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Chapter 3: Sampling Concepts 53

which is simply the product of the individual densities (or mass functions)
evaluated at each sample point.
There are two types of simple random sampling: sampling with replace-
ment and sampling without replacement. When we sample with replace-
ment, we select an object, observe the characteristic we are interested in, and
return the object to the population. In this case, an object can be selected for
the sample more than once. When the sampling is done without replacement,
objects can be selected at most one time. These concepts will be used in Chap-
ters 6 and 7 where the bootstrap and other resampling methods are dis-
cussed.
Alternative sampling methods exist. In some situations, these methods are
more practical and offer better random samples than simple random sam-
pling. One such method, called stratified random sampling, divides the pop-
ulation into levels, and then a simple random sample is taken from each level.
Usually, the sampling is done in such a way that the number sampled from
each level is proportional to the number of objects of that level that are in the
population. Other sampling methods include cluster sampling and system-
atic random sampling. For more information on these and others, see the
book by Levy and Lemeshow [1999].
Sometimes the goal of inferential statistics is to use the sample to estimate
or make some statements about a population parameter. Recall from Chapter
2 that a parameter is a descriptive measure for a population or a distribution
of random variables. For example, population parameters that might be of
interest include the mean (µ), the standard deviation (σ), quantiles, propor-
tions, correlation coefficients, etc.
A statistic is a function of the observed random variables obtained in a
random sample and does not contain any unknown population parameters.
Often the statistic is used for the following purposes:

• as a point estimate for a population parameter,
• to obtain a confidence interval estimate for a parameter, or
• as a test statistic in hypothesis testing.

Before we discuss some of the common methods for deriving statistics, we
present some of the statistics that will be encountered in the remainder of the
, , , of
text. In most cases, we assume that we have a random sample, X 1 … X n
independent, identically (iid) distributed random variables.

aamm pleple
an
and
aancence
VVaarr ii
ple
SSampleample Me eanandSandSa amm pleV Va ar ri iaancence
SampleM
Sample
an
MM eeanandSS
A familiar statistic is the sample mean given by
© 2002 by Chapman & Hall/CRC