Week 5 · Applied Statistics · HEG Genève

If you sampled again tomorrow, how different would the mean be?

The keystone of the course. Your thirty prices gave a mean of about CHF 19.6 — but another thirty would give another mean. This week is about the distribution of the sample mean itself: where it is centred, how wide it is (σ/√n), and why it is bell-shaped even when the population is not. Everything from Week 6 onward rests on it.

🏫 In class: 3 h · lesson ~1 h, then exercises ~2 h 📖 Saylor ch. 6 ▶ The Lecture button shows the slide outline

By the end of Week 5 you will be able to

State the mean and standard deviation of the sample mean: µ_x̄ = µ and σ_x̄ = σ/√n — and explain the √n.
State the Central Limit Theorem and say when it applies.
Compute P(x̄ in a range) for a sample of size n from a population with known µ and σ.
Do the same for a sample proportion p̂, with µ_p̂ = p and σ_p̂ = √(pq/n).
Explain, in the pizzeria's words, why a sample of 30 says something about all of Geneva.

Resources

✓Saylor §6.1 — The mean and standard deviation of the sample mean ↗

✓Saylor §6.2 — The sampling distribution of the sample mean ↗

✓Saylor §6.3 — The sample proportion ↗

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

Where the sample mean lives — the simulator The sampling machine Exam cards c7–c8

← Course page

Section 5.1 · The hook

Another thirty tomorrow

Planned · Hook

Estimate how far a second sample mean would land from the first

Slider: "your thirty averaged CHF 19.6. Another thirty tomorrow — within how many francs would you bet the new mean lands?" Lock 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 5.2 · The sampling machine

Watch the mean of the means

Planned · Keystone

µ_x̄ = µ and σ_x̄ = σ/√n, made visible (§6.1)

The existing sampling-distribution simulator embedded: a skewed population of prices, draw 1 / 100 / 1,000 samples, watch the means pile up. The σ/√n readout against the observed SD of the means is the whole section.
Slider n from 1 to 50: the pile narrows by √n, not by n — the single most-missed fact of the chapter.

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 5.3 · The Central Limit Theorem

Why the bell shows up

Planned · CLT

The sampling distribution of x̄ (§6.2)

Widget: switch the population to bimodal and uniform; the means still go bell-shaped once n ≥ ~30.
Compute P(x̄ > 20) for n = 30 from µ = 19, σ = 4: standardise with σ/√n. Exam card c7.
Classify (6 rows): does the CLT apply? — n = 5 from a skewed population, n = 40 from anything, a normal population at any n, and two traps.

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 5.4 · The sample proportion

Share above CHF 20, again

Planned · Proportion

µ_p̂ = p, σ_p̂ = √(pq/n) (§6.3)

Widget: the proportion of pizzerias over CHF 20 across repeated samples; the pile of p̂ values and its SD against √(pq/n).
P(p̂ in a range) for n = 30, p = 0.35. Exam card c8.

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 5.5 · Checkpoint

Prove it to yourself

Planned · Quiz

Self-check quiz & where this goes next

Ten questions: µ_x̄, σ_x̄, the √n, when the CLT applies, P(x̄ in a range), µ_p̂ and σ_p̂, and one "the population is not normal — so what?" trap.

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 5.6 · Workshop

The pizzeria file · instalment five

Planned · Exercise

Workbook fields

w5_se — the standard error of your sample mean for your n, and what halving it would cost in extra visits.
w5_clt — in two sentences, why thirty pizzerias can speak for Geneva — and the one situation where they could not.

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 · µ_x̄ and σ_x̄ 12 min
from a population with known µ and σ, at three sample sizes
Exercise 2 · The √n, felt 12 min
how much must n grow to halve the standard error?
Exercise 3 · Does the CLT apply? 10 min
five situations, yes or no, with the reason
Exercise 4 · P(x̄ in a range) 18 min
n = 30 from µ = 19, σ = 4 — standardise with σ/√n
Exercise 5 · The same question at n = 10 12 min
why the answer changes, and by how much
Exercise 6 · The sample proportion 15 min
µ_p̂, σ_p̂ and P(p̂ > 0.4)
Exercise 7 · Sample mean vs single value 12 min
the classic confusion: P(X > 22) against P(x̄ > 22)
Exercise 8 · Why thirty can speak for Geneva 12 min
write it in two sentences

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

Course page →