## Content

### Using probability theory to make an inference

In the module Random sampling , we looked at the patterns that occur when taking repeated samples from an underlying distribution, such as a Normal, exponential or uniform distribution. In that module, we generally assumed knowledge of the underlying distribution and its associated parameters. Statements like the following were made:

- 'Suppose we have a random sample from the exponential distribution with mean 7.'
- 'Consider a random sample from the Normal distribution with mean 30 and standard deviation 7.'

But knowing the distribution from which we are sampling is not a common scenario. The opposite is the case. We are often confronted with a situation where we need to make an inference about an unknown population mean, and we do not know the true distribution from which we are sampling.

A remarkable result known as the central limit theorem makes it possible to draw inferences about an unknown population mean, even when the underlying distribution is unknown. This result, the proof of which is beyond the scope of the curriculum, is at the heart of the material covered in this module.

The theory covered in the earlier modules on probability and probability distributions is the foundation for making inferences about unknown population characteristics such as the population mean. In general terms, this is known as **statistical inference**.

Next page - Content - The sample mean \(\bar{X}\) as a point estimate of \(\mu\)