portionp. It is usually either impossible, too costly, or too time consuming to
obtain the entire population data on any variable in order to learn about a
parameter involved in its distribution. Decisions in health science are thus often
made using a small sample of a population. The problem for a decision maker
is to decide on the basis of data the estimated value of a parameter, such as the
population mean, as well as to provide certain ideas concerning errors asso-
ciated with that estimate.
4.1.1 Statistics as Variables
A parameter is a numerical property of a population; examples include popu-
lation meanmand population proportionp. The corresponding quantity ob-
tained from a sample is called astatistic; examples of statistics include the
sample meanxand sample proportionp. Statistics help us draw inferences or
conclusions about population parameters. After a sample has already been
obtained, the value of a statistic—for example, the sample meanx—is known
and fixed; however, if we take a di¤erent sample, we almost certainly have a
di¤erent numerical value for that same statistic. In this repeated sampling con-
text, a statistic is looked upon as a variable that takes di¤erent values from
sample to sample.
4.1.2 Sampling Distributions
The distribution of values of a statistic obtained from repeated samples of the
same size from a given population is called thesampling distributionof that
statistic.
Example 4.1 Consider a population consisting of six subjects (this small size is
impractical, but we need something small enough to use as an illustration here).
Table 4.1 gives the subject names (for identification) and values of a variable
under investigation (e.g., 1 for a smoker and 0 for a nonsmoker). In this case
the population meanm(also the population proportionpfor this very special
dichotomous variable) is 0: 5 ð¼ 3 = 6 Þ. We now consider allpossiblesamples,
without replacement, of size 3; none or some or all subjects in each sample have
TABLE 4.1
Subject Value
A1
B1
C1
D0
E0
F0BASIC CONCEPTS 149