers each ride Route 12 for one week and Route 34 for another (order randomized), recording their average daily minutes on each route. To compare the two routes, the right procedure is —
- A. A paired t-test, because each courier supplies a matched pair of route averages ✅
- B. A two-sample t-test, because two different routes are being compared
- C. A two-proportion z-test, because there are two routes in the study
- D. A one-proportion z-test on the share of couriers who prefer Route 12
Feedback: The same twelve people rode both routes — every measurement in one group matches one measurement in the other, so analyze each courier's difference (Week 14). "Two groups of numbers" is not the tell; matched measurements are.

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item's three required conditions are the only required ones listed; no positional pattern in the key (B A D C A · B A) 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 = 0.04; z = −1.50 and the −1.37 wrong-SE distractor; p = 0.134 = 2 × 0.0668; the germination output z = 1.76 / p = 0.078 from counts 96/150 vs. 81/150 with pooled 0.59; the items-per-order output t = −2.5 / df = 98 / p = 0.014 from the stated summaries (SE = √(4/50 + 4/50) = 0.4); pooled 60/250 = 0.24 vs. the 0.245 average trap) is re-verified in tools/checks/w15_math.py; any output or table value an item needs is stated in the stem; no item asserts a fact outside the Week 15 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces (different contexts and numbers throughout).

Item-bank entries (for variants + the final)

All ten items are tagged week=15 · objective=7 · topic=proportions-two-sample-tests and deposited in Item Bank: Week 15 — Testing Proportions & Two-Sample Inference with idents w15q1–w15q10. The final (Week 18) and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w15q1 one-prop-hypotheses, w15q2 one-prop-z, w15q3 fail-to-reject, w15q4 two-prop-output, w15q5 two-sample-t-output, w15q6 procedure-map, w15q7 one-prop-conditions, w15q8 practical-significance, w15q9 pooled-proportion, w15q10 paired-vs-independent.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 15 Quiz — Testing Proportions & Two-Sample Inference"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true