Your thirty Geneva prices from Week 2 averaged CHF 19.93. Another thirty would give another number. 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 prices are not. It is the keystone of the course: every interval in Week 6 and every test in Week 7 stands on it. This is the textbook's chapter 6.
The session is three hours in two parts. Part 1 (~1 h) is the lesson: your instructor presents the deck and you follow the same numbered sections here — every slide names its section. Part 2 (~2 h) is ten exercises taken straight from the textbook, worked together; solutions open as your instructor reveals them.
In Week 2 you collected thirty Margherita prices: mean CHF 19.93, standard deviation about CHF 4.20. Tomorrow a friend walks into thirty different Geneva pizzerias and computes the same average. Within how many francs of 19.93 would you bet their mean lands — nineteen times out of twenty?
Every sample gives a different mean. So x̄ has a distribution of its own — over all the samples you could have drawn — and that distribution is called the sampling distribution of x̄. It is narrower than the prices themselves, because averaging cancels extremes; this week says exactly how much narrower, and section 5.4 answers the question above.
One pizzeria can be anywhere. The average of thirty cannot.
Below is a population of 300 Margherita prices, skewed to the right like the real market, with mean µ and standard deviation σ printed under it. Each click draws random samples of size n and adds their means to the pile. Watch two things: where the pile is centred, and how wide it is compared with σ/√n.
The centre does not move: on average, a sample mean hits the population mean. The spread shrinks by √n, not by n — so four times as many pizzerias only halves the wobble. That √n is the single most expensive fact in statistics: precision costs the square of what you would hope.
If the population is normal, x̄ is normal for any n. If it is not, the Central Limit Theorem says the distribution of x̄ still becomes approximately normal as n grows; the textbook's working rule is n ≥ 30. Pick an ugly population and watch.
Averaging is a machine for making bells. That is why the normal table keeps working on data that is not normal.
Suppose Geneva prices have µ = CHF 20 and σ = CHF 4.20. How likely is a price over CHF 21? It depends on whether you mean one pizzeria or the average of several. Move n and watch the answer fall.
In Week 2, 11 of your 30 pizzerias charged over CHF 20: p̂ = 0.37. Another thirty would give another p̂. If the true share in Geneva is p:
p̂ is approximately normal when n is large enough. The textbook's check: the interval p ± 3σp̂ must lie entirely inside [0, 1]. If it spills below 0 or above 1, a bell centred at p would put probability on impossible shares, so the normal approximation is not safe.
In Week 1 you met the idea of a test in one breath: assume the claim, ask how surprising your data would be, decide. Now you can do the middle step properly. If the magazine were right (µ = 21, and σ ≈ 4.20), the mean of thirty would be approximately normal with σx̄ = 4.20/√30 = 0.767. How often would it come out as low as 19.93?
The same move works for a proportion: the textbook's Exercise 25 in §6.3 (Part 2, Exercise 10) asks whether a smoking campaign worked, and there the answer is 7 in 10,000 — a very different verdict.
A sampling distribution tells you what "normal wobble" looks like. Evidence is whatever falls outside it — and Week 7 decides where "outside" begins.
This week's exercises are not invented: each one is reproduced exactly as it appears at the end of a section of the textbook, labelled with its section and number, so you can find it in the book. Your instructor puts each exercise on the slide; you work it here, by hand, with a calculator and the normal table. Check marks your number immediately, Hint nudges without giving it away, and the Solution button opens when the room does that exercise together.
About 110 minutes for all ten. Keep σ/√n unrounded, round z to two decimals, give probabilities to four — the book's method. The book writes the sample mean as X̄ and the sample proportion as P̂. Parts that ask you to explain are answered in the worked solution. Your answers are saved and reach your instructor's dashboard. Nothing here is graded.
One attempt per question, every answer explained. Your score syncs to your instructor's dashboard as a self-check but does not count toward your grade. Have a calculator and the table to hand.
In Week 6 the sampling distribution is turned around. This week: "if µ is 20, where will x̄ land?" Next week: "x̄ landed at 19.93 — where is µ?" The answer is a confidence interval, and it is built from exactly the ±1.96·σ/√n of section 5.4. Before that, the Week 5 homework: thirteen more exercises from the book, by hand.
To pin down the Geneva average you pay for each visit — about CHF 20 a pizzeria. Choose the margin you want (95% of the time) and see how many visits it takes and what it costs. Watch what halving the margin does.
Half the margin costs four times the visits. That is σ/√n, paid in Margheritas.
📎 Everything saves automatically and reaches your instructor's dashboard. This is the fifth instalment of the individual project.
Applied Statistics · Week 5 · Sampling Distributions · Jan Erik Meidell · HEG Genève
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