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It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom. Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.
The p-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test, [39] using the chi-squared distribution and notated as capital P. [39] The p-values for the chi-squared distribution (for various values of χ 2 and degrees of freedom), now notated as P, were calculated in (Elderton 1902), collected in (Pearson 1914 ...
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of one parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size ...
If we have a significance level of 5%, then this result (0.02654 < 5%) indicates that we have evidence that is significant enough to reject the null hypothesis that the die is fair. Normally, when we are testing for fairness of a die, we are also interested if the die is biased towards generating fewer 6s than expected, and not only more 6s as ...
The table shown on the right can be used in a two-sample t-test to estimate the sample sizes of an experimental group and a control group that are of equal size, that is, the total number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05. [4]
Thus, when the probability of X occurring in group B is greater than the probability of X occurring in group A, the odds ratio is greater than 1, and the log odds ratio is greater than 0. Suppose that in a sample of 100 men, 90 drank wine in the previous week (so 10 did not), while in a sample of 80 women only 20 drank wine in the same period ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
In information theory, the source coding theorem (Shannon 1948) [2] informally states that (MacKay 2003, pg. 81, [3] Cover 2006, Chapter 5 [4]): N i.i.d. random variables each with entropy H(X) can be compressed into more than N H(X) bits with negligible risk of information loss, as N → ∞; but conversely, if they are compressed into fewer than N H(X) bits it is virtually certain that ...