## Content

• Hypothesis testing for a population mean for a sample drawn from a normal distribution of known variance or for a large sample, including:
• $$P$$-values for hypothesis testing related to the mean.
• Formulation of a null hypothesis and an alternative hypothesis.
• Errors in hypothesis testing.

### Confidence intervals and hypothesis tests

In the module on Inference for means, the idea of confidence intervals is explained. This is one of the ways that we can express uncertainty about an estimated, unknown parameter.

This module deals with the other main way that we express an inference about an unknown parameter: hypothesis testing.

As we shall see, in any of the contexts we consider, we could work out a confidence interval, or we could carry out a hypothesis test. This raises obvious questions of principle and practice. Should we prefer one or the other? Are the two connected somehow, as intuition suggest they must be? If so, what is the connection?

These are all important questions and we will deal them later.

A useful practical observation to make is that in the reporting of statistical inferences, both approaches are common and likely to be seen, often regarding the same parameter in a particular application. That is, a study might report a 95% confidence interval for an unknown population parameter, and also provide the results of a hypothesis test about that parameter.

Since the two approaches are connected, it is not surprising that the underlying ideas and results from probability that support hypothesis testing are the same as those used for confidence intervals. This is an important insight, as it means that there is not a whole new structure to learn about: rather, it is a different way of thinking about inference that uses the same underlying structure. This structure is extensively dealt with in the module on Inference for means, and that should be understood thoroughly for the purposes of this module.

For completeness, we summarise these points in the next sub-section.

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