The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

62 CHAPTER 5 Estimating Means and Proportions

This can easily happen if B 1 is small, and σ 22 is much larger than
σ 12. An estimator is called effi cient if as n gets large, it approaches the
lowest possible mean square error. So if there is an unbiased estimator
that has the smallest possible variance among all consistent estimates,
then it is the best. If a biased estimator is consistent and has its variance
approaching the lowest possible value, then it is effi cient because the
bias approaches zero under these same conditions. This is important
when considering maximum likelihood estimation.

5.3 CONFIDENCE INTERVALS

Point estimates are useful but do not describe the uncertainty associated
with them. Confi dence intervals include the point estimate (often at the
center of the interval), and they express the uncertainty by being an
interval whose width depends on the uncertainty of the estimate.
Formally, confi dence intervals are defi ned as being one - sided or two -
sided, and they have a confi dence level associated with them. For
example, a 95% two - sided confi dence interval for the mean would have
the interpretation that if samples of size n are repeatedly taken, and for
each such sample, a 95% confi dence interval for the mean is calculated,
then approximately 95% of those intervals would include the popula-
tion mean and approximately 5% of the intervals would not. *
As an example, we will show you how to determine a two - sided
95% confi dence interval for the mean, μ , of a normal distribution when
the standard deviation, σ , is assumed known. In that case, the sample
mean X^^ has a normal distribution with mean μ and standard deviation

(^) σ/ n. So let ZX=−()/(/)ˆ μσn. Z has a normal distribution with
mean 0 and standard deviation 1. From the table of the standard normal
distribution, we have P [ − 1.96 ≤ Z ≤ 1.96] = 0.95. We use this fact to

• In contrast for another form of inference called the Bayesian approach, the analogue to
  the confi dence interval is the credible interval. Because it treats parameters like random
  variables, a 95% credible interval is an interval that has probability 0.95 of including the
  parameter. This is not so for confi dence intervals. The Bayesian method takes what is called
  a prior distribution, and based on Bayes ’ rule creates a posterior distribution combining the
  prior with the likelihood function for the data. A credible region is determined by integrating
  the probability density of the posterior distribution until the area under the curve between
  a and b , with a < b , integrates to 0.95.