Testing how well a complete model fits the data

In the previous chapter we learned how to test hypotheses concerning the value of important quantities associated with a population. What we tested was whether a particular model of the chosen type could be supported or should be rejected on the basis of data observed in a random sample. There are times, however, when we might have doubts about the very form of the model, when, for instance, we are uncertain whether it is appropriate for the population in which we are interested to be modelled as a normally distributed population.

Imagine the case where a School District in the USA wishes to identify those children beginning school who should be provided with speech therapy. Rather than involve themselves in the lengthy and expensive business of constructing a new articulation test, they plan to use a test which is already available. One test which seems on the surface to be suitable for this purpose is British. It has been validated and standardised in Glasgow in such a way that for the whole population of 5-year-old children in Glasgow the scores on the test are normally distributed, with a mean score of 50 and standard deviation of 10. In order to discover whether scores on the test will have similar properties when used with 5-year-old children in its own area, the US School District administers it to a random sample of 184 of these children.