Statistical inference

The term statistical inference refers to techniques for obtaining information about a statistical model's parameter (or parameters) based on data that result from the model. In the context of this e-book, the statistical model is usually a standard distribution and the data are a random sample from that distribution. There are two different but related types of question about the 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.

Estimating μ

A manufacturer of muesli bars needs to describe the average fat content of the bars (the mean, \(\mu\), of the distribution 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 mean fat content of the bars, 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.

Testing whether μ is 3.4

A particular brand of muesli bar is claimed by the manufacturer to have a fat content of 3.4 g 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 mean fat content for this type of bar is \(\mu = 3.4\) g.

Uncertainty and strength of evidence

When we studied parameter estimation, we saw that a distribution's 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 parameter value.

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

We will next show how this evidence is obtained and reported.