Chapter 13: Statistical distributions

The chapter problem: the seed packet and the plant heights

A gardener buys a packet of 20 sunflower seeds. The supplier makes two claims on the packet.

• Each seed germinates with probability 0.8, and the seeds germinate independently.
• The height of a mature plant is normally distributed with mean 45 cm and standard deviation 6 cm.

The gardener plants all 20 seeds. Only 13 of them germinate. The plants that grow then have a range of heights.

Two questions follow. How likely is it that 13 or fewer of the 20 seeds germinate, if the first claim is true? And what proportion of mature plants should be taller than 52 cm, if the second claim is true?

By the end of the chapter you will answer both questions. You will also decide which of the two models fits which quantity, and why.

First contact — no working needed yet. Read the two claims. Predict which of these quantities is a count that you can list, and which is a measurement that can take any value in a range. Write your answer down.

Then predict whether 13 germinations out of 20 is unusual. Do not calculate anything yet. Keep both predictions for the Review.

By the end of this chapter you will be able to

• list the values and the probabilities of a discrete random variable, and check that the probabilities sum to 1;
• recognise the conditions for a binomial model and write $X \sim B(n, p)$;
• work out a binomial probability using $P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}$;
• find at-most, at-least and between probabilities for a binomial variable;
• describe the Normal distribution and use it as a model for a measurement;
• standardise a value and read a probability from a standard Normal table;
• explain how a histogram of many values approaches the Normal curve, and why the curve changes from concave to convex at $\mu \pm \sigma$;
• use the Normal distribution to approximate a binomial variable when the conditions are met;
• choose between the binomial and the Normal model for a context, and give a reason for your choice.

Where it is used

Manufacturers test samples to estimate how many items will fail. Doctors use the Normal curve to describe measurements such as blood pressure and birth weight. Insurers model the number of claims in a year. Weather forecasters describe the spread of possible temperatures.

Whenever a number varies from one trial to the next, a distribution describes how it varies. The two models in this chapter, the binomial and the Normal, cover a very large share of those situations.

Your turn to connect it

Think of one situation where you would count how many times something happens in a fixed number of tries, and one situation where you would measure something that varies around an average. Write both down. For each one, say whether the value is a count or a measurement. Keep your answer for the Review.

Roadmap

1. 13.1 Discrete probability distributions and the binomial distribution
2. 13.2 Calculating binomial probabilities
3. 13.3 The Normal distribution as a model
4. 13.4 Links to the binomial, histograms and points of inflection
5. 13.5 Selecting an appropriate distribution
6. Review