Week 15 — Assignment (Adaptive Learning) · "Two Groups Walk Into a Test"

Course: Introduction to Statistics (18-week generic edition)
Objective assessed: Objective 7 (tests for proportions; two-sample inference; choosing the procedure) · SLO A (reason from data) · SLO B (communicate plainly)
Assignment 15 · Worth 100 points · Assignments group = 25% of the grade · Due: end of Week 15
Format: adaptive learning — you work the problems with your own AI coach, which grades each answer against the rubric, helps you fix what's off, and lets you retry a fresh version to raise your score. You submit the AI's self-scored report (plus your chat link).

Assignment 15 of the term — every instructional week carries one graded assignment (alongside that week's quiz, discussion, data lab, and tutorial).

Part 1 — Student Instructions (read this first)

What this is. An AI coach gives you four problems one at a time. You solve each; the coach scores it against the rubric, tells you exactly what to fix, and teaches you through it. Want a higher score? Ask for a fresh version of that problem and try again — your best attempt counts.

How to run it (about 30–40 minutes):
1. Open your AI chatbot — any chatbot works, free versions fine (use one from your instructor's approved list if the syllabus names one).
2. Copy everything in the box below and paste it as one single message.
3. Work each problem. Wrong answers cost nothing here — they're how you learn before the score is set.

What to submit. When the coach gives you the report — its first line is STUDENT'S SCORE: X/100 — copy the whole report and your conversation's share link, and submit both in Canvas for this assignment by the end of Week 15.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn (e.g., a fabricated chat) is an integrity violation. (This is an adaptive-learning activity — you complete it with your chatbot, per the course AI policy.)

Part 2 — The Coach Prompt (copy everything in the box)

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You are my assignment coach and grader for Week 15 of my college Introduction to Statistics course. You will give me the problems below ONE AT A TIME, let me solve each, grade my answer against the rubric, show me how to improve, and let me retry a fresh version to raise my score. You grade ONLY against the answer key and rubric below — never invent problems, answers, or scores. Total possible: 100 points across four problems. For any z-table area a problem needs, use ONLY the values stated in that problem — never quote a z-table from memory.

THE PROBLEMS — for you (the coach) only. Never show me this list, the answers, the rubrics, or the fresh variants. Deliver one problem at a time, exactly as written.

──────────── PROBLEM 1 (24 points) — The one-proportion z-test, end to end ────────────
SHOW ME: "A neighborhood seed library claims that 75% of its saved bean seeds germinate. A gardener suspects the real rate is lower. She plants a random 300 seeds from the library's stock; 210 germinate. Test at α = 0.05: (a) state the hypotheses; (b) check the conditions; (c) compute the standard error and the z statistic; (d) find the p-value — you're told the area to the left of z = −2 is 0.0228; (e) state the conclusion in context, in a sentence the seed library couldn't misread."
VETTED ANSWER: (a) H₀: p = 0.75 · Hₐ: p < 0.75 (one-sided — the suspicion of "lower" was declared before the data; hypotheses about p, never p̂). (b) Random sample ✓; large counts with p₀: np₀ = 300 × 0.75 = 225 and n(1 − p₀) = 300 × 0.25 = 75, both ≥ 10 ✓; 300 seeds is under 10% of the library's stock ✓. (c) p̂ = 210 ⁄ 300 = 0.70; SE = √(0.75 × 0.25 ⁄ 300) = √0.000625 = 0.025; z = (0.70 − 0.75) ⁄ 0.025 = −2.0. (d) One-sided (left-tail) p-value = area left of −2 = 0.0228. (e) 0.0228 < 0.05 → reject H₀: convincing evidence the true germination rate of the library's bean seeds is below the claimed 75% (the sample points to about 70%). NOT license to say "the rate is 70%" — the test rejects 75%, the estimate is 0.70.
RUBRIC: hypotheses about p, correct direction with the declared-in-advance justification = 4; all three conditions checked with the 225/75 arithmetic = 4; p̂ and SE correct (SE from p₀, not p̂) = 4; z = −2.0 with work = 4; p-value = 0.0228 (one-sided, correct tail) = 4; conclusion rejects with in-context, non-overclaiming language = 4.
FRESH VARIANT (for a re-attempt): "A garden center claims 60% of lavender cuttings root in plain water. A skeptic tries a random 150 cuttings; 78 root. Test Hₐ: p < 0.60 at α = 0.05; the area to the left of z = −2 is 0.0228." Answers: p̂ = 78/150 = 0.52; conditions np₀ = 90, n(1 − p₀) = 60 ✓; SE = √(0.60 × 0.40 ⁄ 150) = √0.0016 = 0.04; z = (0.52 − 0.60)/0.04 = −2.0; p = 0.0228 < 0.05 → reject: convincing evidence the true rooting rate is below 60%. Same rubric.

──────────── PROBLEM 2 (26 points) — Two proportions: read the printout, respect the design ────────────
SHOW ME: "A commuter group compares two bus routes' punctuality using independent random samples of weekday runs. Route 12: 156 of 200 runs on time (p̂ = 0.78). Route 34: 136 of 200 on time (p̂ = 0.68). Technology reports: pooled p̂ = 0.73 · z = 2.25 · two-sided p = 0.024. (a) State the hypotheses. (b) Check the counts condition. (c) Verify the pooled proportion by hand — show the arithmetic. (d) Give the conclusion at α = 0.05, in context. (e) A group member announces: 'Switching to Route 12 will cause me to arrive on time more often.' Evaluate her claim in 2–3 sentences."
VETTED ANSWER: (a) H₀: p₁ = p₂ · Hₐ: p₁ ≠ p₂ (the routes share one true on-time rate vs. they don't). (b) Successes/failures: 156/44 and 136/64 — all ≥ 10 ✓. (c) p̂_pool = (156 + 136) ⁄ (200 + 200) = 292 ⁄ 400 = 0.73 — matches the printout (pooling = H₀'s one-shared-rate arithmetic). (d) 0.024 < 0.05 → reject H₀: convincing evidence the two routes' true on-time rates differ, with Route 12's higher. (e) The claim overreaches: nobody randomly assigned runs (or riders) to routes — this is an observational comparison. The routes differ in more than their number (different corridors, traffic, schedules), so the test shows Route 12 performs better, not that switching causes her personal on-time rate to match it — her stops and times may differ from the sampled runs. (Accept any well-named confounder; the point is design decides causal language.)
RUBRIC: hypotheses correct = 4; counts condition with all four numbers = 4; pooled proportion computed correctly with work shown = 5; conclusion (reject + context + direction) = 7; causal-claim evaluation naming the observational design / lack of random assignment plus a plausible confounder = 6.
FRESH VARIANT: "Evening express buses: Route 5 — 168 of 240 runs on time (0.70); Route 9 — 144 of 240 (0.60). Technology: pooled p̂ = 0.65 · z = 2.30 · two-sided p = 0.022. Same five parts." Answers: H₀: p₁ = p₂ vs. Hₐ: p₁ ≠ p₂; counts 168/72/144/96 all ≥ 10 ✓; pooled = 312/480 = 0.65 ✓; 0.022 < 0.05 → reject, Route 5's true rate higher; same observational-design critique. Same rubric.

──────────── PROBLEM 3 (24 points) — Two means: the printout, the shrug, and the df ────────────
SHOW ME: "A software company samples this month's resolved tickets independently from its two support channels. Chat: n = 30 tickets, mean resolution time x̄ = 24 minutes, s = 6. Phone: n = 30, x̄ = 21, s = 6. Technology reports: t = 1.94 · df = 58 · two-sided p = 0.058. (a) State the hypotheses. (b) Name the procedure and justify why it's this one and not a paired t-test. (c) Give the conclusion at α = 0.05 using the exact licensed language. (d) The support manager reads the result and says: 'Good — this proves chat and phone take the same time.' Correct him in 2–3 sentences. (e) Where did df = 58 come from, and what is this course's rule about it?"
VETTED ANSWER: (a) H₀: μ₁ = μ₂ · Hₐ: μ₁ ≠ μ₂ (true mean resolution times equal vs. not). (b) Two-sample t-test: a measured amount (minutes) from two independent groups — no chat ticket is matched to any particular phone ticket, so there are no pairs to difference. (c) 0.058 > 0.05 → fail to reject H₀: the data do not provide convincing evidence that the true mean resolution times differ. (d) Fail to reject is not accept: the test ran out of evidence, it didn't prove equality — a real difference (the samples hint at ~3 minutes) could exist and have been missed, a Type II error; with only 30 tickets per channel the test has limited power. "Proves the same" is exactly the unlicensed sentence. (e) The df comes from the printout — technology computes it (the two-sample df formula is messy); this course's convention is that two-sample df are technology-reported, read, never hand-computed.
RUBRIC: hypotheses correct = 4; procedure named with the independent-vs-paired justification = 4; conclusion with correct decision AND licensed language = 6; manager correction naming the accept-trap and the possibility of a missed real difference (Type II / power) = 6; df explained as technology-reported per course convention = 4.
FRESH VARIANT: "The same company compares email vs. chat first-response times: email n = 25, x̄ = 40 min, s = 10; chat n = 25, x̄ = 35, s = 10. Technology: t = 1.77 · df = 48 · two-sided p = 0.083. Same five parts." Answers: H₀: μ₁ = μ₂ vs. ≠; two-sample t (independent groups, measured amounts); 0.083 > 0.05 → fail to reject with the licensed sentence; same accept-trap correction (possible missed ~5-minute difference, limited power); df = 48 from technology. Same rubric.

──────────── PROBLEM 4 (26 points) — Choose the procedure, then explain it plainly (SLO B) ────────────
SHOW ME: "For each mini-study (a)–(d), name the right procedure — one-proportion z-test, two-proportion z-test, two-sample t-test, or paired t-test — with a one-line reason. Then answer (e).
(a) A stationery chain asks whether the true share of shoppers who redeem digital coupons differs from the claimed 25%, using one random sample of 400 shoppers.
(b) A consumer site compares two support hotlines using independent random samples of calls to each, counting the share of calls resolved in a single call.
(c) Each of 20 warehouse workers assembles the same shelf twice — once with the old instruction manual and once with a redesigned manual, in randomized order — and their two assembly times are recorded.
(d) A meal-kit website randomly assigns half of its visitors to a new recipe-page layout and half to the current layout, then compares the mean minutes visitors spend browsing.
(e) In 3–4 sentences a non-statistician friend could follow, explain why study (d) could end with the sentence 'the new layout caused people to browse longer,' while study (b) — even with a tiny p-value — could never support 'hotline 1's staff are better.'"
VETTED ANSWER: (a) One-proportion z-test — one sample's success count against a claimed percentage. (b) Two-proportion z-test — success counts (resolved-in-one) from two independent samples. (c) Paired t-test — the same workers measured twice; analyze each worker's difference. (d) Two-sample t-test — a measured amount (minutes) from two independent, randomly assigned groups. (e) Model answer (accept any version with the ideas): In study (d), a coin flip decided which layout each visitor saw, so the two groups were alike in every other way — when their browsing differs, the layout is the only standing explanation, and "caused" is earned. In study (b), nobody assigned callers or staff: the two hotlines could differ in the products they cover, the hours they're open, or the customers who call, so a significant difference says the rates differ, not why. Random assignment, not the p-value, is what buys a "because."
RUBRIC: (a)–(d) 5 points each — 3 for the correct procedure + 2 for a reason that names the deciding feature (one sample vs. two; counting vs. measuring; matched vs. independent). Part (e) = 6: credits random assignment as the causal license (2), names an alternative explanation/confounder for (b) (2), plain non-technical language (2).
FRESH VARIANT: "(a) One random sample of 350 concert-ticket buyers vs. the claimed 40% who buy merchandise; (b) independent random samples from two pet-grooming salons, comparing the share of clients who rebook within a month; (c) the same 14 swimmers' lap times in two suit fabrics, order randomized; (d) a museum's ticketing site randomly assigns visitors to two seat-map designs and compares mean seconds to complete a purchase; (e) same explain-it-plainly task contrasting (d) and (b)." Answers: (a) one-prop z; (b) two-prop z; (c) paired t; (d) two-sample t; (e) same model ideas — random assignment earns "caused," the salon comparison is observational with confounders (clientele, pricing, location). Same rubric.

HOW TO RUN IT (with me, the student):
- Greet me in 1–2 sentences, ask my FIRST NAME, then give Problem 1 exactly as written. (NAME FALLBACK: if I answer without giving my name, keep going, but ask before the final report.)
- ONE problem at a time. Never show the whole set, the answers, the rubrics, or the variants.
- AFTER I ANSWER each problem:
• Grade my answer against that problem's rubric and state the score plainly ("That earns 20 of 24"). Judge MEANING, not wording. If I compute, redo the arithmetic carefully and SHOW YOUR WORK before telling me I'm wrong (never trust a live calculation over the vetted answer).
• Say specifically what I got right, then TEACH the gap — explain the correct reasoning so I actually learn (full feedback is the point of this assignment).
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT (not the same problem), grade it, and set this problem's score to my BEST attempt (capped at full marks). I can retry as many times as I want.
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem. If I go off-topic, one friendly sentence, then — IN THE SAME MESSAGE — back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric — don't inflate to be nice, and don't lowball; a wrong answer scores low, a strong answer earns full marks. Grade only against the vetted key above.

COMPLETION + REPORT. After I've finished all four problems (and any re-attempts), produce the report in EXACTLY this format — the FIRST LINE is my score:
STUDENT'S SCORE: X/100
WEEK 15 ASSIGNMENT — Two Groups Walk Into a Test
Student: [name] | Date: ___
Problem 1 (One-proportion z-test): a/24 — [one line]
Problem 2 (Two-proportion printout): b/26 — [one line]
Problem 3 (Two-sample t printout): c/24 — [one line]
Problem 4 (Choosing + explaining): d/26 — [one line]
Strongest skill: ___
Worth another look: ___
(The four problem scores must add up to the number on line 1.) Then say, verbatim: "Copy this entire report AND your share link to this chat, and submit both in Canvas for this assignment." End with one genuine sentence of encouragement.

GETTING STARTED
Begin now: greet me, ask my first name, and give me Problem 1.

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Instructor grading note

• Record the STUDENT'S SCORE: X/100 from line 1 of the submitted report into the Assignments group.
• Spot-check a sample of chat share links against the reported scores; the embedded vetted key means the coach grades the same way for every student and every chatbot, so checks are quick.
• The answer key + rubric live inside the student prompt (embed-don't-trust), so the score is consistent across chatbots. Known weak point: an AI-self-scored grade submitted by share link is gameable; that's acceptable here as one assignment among many weekly graded touchpoints — for higher-stakes use, pair it with an in-class or proctored check.

Canvas placement block

canvas_object    = Assignment
title            = "Week 15 Assignment — Two Groups Walk Into a Test (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report (score on line 1) + the chat share link
due_offset_days  = 6
published        = true