Inferential Statistics 371
of error α. Then, with probability 1 – α, the true parameter is covered by the
interval [l(X), u(X)]. The probability
Pl([θ∈(),()]) 1XuX =−αis called confidence level.
It may not be possible to find bounds to obtain a confidence level exactly.
We, then, simply postulate for the confidence level 1 – α that
Pl([θ∈(),()]) 1XuX =−αis satisfied, no matter what the value θ may be.
definition and interpretation of a Confidence interval Given the definition of the
confidence level, we can refer to an interval [l(X), u(X)] as 1 – α confidence
interval if
Pl([θ∈(),()]) 1XuX =−αholds no matter what is the true but unknown parameter value θ.^4
The interpretation of the confidence interval is that if we draw an
increasing number of samples of constant size n and compute an interval,
from each sample, 1 – α of all intervals will eventually contain the true
parameter value θ.
The bounds of the confidence interval are often determined by some stan-
dardized random variable composed of both the parameter and point estima-
tor, and whose distribution is known. Furthermore, for a symmetric density
function such as that of the normal distribution, it can be shown that with
given α, the confidence interval is the tightest if we have pl = α/2 and pu = α/2
with pl and pu as defined before. That corresponds to bounds l and u with dis-
tributions that are symmetric to each other with respect to the the true param-
eter θ. This is an important property of a confidence interval since we seek
to obtain the maximum precision possible for a particular confidence level.
Often in discussions of confidence intervals the statement is made that
with probability 1 – α, the parameter falls inside of the confidence inter-
val and is outside with probability α. This interpretation can be misleading
in that one may assume that the parameter is a random variable. Recall
that only the confidence interval bounds are random. The position of the
(^4) Note that if equality cannot be exactly achieved, we take the smallest interval for
which the probability is greater than 1 – α.