Page 66 - Computational Statistics Handbook with MATLAB

52 Computational Statistics Handbook with MATLAB

using statistical techniques, we can measure and manage the degree of uncer-
tainty in our results.
Inferential statistics is a collection of techniques and methods that enable
researchers to observe a subset of the objects of interest and using the infor-
mation obtained from these observations make statements or inferences
about the entire population of objects. Some of these methods include the
estimation of population parameters, statistical hypothesis testing, and prob-
ability density estimation.
The target population is defined as the entire collection of objects or indi-
viduals about which we need some information. The target population must
be well defined in terms of what constitutes membership in the population
(e.g., income level, geographic area, etc.) and what characteristics of the pop-
ulation we are measuring (e.g., height, IQ, number of failures, etc.).
The following are some examples of populations, where we refer back to
those described at the beginning of Chapter 2.

• For the piston ring example, our population is all piston rings
contained in the legs of steam-driven compressors. We would be
observing the time to failure for each piston ring.
• In the glucose example, our population might be all pregnant
women, and we would be measuring the glucose levels.
• For cement manufacturing, our population would be batches of
cement, where we measure the tensile strength and the number of
days the cement is cured.
• In the software engineering example, our population consists of all
executions of a particular command and control software system,
and we observe the failure time of the system in seconds.

In most cases, it is impossible or unrealistic to observe the entire popula-
tion. For example, some populations have members that do not exist yet (e.g.,
future batches of cement) or the population is too large (e.g., all pregnant
women). So researchers measure only a part of the target population, called
a sample. If we are going to make inferences about the population using the
information obtained from a sample, then it is important that the sample be
representative of the population. This can usually be accomplished by select-
ing a simple random sample, where all possible samples are equally likely to
be selected.
A random sample of size n is said to be independent and identically dis-
tributed (iid) when the random variables X X … X, 2 , , n each have a common
1
probability density (mass) function given by f x() . Additionally, when they
are both independent and identically distributed (iid), the joint probability
density (mass) function is given by

,
(
(
(
,
f x 1 … x n ) = fx 1 ) × … × fx n , )
© 2002 by Chapman & Hall/CRC