ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 2 ST 520, A. TSIATIS and D. Zhang
We start by reviewing the definition of a confidence interval and then show how to construct an
exact confidence interval for the parameter π of a binomial distribution.
Definition: The definition of a (1 − α)-th confidence region (interval) for the parameter π is as
follows:
For each realization of the data X = k, a region of the parameter space, denoted by C(k) (usually
an interval) is defined in such a way that the random region C(X) contains the true value of
the parameter with probability greater than or equal to (1 − α) regardless of the value of the
parameter. That is,
Pπ{C(X) ⊃ π} ≥ 1 − α, for all 0 ≤ π ≤ 1,
where Pπ(·) denotes probability calculated under the assumption that X ∼ b(n, π) and ⊃ denotes
“contains”. The confidence interval is the random interval C(X). After we collect data and obtain
the realization X = k, then the corresponding confidence interval is defined as C(k).
This definition is equivalent to defining an acceptance region (of the sample space) for each value
π, denoted as A(π), that has probability greater than equal to 1 − α, i.e.
in which case C(k) = {π : k ∈ A(π)}.
Pπ{X ∈ A(π)} ≥ 1 − α, for all 0 ≤ π ≤ 1,
We find it useful to consider a graphical representation of the relationship between confidence
intervals and acceptance regions.
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