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In statistics, minimum chi-square estimation is a method of estimation of unobserved quantities based on observed data. [1]In certain chi-square tests, one rejects a null hypothesis about a population distribution if a specified test statistic is too large, when that statistic would have approximately a chi-square distribution if the null hypothesis is true.
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation ( MSWD ) in isotopic dating [ 1 ] and variance of unit weight in the context of weighted least squares .
The simplest chi-squared distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-squared distribution could be used. Suppose that Z {\displaystyle Z} is a random variable sampled from the standard normal distribution, where the mean is 0 {\displaystyle 0} and the ...
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
It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise. Alternatively it can be seen via the interpretation in the background section above as sums of squares of independent normally distributed random variables with variances of 1 and the specified means.
Specific activity (symbol a) is the activity per unit mass of a radionuclide and is a physical property of that radionuclide. [1] [2] It is usually given in units of becquerel per kilogram (Bq/kg), but another commonly used unit of specific activity is the curie per gram (Ci/g).
Stochastic health effects are those that occur by chance, and whose probability is proportional to the dose, but whose severity is independent of the dose. [3] The LNT model assumes there is no lower threshold at which stochastic effects start, and assumes a linear relationship between dose and the stochastic health risk.
If are k independent, normally distributed random variables with means and variances , then the statistic = = is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables b