*Section author: Danielle J. Navarro and David R. Foxcroft*

# Estimating unknown quantities from a sample

At the start of the last chapter I highlighted the critical distinction
between *descriptive statistics* and *inferential statistics*. As discussed
in Descriptive statistics, the role of descriptive statistics is to
concisely summarise what we *do* know. In contrast, the purpose of inferential
statistics is to “learn what we do not know from what we do”. Now that we have
a foundation in probability theory we are in a good position to think about
the problem of statistical inference. What kinds of things would we like to
learn about? And how do we learn them? These are the questions that lie at the
heart of inferential statistics, and they are traditionally divided into two
“big ideas”: estimation and hypothesis testing. The goal in this chapter is
to introduce the first of these big ideas, estimation theory, but I’m going
to witter on about sampling theory first because estimation theory doesn’t
make sense until you understand sampling. As a consequence, this chapter
divides naturally into two parts Samples, populations and sampling through
Sampling distributions and the central limit theorem are focused on sampling theory, and
Estimating population parameters and Estimating a confidence interval make use of
sampling theory to discuss how statisticians think about estimation.