Introduction to Statistical inference

Statistical inference is the act of using observed data to infer unknown properties and characteristics of the probability distribution from which the observed data have been generated. The set of data that is used to make inferences is called sample.

        In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation. More substantially, the terms statistical inference, statistical induction and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from data sets arising from systems affected by random variation. Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to well-defined situations and that it should be general enough to be applied across a range of situations.

Scope

For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:

• a statistical model of the random process that is supposed to generate the data, and
• a particular realization of the random process; i.e., a set of data.

The conclusion of a statistical inference is a statistical proposition. Some common forms of statistical proposition are:

• an estimate; i.e., a particular value that best approximates some parameter of interest,
• a confidence interval (or set estimate); i.e., an interval constructed from the data in such a way that, under repeated sampling of datasets, such intervals would contain the true parameter value with the probability at the stated confidence level,
• a credible interval; i.e., a set of values containing, for example, 95% of posterior belief,
• rejection of a hypothesis
• clustering or classification of data points into groups

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Introduction of Statistical inference(PDF)
An Introduction to Statistical. Inference and Its Applications