Page 465 - Probability and Statistical Inference

442 9. Confidence Interval Estimation

The confidence coefficient corresponding to the confidence interval J is de-
fined to be

However, the coverage probability P {θ ∈ (T (X), T (X))} will not in-
θ
U
L
volve the unknown parameter θ in many standard applications. In those situ-
ations, it will be easy to derive Inf θ∈T θ L U
P {θ ∈ (T (X), T (X))} because then it
will coincide with the coverage probability itself. Thus, we will interchange-
ably use the two phrases, the confidence coefficient and the coverage prob-
ability when describing a confidence interval.
Customarily, we fix a small preassigned number α ∈ (0, 1) and require a
confidence interval for θ with the confidence coefficient exactly (1 – α). We
refer to such an interval as a (1 – θ) or 100(1 – α)% confidence interval
estimator for the unknown parameter θ.
Example 9.1.1 Suppose that X , X are iid N(µ, 1) where µ(∈ ℜ) is the
2
1
unknown parameter.

Figure 9.1.1. Standard Normal PDF: The Shaded Area Between
–z and z = 1.96 Is 1 – α Where α = 0.05
α/2 α/2
First consider T (X) = X – 1.96, T (X) = X + 1.96, leading to the confi-
U
1
L
1
dence interval
The associated coverage probability is given by

which is .95 and it does not depend upon µ. So, the confidence coefficient
associated with the interval J is .95.
1