Advanced High-School Mathematics

380 CHAPTER 6 Inferential Statistics

asn → ∞becomes arbitrarily close to the normal distribution with

meanμand variance

σ^2

n

.

Perhaps a better way to state the Central Limit Theorem is as fol-
lows. IfZis the normal random variable with mean 0 and variance 1,
then for any real numberz,

nlim→∞P

Ñ

X−μ

σ/

√

n

< z

é

=P(Z < z).

6.4 Confidence Intervals for the Mean of a Popu-

lation

A major role of statistics is to provide reasonable methods by which
we can make inferences about the parameters of a population. This
is important as we typically never know the parameters of a given
population.^23 When giving an estimate of the mean of a population, one
often gives an interval estimate, together with alevel of confidence.
So, for example, I might collect a sample from a population and measure
that the meanxis 24.56. Reporting this estimate by itself is not terribly
useful, as it is highly unlikely that this estimate coincides with the
population mean. So the natural question is “how far off can this
estimate be?” Again, not knowing the population mean, this question is
impossible to answer. In practice what is done is to report aconfidence
intervaltogether with aconfidence level. Therefore, in continuing
the above hypothetical example, I might report that

“The 95% confidence interval for the mean is 24.56±2.11.”

or that

“The mean falls within 24.56±2.11 with 95% confidence.”

(^23) In fact we almost never even know the population’s underlying distribution. However, thanks
to the Central Limit Theorem, as long as we take large enough samples, we can be assured of being
“pretty close” to a normal distribution.