We describe two new derivations of the chi-square distribution. The first derivation\nuses the induction method, which requires only a single integral to\ncalculate. The second derivation uses the Laplace transform and requires\nminimum assumptions. The new derivations are compared with the established\nderivations, such as by convolution, moment generating function, and\nBayesian inference. The chi-square testing has seen many applications to\nphysics and other fields. We describe a unique version of the chi-square test\nwhere both the variance and location are tested, which is then applied to environmental\ndata. The chi-square test is used to make a judgment whether a\nlaboratory method is capable of detection of gross alpha and beta radioactivity\nin drinking water for regulatory monitoring to protect health of population.\nA case of a failure of the chi-square test and its amelioration are described.\nThe chi-square test is compared to and supplemented by the t -test.
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