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A chi-squared test (also chi-square or χ 2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables ( two dimensions of the contingency table ) are independent in influencing the test statistic ...
Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value.
The chi-square distribution has (k − c) degrees of freedom, where k is the number of non-empty bins and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one. For example, for a 3-parameter Weibull distribution, c = 4.
For the test of independence, also known as the test of homogeneity, a chi-squared probability of less than or equal to 0.05 (or the chi-squared statistic being at or larger than the 0.05 critical point) is commonly interpreted by applied workers as justification for rejecting the null hypothesis that the row variable is independent of the ...
In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. It is the distribution of the positive square root of a sum of squared independent Gaussian random variables .
The following example is adapted and abridged from Stuart, Ord & Arnold (1999, §22.2). Suppose that we have a random sample, of size n, from a population that is normally-distributed. Both the mean, μ, and the standard deviation, σ, of the population are unknown. We want to test whether the mean is equal to a given value, μ 0.
To be a real statistical hypothesis test, this example requires the formalities of a probability calculation and a comparison of that probability to a standard. A simple generalization of the example considers a mixed bag of beans and a handful that contain either very few or very many white beans. The generalization considers both extremes.
The demonstration of the t and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models , including linear regression and analysis of variance .