4.2. Confidence Intervals 239That is, the probability that the interval includesθis 1 −α, which is called the
confidence coefficientor theconfidence levelof the interval.Once the sample is drawn, the realized value of the confidence interval is (l, u),
an interval of real numbers. Either the interval (l, u)trapsθor it does not. One way
of thinking of a confidence interval is in terms of a Bernoulli trial with probability
of success 1−α. If one makes, say,Mindependent (1−α)100% confidence intervals
over a period of time, then one would expect to have (1−α)Msuccessful confidence
intervals (those that trapθ) over this period of time. Hence one feels (1−α)100%
confident that the true value ofθlies in the interval (l, u).
A measure of efficiency based on a confidence interval is its expected length.
Suppose (L 1 ,U 1 )and(L 2 ,U 2 ) are two confidence intervals forθthat have the same
confidence coefficient. Then we say that (L 1 ,U 1 ) is more efficient than (L 2 ,U 2 )if
Eθ(U 1 −L 1 )≤Eθ(U 2 −L 2 ) for allθ∈Ω.
There are several procedures for obtaining confidence intervals. We explore one
of them in this section. It is based on a pivot random variable. The pivot is usually
a function of an estimator ofθand the parameter and, further, the distribution of
the pivot is known. Using this information, an algebraic derivation can often be
used to obtain a confidence interval. The next several examples illustrate the pivot
method. A second way to obtain a confidence interval involves distribution free
techniques, as used in Section 4.4.2 to determine confidence intervals for quantiles.
Example 4.2.1(Confidence Interval forμUnder Normality).Suppose the random
variablesX 1 ,...,Xnare a random sample from aN(μ, σ^2 ) distribution. LetXand
S^2 denote the sample mean and sample variance, respectively. Recall from the last
section thatXis the mle ofμand [(n−1)/n]S^2 is the mle ofσ^2. By part (d) of
Theorem 3.6.1, the random variableT=(X−μ)/(S/
√
n)hasat-distribution with
n−1 degrees of freedom. The random variableTis our pivot variable.
For 0 <α<1, definetα/ 2 ,n− 1 to be the upperα/2 critical point of at-
distribution withn−1 degrees of freedom; i.e.,α/2=P(T>tα/ 2 ,n− 1 ). Using
a simple algebraic derivation, we obtain
1 −α = P(−tα/ 2 ,n− 1 <T <tα/ 2 ,n− 1 )= Pμ(
−tα/ 2 ,n− 1 <
X−μ
S/√
n<tα/ 2 ,n− 1)= Pμ(
−tα/ 2 ,n− 1
S
√
n<X−μ<tα/ 2 ,n− 1
S
√
n)= Pμ(
X−tα/ 2 ,n− 1S
√
n<μ<X+tα/ 2 ,n− 1S
√
n). (4.2.2)
Once the sample is drawn, letxandsdenote the realized values of the statisticsX
andS, respectively. Then a (1−α)100% confidence interval forμis given by
(x−tα/ 2 ,n− 1 s/√
n,x+tα/ 2 ,n− 1 s/√
n). (4.2.3)This interval is often referred to as the (1−α)100%t-intervalforμ. The estimate
of the standard deviation ofX,s/
√
n,isreferredtoasthestandard errorofX.