Applied Statistics Using SPSS, STATISTICA, MATLAB and R

3 Estimating Data Parameters
Making inferences about a population based upon a random sample is a major task
in statistical analysis. Statistical inference comprehends two inter-related
problems: parameter estimation and test of hypotheses. In this chapter, we describe
the estimation of several distribution parameters, using sample estimates that were
presented as descriptive statistics in the preceding chapter. Because these
descriptive statistics are single values, determined by appropriate formulas, they
are called point estimates. Appendix C contains an introductory survey on how
such point estimators may be derived and which desirable properties they should
have. In this chapter, we also introduce the notion and methodology of interval
estimation. In this and later chapters, we always assume that we are dealing with
random samples. By definition, in a random sample x1, …, xn from a population
with probability density function fX(x), the random variables associated with the
sample values, X1, …, Xn, are i.i.d., hence the random sample has a joint density
given by:
f x , x ,..., x ) = f ( x ) f ( x )... f ( x ) .
X (
1,
X 2 ,..., X n 1 2 n X 1 X 2
A similar result applies to the joint probability function when the variables are
discrete. Therefore, we rule out sampling from a finite population without
replacement since, then, the random variables X1, …, Xn are not independent.
Note, also, that in the applications one must often carefully distinguish between
target population and sampled population. For instance, sometimes in the
newspaper one finds estimation results concerning the proportion of votes on
political parties. These results are usually presented as estimates for the whole
population of a given country. However, careful reading discloses that the sample
(hopefully a random one) was drawn using a telephone enquiry from the
population residing in certain provinces. Although the target population is the
population of the whole country, any inference made is only legitimate for the
sampled population, i.e., the population residing in those provinces and that use
telephones.
3.1 Point Estimation and Interval Estimation
Imagine that someone wanted to weigh a certain object using spring scales. The
object has an unknown weight, ω. The weight measurement, performed with the
scales, has usually two sources of error: a calibration error, because of the spring’s
X
n