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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]
Most test statistics have the form t = Z/s, where Z and s are functions of the data. Z may be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined. As an example, in the one-sample t-test
The program provides methods that are appropriate for matched and independent t-tests, [2] survival analysis, [5] matched [6] and unmatched [7] [8] studies of dichotomous events, the Mantel-Haenszel test, [9] and linear regression. [3] The program can generate graphs of the relationships between power, sample size and the detectable alternative ...
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals. The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these ...
Thanks to t-test theory, we know this test statistic under the null hypothesis follows a Student t-distribution with degrees of freedom. If we wish to reject the null at significance level α = 0.05 {\displaystyle \alpha =0.05\,} , we must find the critical value t α {\displaystyle t_{\alpha }} such that the probability of T n > t α ...
Given an r-sample statistic, one can create an n-sample statistic by something similar to bootstrapping (taking the average of the statistic over all subsamples of size r). This procedure is known to have certain good properties and the result is a U-statistic. The sample mean and sample variance are of this form, for r = 1 and r = 2.
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 .
Each row of points is a sample from the same normal distribution. The colored lines are 50% confidence intervals for the mean, μ. At the center of each interval is the sample mean, marked with a diamond. The blue intervals contain the population mean, and the red ones do not.
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