Statistical inference

The term statistical inference describes statistical techniques that obtain information about a population parameter (or parameters) based on a single random sample from that population. There are two different but related types of question about the population parameter (or parameters) that we might ask:

What parameter values would be consistent with the sample data?

This branch of inference is called estimation and its main tool is a confidence interval. We described confidence intervals in the previous chapter.

A manufacturer of muesli bars needs to describe the average fat content of the bars (the mean of the hypothetical population of fat contents that would be produced using the recipe). Several bars are analysed and their fat contents are measured.

The sample mean is a point estimate of the population mean, and a 95% confidence interval can also be found.

Are the sample data consistent with some statement about the parameters?

This branch of inference is called hypothesis testing and is the focus of this chapter.

A particular brand of meusli bar is claimed by the manufacturer to have a fat content of 3.4g per bar. A consumer group suspects that the manufacturer is understating the fat content, so a random sample of bars is analysed.

The consumer group must assess whether the data are consistent with the statement (hypothesis) that the underlying population mean is 3.4g.

Uncertainty and strength of evidence

When we studied parameter estimation, we saw that a population parameter cannot be determined exactly from a single random sample — there is a 5% chance that a 95% confidence interval will not include the true population parameter.

In a similar way, a single random sample can rarely provide enough information about a population parameter to allow us to be sure whether or not any statement about that parameter will be true. The best we can hope for is an indication of the strength of the evidence against the statement.

The remainder of this chapter explains how this evidence is obtained and reported.