Week 11 — Quiz (auto-graded) · Confidence Intervals for a Mean

Course: Introduction to Statistics (18-week generic edition)
Objective tested: Objective 6 — construct and interpret confidence intervals for a population mean.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Week 11 · 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-11-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file. Any t* value a question needs is supplied in its stem — nothing beyond the course's friendly table is required.

Blueprint

No trick questions; distractors target the Week 11 misconceptions named in the lecture outline (s-vs-SE, df = n, z-instead-of-t, individuals-vs-mean, probability-vs-confidence).

Questions, key, and feedback

Q1 (MC). A sleep-research lab tracks the nightly sleep of a random sample of 40 adults for a month; the sample averages 6.8 hours. As a point estimate of the mean nightly sleep of all adults in the study population, the lab should report —
- A. 8 hours, the commonly recommended amount of sleep
- B. 40, the number of adults who were sampled
- C. Nothing yet — a point estimate requires knowing σ
- D. 6.8 hours — the mean of the 40 sampled adults ✅
Feedback: The point estimate for a population mean is the sample mean, full stop — no σ needed. This week's work is giving that single number its honest cushion.

Q2 (MC). When building a confidence interval for a mean from a small sample, we use a t* multiplier instead of z* because —
- A. the sample mean x̄ is unknown until the data are collected
- B. σ is unknown, so s stands in for it — and that estimate adds extra wobble ✅
- C. the t-table is faster to use than the normal z-table
- D. sample means never follow a bell-shaped curve at any size
Feedback: Estimating σ with s adds a second layer of uncertainty; t's heavier tails budget for it. "t is z with humility."

Q3 (MC). A drive-through coffee stand samples n = 16 medium drip coffees to build a 95% confidence interval for its mean fill. Using the course t-table (df 15 row: 1.753 / 2.131 / 2.947 for 90% / 95% / 99%; df 24 row: 1.711 / 2.064 / 2.797), the correct t* is —
- A. 1.753
- B. 2.064
- C. 2.131 ✅
- D. 2.947
Feedback: Two steps: df = n − 1 = 15 (not 16), then the 95% column → 2.131. (Distractors are real table values reached by the classic wrong turns: the 90% column, the df 24 row, the 99% column.)

Q4 (MC). A tech site measures the battery life of a random sample of 25 laptops of one model: x̄ = 10.4 hours, s = 1.5 hours. The standard error of the sample mean is —
- A. 0.06 hours
- B. 0.3 hours ✅
- C. 1.5 hours
- D. 7.5 hours
Feedback: SE = s/√n = 1.5/√25 = 1.5/5 = 0.3. (0.06 divides by n; 1.5 forgets to divide; 7.5 multiplies by √n.)

Q5 (MC). Continuing with the 25 laptops (x̄ = 10.4 hours, SE = 0.3, df = 24, t* = 2.064): the 95% confidence interval for the mean battery life is —
- A. (9.78, 11.02) hours ✅
- B. (10.1, 10.7) hours
- C. (8.9, 11.9) hours
- D. (9.81, 10.99) hours
Feedback: ME = 2.064 × 0.3 = 0.6192 ≈ 0.62; 10.4 ∓ 0.62 → (9.78, 11.02). (B forgot the t* multiplier — ± SE only; C used ± s, the individuals' spread; D used z* = 1.96 — the sneaked-in z error.)

Q6 (Matching). Match each piece of a confidence interval's anatomy to its description.

The table below pairs each symbol with its correct description.

Q7 (Multiple answer — select all that apply). A team is building a 95% confidence interval for a mean. Which changes would make the interval wider, other things staying equal?
- A. Raising the confidence level from 95% to 99% ✅
- B. Using a smaller sample (n = 10 instead of n = 25) ✅
- C. Using a much larger sample (n = 100 instead of n = 25)
- D. Lowering the confidence level from 95% to 90%
- E. Sampling from a population with a larger standard deviation ✅
Feedback: ME = t* × s/√n: more confidence → bigger t*; smaller n → bigger SE (and a bigger-t* row); larger s → bigger SE. Bigger n and lower confidence both narrow the interval. Certainty costs width.

Q8 (True / False). A sleep clinic reports a 95% confidence interval of (6.6, 7.4) hours for the mean nightly sleep of adults with insomnia. This means about 95% of these adults sleep between 6.6 and 7.4 hours a night.
- True
- False ✅
Feedback: False — the interval describes the mean, not individuals. Individual sleepers spread far more widely than means do; most individuals sit outside a good CI for μ, and that's healthy.

Q9 (MC). An office audits its breakroom coffee machine: from a random sample of cups, the 95% confidence interval for the machine's mean fill is (284, 296) mL. Which statement interprets this correctly?
- A. 95% of all the cups this machine pours contain between 284 and 296 mL
- B. There is a 95% probability that any single cup poured tomorrow lands in this range
- C. We are 95% confident that the machine's mean fill, across all cups, is between 284 and 296 mL ✅
- D. 95% of future samples of cups will have their sample mean land inside this exact range
Feedback: The licensed sentence names the mean and the population. A and B confuse individuals with the mean; D shifts the 95% onto future sample means instead of the method's capture of μ.

Q10 (MC). A corner market times a sample of n = 12 checkout transactions; the times are strongly right-skewed with two extreme outliers. Before trusting a t-interval here, the owner should recognize that —
- A. the Central Limit Theorem guarantees normality, so the interval is fine at any sample size
- B. switching from the t* multiplier to z* will fix problems caused by skewed data
- C. raising the confidence level from 95% to 99% corrects for the skewness
- D. at this small n, the t-interval needs roughly normal data — the skew and outliers undermine it ✅
Feedback: The conditions are the password: random sample, and for small n, no strong skew or wild outliers. The CLT needs larger samples to flatten skew; neither z* nor a higher confidence level repairs a shape problem.

Answer key (quick reference)

Quality gate (self-checked): each single-answer item has exactly one correct option; the multiple-answer item's three wideners (A, B, E) are the only correct options listed; every t* value needed is supplied in the stem (nothing beyond the friendly table); all arithmetic re-verified by script (tools/checks/w11_math.py) — SE 0.3, ME 0.6192, interval (9.78, 11.02), and every distractor's wrong-turn arithmetic (0.06, 7.5, ±SE, ±s, z* 1.96 → (9.81, 10.99)); no positional pattern in the MC key (D B C B A C D) and no length giveaway (each key within the 0.55–1.45 length band of its distractors); no item asserts a fact outside the Week 11 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=11 · objective=6 · topic=confidence-intervals-mean and deposited in Item Bank: Week 11 — Confidence Intervals for a Mean with idents w11q1–w11q10. The final (Week 18) and per-term variant updates draw fresh variants from this bank's concepts — never these live stems. (Tags: w11q1 point-estimate, w11q2 why-t, w11q3 df-tstar, w11q4 standard-error, w11q5 t-interval, w11q6 interval-anatomy, w11q7 width-drivers, w11q8 individuals-vs-mean, w11q9 interpretation, w11q10 conditions.)

Canvas placement block

canvas_object    = Quizzes::Quiz
title            = "Week 11 Quiz — Confidence Intervals for a Mean"
assignment_group = "Quizzes"
points_possible  = 10
grading_type     = points
due_offset_days  = 6        # end of the module's week
published        = true
shuffle_answers  = true