Last week you counted: Margheritas, desserts, warranties. This week you measure: delivery time, dough weight, the fill of a can. A measured quantity can take any value in a range, so probability is no longer a bar you add up — it is an area under a curve. The curve that matters most is the normal distribution, and by the end of the session you will read its table forwards and backwards and turn it into a promise the pizzeria can print on its menu. This is the textbook's chapter 5.
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.
You timed a month of Friday service. The time from order to plate-on-the-table averages 14 minutes with a standard deviation of 3 minutes, and the times pile up in a bell. A food blogger will write about any place that serves in under 12 minutes. What share of your pizzas make it?
Last week's random variables took values you could list — 0, 1, 2 Margheritas — and a probability was a bar. A time can be 11.7 minutes, or 11.73, or 11.7342…: there are too many values to list, and the chance of any exact one is zero. What has a probability is an interval — "under 12 minutes" — and that probability is an area. Section 4.4 computes the area you just guessed.
Counting gave you bars. Measuring gives you a curve, and the answer is the area under it.
The density function of X is a curve with two properties: it never goes below the axis, and the total area under it is 1. The probability that X falls between a and b is the area under the curve between a and b. A single point has no width, so no area: P(X = x) = 0, and P(X < 12) and P(X ≤ 12) are the same number.
The simplest density is flat: the uniform distribution. The oven takes between 10 and 16 minutes to reach temperature, any time in that range equally likely. The density is a rectangle of width 6, so its height must be 1/6 — the area has to come to 1.
A normal distribution is the bell you met in Week 2's Empirical Rule, now with exact areas. Every normal curve is the same shape, stretched and shifted; the one with µ = 0 and σ = 1 is called the standard normal, its variable is written Z, and its areas are printed once, in a table. The table gives the area to the left: P(Z < z).
The table only knows left tails. Every other area is one subtraction, or one use of symmetry, away.
The table is for Z only. To use it for delivery times, turn each time into a z-score — how many standard deviations it sits from the mean, exactly Week 2's z. Then P(X < x) = P(Z < z). Round z to two decimals, because that is what the table has.
So far: value in, area out. The other question is just as common: which time do 95% of pizzas beat? Find the area inside the table, read off z, then undo the standardisation: x = µ + z·σ. The z that leaves an area c in the right tail is written zc. Two of them you will use every week from now on: z0.05 = 1.645 and z0.025 = 1.96.
A promise is a percentile. "Within 19 minutes, 95% of the time" is z0.05 = 1.645 in the kitchen's own units.
Last week the chance of selling out, P(X ≥ 121) for X binomial with n = 140 and p = 0.8, took the site twenty terms of the formula: 3.2%. Look at those bars again — they are a bell. When n is large, a binomial is close to a normal with the same mean and standard deviation: µ = np = 112 and σ = √(npq) = 4.73.
The half-unit correction uses 120.5 instead of 120, because the bar for 121 starts at 120.5 on a continuous scale. It is a refinement, not a rule you will be examined on — but it shows why the plain line is a little high.
One line, one table lookup, and within a percentage point of twenty terms. That is why the normal curve runs the rest of this course.
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. Round z to two decimals and give probabilities to four, as the book does. When the book says "Figure 12.2", it means the cumulative normal table ↗. Parts that ask you to sketch or 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 5 the normal curve turns up somewhere you did not put it: in the average of a sample. Take thirty pizzerias, compute their mean price, do it again and again — the means pile up in a bell, whatever the prices themselves look like. That is the Central Limit Theorem, and today's table is how you use it. Before that, the Week 4 homework: thirteen more exercises from the book, by hand.
A menu line like "your pizza in 20 minutes or it's free" is a percentile with a price tag. Choose the minutes; see the share of pizzas you would give away, and what that costs over a Friday of 140 pizzas at CHF 19.
Every guarantee is a tail area. Knowing its size is the difference between a marketing idea and a loss.
📎 Everything saves automatically and reaches your instructor's dashboard. This is the fourth instalment of the individual project.
Applied Statistics · Week 4 · Continuous Random Variables · Jan Erik Meidell · HEG Genève
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