Confidence Intervals

Abstract

We discussed interval estimation briefly in Chapters 3 and 4. We shall recall briefly how interval estimation differs from the considerations of Chapters 5 and 7, in which D consisted of the possible values of some function ϕ of the unknown parameter value θ, the decision d₀ meaning “my guess is that the true value of ϕ(θ) is d₀.” Thus d₀ was typically a single point of an interval D (i.e., for simplicity we think of the set of decisions as a real number interval), and these problems were termed ones of point estimation. If the decision is to be an interval of real numbers rather than a single point, the problem is called one of interval estimation. In a problem of this kind, a decision may be “my guess is that the table is between 4 and 4.3 feet long” rather than “my guess is that the table is 4.1 feet long.” Thus, if were discussing confidence intervals, D consists of a collection of intervals, and we may denote by [a, b] that decision which consists of the set of real numbers between a and b inclusive; the decision in the example of the table would be [4.0,4.3].