Probability space and Chi-squared test: Difference between pages

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[[File:Probability-measure.svg|thumb|Modelling a wheel of fortune using probability space]]
'''Chi-squared test''' (or <math>\Chi^2</math> test) is a [[statistical hypothesis test]]. It usually tests the [[hypothesis]] that "the experimental data does not differ from untreated data". That is a [[null hypothesis]]. The distribution of the test statistic is a [[chi-squared distribution]] when the null hypothesis is true.
'''Probability space''' is a [[mathematical model]] used to describe [[scientific]] [[experiments]] A probability space consists of three parts:
# A [[sample space]] which lists all possible outcomes
# A [[set]] of events. Each event associates zero or more outcomes
# A [[Function (mathematics)|function]] that assigns probabilities to each event


An ''outcome'' is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex ''events'' are used to characterize groups of outcomes. The collection of all such events is a ''[[σ-algebra]]'' <math>\scriptstyle \mathcal{F}</math>. Finally, there is a need to specify each event's likelihood of happening. This is done using the ''[[probability measure]]'' function, ''P''.
The test results are regarded as 'significant' if there is only one chance in 20 that the result could be got by chance.


Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ''ω'', from the sample space Ω. All the events in <math>\scriptstyle \mathcal{F}</math> that contain the selected outcome ''ω'' (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function ''P''.
There are three main groups of tests:
*Tests for distribution check that the values follow a given [[probability distribution]].
*Tests for independence check that the values are [[Independence (statistics)|independent]]; if this is the case, no value can be left out without losing information.
*Tests for homogeneity: These check that all samples taken have the same probability distribution, or are from the same set of values.


The prominent Soviet mathematician [[Andrey Kolmogorov]] introduced the notion of probability space, together with other [[axioms of probability]], in the 1930s.


[[Category:Mathematics]]
{{math-stub}}
 
[[Category:Statistics]]

Latest revision as of 02:25, 5 October 2016

Chi-squared test (or [math]\displaystyle{ \Chi^2 }[/math] test) is a statistical hypothesis test. It usually tests the hypothesis that "the experimental data does not differ from untreated data". That is a null hypothesis. The distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

The test results are regarded as 'significant' if there is only one chance in 20 that the result could be got by chance.

There are three main groups of tests:

  • Tests for distribution check that the values follow a given probability distribution.
  • Tests for independence check that the values are independent; if this is the case, no value can be left out without losing information.
  • Tests for homogeneity: These check that all samples taken have the same probability distribution, or are from the same set of values.


Template:Math-stub

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