## Assumed knowledge

The contents of the modules:

- Inference for means, which assumes
- Continuous probability distributions;
- Random sampling;
- Exponential and Normal distributions;
- Inference for proportions.

## Motivation

- Why can we rely on random samples to test claims about population means?
- We get a different result every time we take a sample. How consistent are the sample results with a claim about the population mean?
- How can we quantify the consistency of the results from a sample with a claim about a population mean?

Some material from the Inference for means "Motivation" is equally applicable here, and in order to create a self-contained module, this material is repeated below (in the next four paragraphs).

In the Random sampling module students were introduced to random sampling from a variety of distributions. In that module, the distribution from which the samples were taken was consistently assumed to be known.

In practice, 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 was given in the Random sampling module, using mainly visual techniques.

One important inference in many different contexts is about the unknown population mean.

A random sample can be used to provide a point estimate of the unknown population mean \(\mu\): the sample mean \(\bar{x}\) is an estimate of the population mean \(\mu\).

In the Inference for means module we discussed quantification of the uncertainty in the estimate of a population mean with a confidence interval for the (unknown) population mean. This is one important approach to inference about the unknown population mean.

There is a different approach to making inferences about the unknown population mean. In this second approach, we assert a possible value of the (unknown) population mean.The random sample provides an estimate of the unknown mean, and we evaluate the plausibility of the sample mean we have observed considering the value we postulated for the population mean. If our assertion about the population mean is correct, how likely is the result we have observed?

This provides methods for addressing questions such as these:

- A manufacturer claims the average weight of chocolate bars is 50 grams. How consistent are the bars produced, with this claim?
- The Australian Government's Average Quantity System specifies rules for deciding if a sample of products of a particular type is consistent with the quantity stated on the packaging. If a sample of jars of honey weighs 249 grams on average, what quantity could be stated on the packaging?
- The Australian Government's Department of Health recommends that parents of children aged 5 to 12 years "limit use of electronic media for entertainment to no more than two hours a day". Is there evidence that the average amount of electronic media use is consistent with the upper limit of two hours per day?

Next page - Content - Confidence intervals and hypothesis tests