We can use what we have learnt to start making some inferences about data. Maybe we have collected measurements of a quantity and wish to see if these are consistent with some theoretical expectation. We don't just want to compute the sample mean but to compare it with something else. Perhaps we have two samples, taken under different conditions (such as a ‘treatment’ and ‘control’ group) and wish to see if their mean responses differ. Another very common situation is that we have measurements of some response (y) taken at different values of some explanatory variable (x) and wish to quantify the way that y responds. We can go some way to getting useful inferences out of such data using numerical and graphical summaries (Chapter 2). These can be refined once we have studied some probability theory (Chapters 4 and 5).

Inference about the mean of a sample

We take repeated measurements of a single quantity, or measure the same quantity for each member of a finite sample, and wish to discover whether these data are consistent with a predetermined theoretical value. We want to know if our sample is consistent with being randomly drawn from a theoretical population, with some particular population mean. As an example, let's consider the first ‘experiment’ (batch of 20 runs) of Michelson's dataset (see Appendix B, section B.1).