SECTION 6.4 Confidence Intervals 381
A very common misconception is that the above two statements
mean that the population mean lies within the above reported inter-
val with probability 95%. However, this is meaningless: either the
population mean does or doesn’t lie in the above interval; there is no
randomness associated with the interval reported! As we’ll see, the
randomness is associated with the process of arriving at the interval
itself. If 100 statisticians go out and compute 95% confidence intervals
for the mean, then roughly 95 of the computed confidence intervals will
actually contain the true population mean. Unfortunately, we won’t
know which ones actually contain the true mean!
6.4.1 Confidence intervals for the mean; known population
variance
While it is highly unreasonable to assume that we would know the
variance of a population but not know the mean, the ensuing discussion
will help to serve as a basis for more practical (and realistic) methods
to follow. Therefore, we assume that we wish to estimate the meanμof
a population whose varianceσ^2 is known. It follows then, that ifX is
the random variable representing the mean ofnindependently-selected
samples, then
- the variance ofXis
σ^2
n
,and- (ifnis “large”)^24 the random variableXis approximately normally
distributed.
In the ensuing discussion, we shall assume either that we are sam-
pling from an (approximately) normal population or thatnis relatively
large. In either case,X will be (approximately) normally distributed.
We have that
E(X) =μ, and Var(X) =σ^2
n;
(^24) A typical benchmark is to use sample sizes ofn≥30 in order for the normality assumption to
be reasonable. On the other hand, if we know—or can assume—that we are sampling from a normal
population in the first place, thenXwill be normally distributed for anyn.