Week 6 · Applied Statistics · HEG Genève

The Geneva price — with a margin

Week 5 told you how far a sample mean wanders. Now turn that around: from one sample, build an interval that captures the true Geneva price with a stated confidence. Large samples use z; small samples use t; proportions have their own formula; and the last section asks the question a budget asks — how many pizzerias must we visit to be within one franc?

🏫 In class: 3 h · lesson ~1 h, then exercises ~2 h 📖 Saylor ch. 7 ▶ The Lecture button shows the slide outline

By the end of Week 6 you will be able to

Build and interpret a confidence interval for a mean with a large sample: x̄ ± z·σ/√n.
Build one for a small sample with t and df = n − 1 — and say why t is wider.
Build a confidence interval for a proportion: p̂ ± z·√(p̂q̂/n).
Compute the sample size needed for a target margin of error.
Say what "95% confident" means — and what it does not.

Resources

✓Saylor §7.1 — Large-sample estimation of a population mean ↗

✓Saylor §7.2 — Small-sample estimation of a population mean ↗

✓Saylor §7.3 — Large-sample estimation of a population proportion ↗

✓Saylor §7.4 — Sample size considerations ↗

Homework for this week is written with the week itself, on the Week 1–2 pattern: a fresh dataset, hints and worked solutions.

Tools for this week

Catch the mean — intervals that catch µ Exam cards c9–c12

← Course page

Section 6.1 · The hook

Give me a margin

Planned · Hook

Estimate the ± you would put on the Geneva price

Slider: "the true Geneva Margherita price is CHF 19.6, plus or minus how much?" Lock it; section 6.2 computes it.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.2 · Large samples

x̄ ± z·σ/√n

Planned · CI (z)

The confidence interval for a mean (§7.1)

Widget: the interval on the number line, confidence level 90/95/99 as a toggle, n as a slider; watch the interval widen with confidence and shrink with n.
Catch the mean: 100 samples, 100 intervals, about 95 of them capture µ — the meaning of "95% confident" as a picture. Exam card c9.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.3 · Small samples

When n is small, use t

Planned · CI (t)

df = n − 1 and the t table (§7.2)

Widget: z versus t curves for df = 5, 10, 29 — the fatter tails and the wider interval.
Classify (6 rows): z or t? — n = 8 with unknown σ, n = 50, a normal population with known σ, and traps. Exam card c10.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.4 · Proportions

Share above CHF 20, with a margin

Planned · Proportion

p̂ ± z·√(p̂q̂/n) (§7.3)

Widget: the interval for the proportion over CHF 20 from 30 pizzerias, and why 30 is barely enough (np̂ ≥ 10 and nq̂ ≥ 10 check). Exam card c11.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.5 · Sample size

How many pizzerias for ± 1 franc?

Planned · Sample size

n = (z·σ/E)² and n = z²·p̂q̂/E² (§7.4)

Widget: choose the margin E and confidence; read n. The budget question: each visit costs a Margherita — what does ±1 franc cost? Exam card c12.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.6 · Checkpoint

Prove it to yourself

Planned · Quiz

Self-check quiz & where this goes next

Ten questions: a z-interval, a t-interval, a proportion interval, a sample-size calculation, two interpretation traps ("95% of pizzerias are inside the interval" is wrong), and the z-or-t choice.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Section 6.7 · Workshop

The pizzeria file · instalment six

Planned · Exercise

Workbook fields

w6_ci — your 95% confidence interval for the Geneva price, written as a sentence a bank would accept.
w6_n — the sample size for the margin you would want, and whether it is worth the visits.

Status → this section is an outline. When the week is written it becomes an interactive screen on the Week 1–2 pattern: an earned completion, a classify exercise or lab where one is listed, and its own share of the progress bar.

Course page →

Part 2 · Exercises

8 exercises, worked together

Planned · ~103 min

The second half of the session

When this week is written, each exercise below becomes a form on this page and a slide in the deck, generated from one array so the two can never disagree — and its solution opens when the instructor reveals it in class. Same engine as Weeks 1–2 (exercises.js).

Exercise 1 · A 95% interval for the mean 15 min
x̄ ± z·σ/√n on the Week 2 data
Exercise 2 · Change the confidence 10 min
90, 95, 99 — what widens, and what it buys
Exercise 3 · z or t? 10 min
six situations, the decision and the reason
Exercise 4 · A small-sample interval 18 min
n = 8 lakeside pizzerias, t with df = 7
Exercise 5 · An interval for a proportion 15 min
p̂ ± z·√(p̂q̂/n), with the np̂ ≥ 10 check
Exercise 6 · What does 95% mean? 10 min
four statements, one correct — the misinterpretation clinic
Exercise 7 · Sample size for ± 1 franc 15 min
n = (z·σ/E)², and what it costs in visits
Exercise 8 · Your interval, in a sentence 10 min
the one a bank would accept

Status → outline only. Nothing here is interactive or counted yet.

Course page →