Assumed knowledge

The content of the modules:

• Continuous probability distributions
• Exponential and normal distributions
• Random sampling
• Inference for proportions .

Motivation

• Why can we rely on random samples to provide information about population means?
• Should we worry that different random samples taken from the same population will give different results?
• How variable are sample means obtained from different random samples?
• How can we quantify the uncertainty (imprecision) in the results from a sample?

The module Random sampling discusses sampling from a variety of distributions. In that module, it is assumed that we know the distribution from which the samples are taken. In practice, however, we typically do not know the underlying or parent distribution. We may wish to use a random sample to infer something about this parent distribution. An impression of how this might be possible is given in the module Random sampling , using just visual techniques.

One important inference in many different contexts is about the unknown population mean \(\mu\). A random sample can be used to provide a point estimate of the unknown population mean: the sample mean \(\bar{x}\) is an estimate of the population mean \(\mu\). There will be some imprecision associated with a single point estimate, and we would like to quantify this sensibly.

In this module, we discuss the distribution of the sample mean to illustrate how it serves as a basis for using a sample mean to estimate an unknown population mean \(\mu\). By considering the approximate distribution of sample means, we can provide a quantification of the uncertainty in an estimate of the population mean. This is a confidence interval for the unknown population mean \(\mu\).

This provides methods for answering questions like:

• What is our best estimate of the average number of hours per week of internet use by Australian children aged 5–8?
• What is the uncertainty in this estimate of the average number of hours per week of internet use by Australian children aged 5–8?

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