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More precisely, a study's defined significance level, denoted by , is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; [4] and the p-value of a result, , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. [5]
The significance level is 5% and the number of cases is 60. Power of unpaired and paired two-sample t-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1 and a deviation of the expected value of 0.4. The significance level is 5% and the number of cases is 60.
Suppose the data can be realized from an N(0,1) distribution. For example, with a chosen significance level α = 0.05, from the Z-table, a one-tailed critical value of approximately 1.645 can be obtained. The one-tailed critical value C α ≈ 1.645 corresponds to the chosen significance level.
A desired significance level α would then define a corresponding "rejection region" (bounded by certain "critical values"), a set of values t is unlikely to take if was correct. If we reject H 0 {\displaystyle H_{0}} in favor of H 1 {\displaystyle H_{1}} only when the sample t takes those values, we would be able to keep the probability of ...
For the statistic t, with ν degrees of freedom, A(t | ν) is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t ≥ 0). It can be easily calculated from the cumulative distribution function F ν (t) of the t distribution:
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 ...
For a given significance level in a two-tailed test for a test statistic, the corresponding one-tailed tests for the same test statistic will be considered either twice as significant (half the p-value) if the data is in the direction specified by the test, or not significant at all (p-value above ) if the data is in the direction opposite of ...
[13] [14] [15] The apparent contradiction stems from the combination of a discrete statistic with fixed significance levels. [16] [17] Consider the following proposal for a significance test at the 5%-level: reject the null hypothesis for each table to which Fisher's test assigns a p-value equal to or smaller than 5%. Because the set of all ...