## Content

### Population parameters and sample estimates

In the module Random sampling, the distinction between a population and a sample is described. Over the previous sections, we have considered taking samples of size \(n\) from a population in which the true proportion of people preferring Labor is 0.5. In considering this example, and others from probability, it is common to see the following reaction: 'But how do you know that the true proportion preferring Labor is 0.5?'

In many examples in the module Probability, there were assumptions made about specific probabilities, and the implications were explored. This is important, because we do need to understand the rules of probability and the nature of random variables and distributions. One of the main reasons for understanding this theoretical material is that it is the foundation for making inferences in real-life situations that we care about, such as 'What is the true proportion preferring Labor?' and 'How precise is our estimate?'

The actual proportion of people in the population preferring Labor is an example of a **population parameter**. It is important to make the distinction between this population parameter and a sample estimate. In practice (unlike in our voting-preference example), we are interested in finding out about an unknown population parameter; this is a proportion of the population, and has a fixed value. We collect data from a random sample in order to obtain a sample estimate of the population parameter. As we have seen, different samples from the same population do not all give the same estimate: rather, they will vary.

The unknown population parameter, the true proportion, is \(p\). An estimate we obtain from a single sample, the observed sample proportion, is the point estimate \(\hat{p}\). The aim of the methods we describe later in this module is to infer something about the parameter of a population from the sample. This is an **inference** because there is uncertainty about the parameter. We can, however, quantify this uncertainty.

The uncertainty involved in using sample proportions to estimate population proportions can be understood by considering the distribution of sample proportions when we sample repeatedly from the same population. Here we think of the sample proportion as a random variable. It varies from sample to sample and has a distribution. By understanding the distributional properties of the sample proportion \(\hat{P}\) as an estimator of the population proportion, we can quantify the uncertainty in a sample estimate of a population parameter.

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