Statistical inference is the process of making conclusions about a population
based on information from sample data. In order for the sample data to be
meaningful, the sample must be random, i.e., the selection of a particular
sample must have the same probability as that of any other possible sample
out of a given population. The following are some terms relevant to the
concept of random sampling:
Population: collection of all conceivable observations of a process or
attribute of a component.
Sample: sub-set of a population.
Random sample: a sample representative of the population.
Random variable: real-valued function defined on a sample space. Could
be discrete or continuous.
If the population follows a certain probability distribution that depends on
a parameter θ, a random sample of observations (X
n, can be used to estimate θ.
Sampling distribution: the joint probability distribution of X
A statistic: any function of the observations that is quantifiable and does
not contain any unknown parameters. A statistic is a random variable
that provides a means of estimation.
Point estimation: when a single value of the parameter θ is provided.
Confidence interval: a numerical interval that contains the parameter θ at
a given level of probability.
Estimator: rule or method of estimation of the parameter θ.
Estimate: value that the estimator yields in a particular application.
Example 1 -- Let X represent the time (hours) required by a specific
manufacturing process to be completed. Given the following sample of values
of X: 2.2 2.5 2.1 2.3 2.2. The population from where this sample is
taken is the collection of all possible values of the process time, therefore, it is
an infinite population. Suppose that the population parameter we are trying
,... , X
), of size
,... , X