ABSTRACT
A frequently occurring problem of no small importance is to determine whether or not a product meets a standard. The quality of the product is usually measured by a quantitative variable X defined on a population. Since the quality of the product is always subject to some kind of random variation, the standard is usually expressed as an assertion about the distribution of the variable X. For example, suppose X denotes the fuel efficiency of a car [measured in miles per gallon (mpg)], and the fuel efficiency standard is that the mean mpg µ be greater than µ0. The assertion that µ > µ0 is an example of a statistical hypothesis; it is a claim about the distribution of the variable X . For another example, consider the widely used Black-Scholes options pricing formula, which assumes that returns on a stock are normally distributed (see Section 5.9.1). Is this hypothesis consistent with the kinds of actual data listed in Tables 1.19 and 1.20? Problems of this sort are discussed in Section 8.4. In this chapter we learn how to use statistics such as the sample mean and sample variance to test the truth or falsity of a statistical hypothesis. We then extend these methods to testing hypotheses about two distributions where the goal is to study the difference between their means.
