Single Sample Problems

Abstract

In this chapter, we consider various problems of inference which arise when we are presented with a single set of data, as in the measuring equipment problem, Example 4.1. Following the reasoning given in § 2.6(iv) and § 3.5, it is frequently possible to assume that our data are independently and normally distributed with some (unknown) expectation and variance, μ and σ ². All the methods of this chapter are based on these assumptions. A simple check of normality can be carried out by plotting the data on normal probability paper, but this technique is unlikely to be fruitful with less than 20 or 30 observations. Wilk and Shapiro (1965) have devised a more elaborate test for normality which achieves quite high power even in small sample sizes, see the reference for details. Simple checks of statistical independence can also be made, but these depend upon the form of dependence thought to be present, and will be discussed later. The vital question here is not whether the assumptions are valid, but whether possible deviations from them are important. This problem is considered in § 5.7.