Week 14 — Quiz (auto-graded) · Testing Claims About Means

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 7 — one-sample and paired t procedures (plus the test ↔ interval duality, Objective 6).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 14 · 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-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. Every critical value needed is stated in the stem — the quiz is closed-book-friendly.

Questions, key, and feedback

Q1 (MC). A gym chain's app claims "the average member workout is 45 minutes." A researcher plans a t-test of whether the true average differs from the claim in either direction. The hypotheses are —
- A. H₀: x̄ = 45 vs. Hₐ: x̄ ≠ 45
- B. H₀: μ ≠ 45 vs. Hₐ: μ = 45
- C. H₀: μ = 45 vs. Hₐ: μ < 45
- D. H₀: μ = 45 vs. Hₐ: μ ≠ 45 ✅
Feedback: Hypotheses are statements about the population mean μ — never the sample's x̄ — the null carries the claim, and "differs in either direction" is the two-sided alternative. (A = sample-notation trap; B = swapped roles; C = an uninvited one-sided test.)

Q2 (MC). A sedan model is advertised at 50 mpg highway. A testing lab runs 16 standardized drives: x̄ = 48.5 mpg, s = 4. The t-statistic is —
- A. t = −0.375
- B. t = −1.5 ✅
- C. t = −6.0
- D. t = +1.5
Feedback: SE = 4 ⁄ √16 = 1, so t = (48.5 − 50) ⁄ 1 = −1.5. (A divides by s instead of SE; C divides by s∕n; D loses the direction.)

Q3 (MC). For the sedan test above (df = 15, two-sided, α = 0.05, cutoff t* = 2.131), the correct decision and conclusion are —
- A. Fail to reject H₀ — the sample does not give convincing evidence that the true mean differs from 50 mpg ✅
- B. Reject H₀ — the t-statistic proves the true average is really 48.5 mpg, the sample's own value
- C. Fail to reject H₀ — the data prove the sedan genuinely does average 50 mpg on the highway
- D. Reject H₀ — because the size of the t-statistic, 1.5, is greater than the significance level 0.05
Feedback: |−1.5| < 2.131 → inside the cutoff → fail to reject, worded without "prove." C is the accept-trap; D compares a t to an α (different animals).

Q4 (MC). A quality team tests a claimed mean and reports p = 0.03. The correct reading of that number is —
- A. There is a 3% chance that the company's claimed mean is actually true
- B. Only 3% of the individual items measured happened to meet the claim
- C. If the claimed mean were true, samples at least this far from it would occur about 3% of the time ✅
- D. The true mean must differ from the company's claimed value by exactly 3 percent
Feedback: The p-value judges the data, assuming the claim: "how surprising is a sample like ours in a world where H₀ holds?" It is never the probability that a hypothesis is true.

Q5 (Multiple answer — select all that apply). Which of the following are conditions to check before trusting a one-sample t-test?
- A. The sample was collected by a random or otherwise representative method ✅
- B. The population standard deviation σ is known exactly
- C. The observations are independent of one another ✅
- D. The population is roughly normal, or the sample is large enough ✅
- E. The sample mean landed within 2 standard errors of the claimed mean
Feedback: Random/representative, independent, and roughly-normal-or-large-n are the three checks. The t procedure exists precisely because σ is unknown (B), and E describes a possible outcome of the test, not a requirement for running it.

Q6 (Matching). Match each item to its description.
| Item | Correct description |
|---|---|
| Degrees of freedom (one-sample t) | n − 1 — it selects the row of the t-table the test uses |
| Standard error of the sample mean | s ⁄ √n — the typical sample-to-sample wobble of x̄ |
| Paired data | Two linked measurements per individual, analyzed as one sample of differences |
| p-value | The probability, assuming H₀ is true, of a sample at least as extreme as the one observed |
Feedback: df picks the table row; SE is the ruler; pairs collapse to differences; p assumes the null and judges the data.

Q7 (MC). A 95% confidence interval for the mean session length at a climbing gym is (41.3, 44.9) minutes. Using only this interval, a two-sided test at α = 0.05 of the claim "the average session is 45 minutes" —
- A. Fails to reject the claim, because 45 sits within half a minute of the interval's edge
- B. Rejects the claim, because 45 lies outside the interval of plausible mean session lengths ✅
- C. Cannot be decided from the interval alone without computing the t-statistic from raw data
- D. Rejects the claim, because the interval is wider than three minutes from end to end
Feedback: The duality: a value outside the 95% CI is exactly a value the two-sided α = 0.05 test rejects — no further computation needed. Width (D) and closeness (A) are red herrings.

Q8 (True / False). "If a two-sided test at α = 0.05 fails to reject H₀: μ = 30, the data have proven that the true mean equals 30."
- True
- False ✅
Feedback: False. Fail to reject = the evidence didn't clear the bar — a not-guilty verdict, not a certificate of innocence. The same data would also fail to reject many nearby values (the whole confidence interval's worth).

Q9 (MC). A consumer group suspected — before collecting any data — that a pickup truck's true average mpg is lower than advertised. Their sample gives t = −1.9, and technology reports the two-sided p-value as 0.08. For their stated one-sided alternative, the p-value and the α = 0.05 decision are —
- A. p = 0.16 and fail to reject H₀
- B. p = 0.08 and reject H₀
- C. p = 0.04 and fail to reject H₀
- D. p = 0.04 and reject H₀ ✅
Feedback: The data lean the suspected way, so the one-sided p is half the two-sided one: 0.08 ⁄ 2 = 0.04 < 0.05 → reject. Legitimate only because the direction was declared before the data — pick your tail before you peek.

Q10 (MC). Twelve data-entry staffers each have their typing speed measured on the office's old keyboard layout and again on a new layout. To test whether the layout changed average speed, the appropriate analysis is —
- A. A two-proportion z procedure comparing the old and new layouts
- B. Two separate confidence intervals, one per layout, checked for overlap
- C. A one-sample t-test on each staffer's difference between the two layouts ✅
- D. No valid test exists, because the same people used both keyboard layouts
Feedback: Same person measured twice = paired data: subtract first, then one-sample t on the differences (df = 11 here — technology supplies off-table cutoffs). The linkage is a feature — each staffer is their own control — not a flaw (D).

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option (q2's arithmetic and q9's halving verified in w14_math.py; q7's 45 really lies outside 41.3–44.9); the multiple-answer item's three conditions are the only true conditions listed; no positional pattern in the MC key (D B A C · B · D C) and no letter appears more than twice among the seven MC items; option lengths within each item are comparable (no length giveaway); every needed critical value is stated in its stem; no item asserts a fact outside the Week 14 course definitions; no scenario reuses the tutorial, practice, chapter, lab, or assignment surfaces.

Item-bank entries (for variants + the final)

All ten items are tagged week=14 · objective=7 · topic=testing-means and deposited in Item Bank: Week 14 — Testing Claims About Means with idents w14q1–w14q10. The final (Week 18) and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w14q1 hypotheses-setup, w14q2 t-statistic, w14q3 decision-conclusion, w14q4 p-value-meaning, w14q5 t-conditions, w14q6 df-se-paired-p, w14q7 test-interval-duality, w14q8 fail-to-reject-trap, w14q9 one-vs-two-sided, w14q10 paired-recognition.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 14 Quiz — Testing Claims About Means"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true