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Chi-Square Test

Chi-Square Test:

Chi-square test (or Math image -test) is a statistical test for testing the null hypothesis that the distribution of a discrete random variable coincides with a given distribution. It is one of the most popular goodness-of-fit tests .

For example, in a supermarket, relative frequencies of purchasing 4 brands of tee have been 0.1, 0.4, 0.2, and 0.3 during the last year; during the last week the number of packets sold have been 31, 41, 22, 18 for the 4 brands, respectively. Has the preference changed – i.e. probabilities of purchasing now differs from the last year average preferences, or the deviations in the observed relative frequencies is caused by chance alone?

The chi-square test, besides discrete variables, is often applied to problems involving continuous random variables . In this case, the values of a continuous variable are transformed to a discrete variable with a finite number of values – e.g. the whole range of possible values is split into a finite number of intervals, and every such interval is considered as a discrete value (e.g. age groups “20…29”, “30…39”, etc). Then the chi-square test is applied to the new discrete variable.

For small samples, the classical chi-square test is not very accurate – because the sampling distribution of the statistic of the test differs from the chi-square distribution . In such cases, Monte Carlo simulation is a more reasonable approach. In many cases such simulation can be carried out by creating an artificial sample with the given proportion of values and applying a resampling procedure to this sample. Besides the one-sample chi-square test, there are variants of the test for comparison of the distribution of two or several samples. For these variants, a permutation version of the test is more accurate when at least one sample is small. See more on the use of resampling and permutation in short online courses

Resampling ,

and in the online book Resampling: The New Statistics

The chi-square test is typically used in categorical data analysis , e.g. to check if two such variables are independent random variables ).

The chi-square test is based on the chi-square statistic .

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