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The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in finding the confidence interval for estimating the population standard deviation of a normal distribution from a sample standard ...
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 ...
The chi-squared statistic can then be used to calculate a p-value by comparing the value of the statistic to a chi-squared distribution. The number of degrees of freedom is equal to the number of cells , minus the reduction in degrees of freedom, . The chi-squared statistic can be also calculated as
This reduces the chi-squared value obtained and thus increases its p-value. The effect of Yates's correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5.
The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics. Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution. The same prior in alternative parametrization is given by the inverse-gamma distribution.
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 .
[1] [2] The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should ...
Classically, a confidence distribution is defined by inverting the upper limits of a series of lower-sided confidence intervals. [15] [16] [page needed] In particular, For every α in (0, 1), let (−∞, ξ n (α)] be a 100α% lower-side confidence interval for θ, where ξ n (α) = ξ n (X n,α) is continuous and increasing in α for each sample X n.