Methods of statistical inference enable the investigator to argue from the particular observations in a sample to the general case. A statistical inference can sometimes be incorrect. Nevertheless, one of the great intellectual advances of the twentieth century is the realization that strong scientific evidence can be developed on the basis of many, highly variable, observations.
The subject of statistical inference deals with collecting informative data, interpreting these data, and drawing conclusions. Statistical inference includes all processes of acquiring knowledge that involve fact finding through the collection and examination of data. These data collection processes are as diverse as opinion polls, agricultural field trials, clinical trials of new medicines, and the studying of properties of exotic new materials. Statistical inference has permeated all fields of human endeavor in which the data-based evidence is being used.
A few characteristics are common to statistical inference projects. First, relevant data must be collected. Second, variability is unavoidable in observations even when observations are made under the same or very similar conditions. The third, is that access to a complete set of data of the population is either not feasible from a practical standpoint or is physically impossible to obtain.
The first step in making a statistical inference is to model the population(s) by a probability distribution which has a numerical feature of interest called a parameter. The problem of statistical inference arises when we want to make generalizations about the population based on sample data.
A statistic, based on a sample, must serve as the source of information about a parameter. The salient points regarding statistics.
Because a sample is only part of the population, the numerical value of the statistic will not be the exact value of the parameter. The observed value of the statistic depends on the particular
sample selected. Some variability in the values of a statistic, over different
samples, is unavoidable.
The two main classes of inference problems are estimation of parameter(s) and testing hypotheses about the value of the parameter(s).
The first class consists of point estimators, a single number estimate of the value of the parameter, and interval estimates. Typically, the interval estimate specifies an interval of plausible values for the parameter but the subclass also includes prediction intervals for future observations.
A test of hypotheses provides a yes/no answer as to whether the parameter lies in a specified region of values.
Because statistical inferences are based on a sample, they will sometimes be in error. Because the actual value of the parameter is unknown, even a test of hypotheses may yield the wrong yes/no answer and the interval of plausible values may not contain the true value of the parameter.
Statistical inferences, or generalizations from the sample to the population, are founded on an understanding of the manner in which variation in the population is transmitted, via sampling, to variation in a statistic.
There are two primary approaches, frequentist and Bayesian, for making statistical inferences. Both are based on the likelihood but their frameworks are entirely different.
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