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At α = 0.05 and DOF = 176, null hypothesis is accepted

The critical value or values are used to locate the areas under the curve of a distribution that are too extreme to be consistent with the null hypothesis. For a two-tailed test, the value of the level of significance

However when I use the union intersection test I reject the null hypothesis.

I have to compare several distributions, but I don't want to use Kolmogorov-Smirnov test, since I'm dealing with big data ($N > 10^$) and traditional hypothesis tests end up with detecting the smallest yet meaningless difference between the two distributions --- i.e. I always reject the null hypothesis, even if a small difference between the two distributions is acceptable considering that I'm analyzing social dynamics/interactions and not physical/biological phenomena.

At α = 0.05 and DOF = 128, null hypothesis is accepted

At α = 0.05 and degrees of freedom (DOF) = 138, null hypothesis is accepted

When testing a hypothesis for a small sample where you have to find the appropriate critical right-tail value, this value depends on certain criteria. In addition to being positive, the value also depends on the sample size and whether or not the population standard deviation is known.

After you calculate a test statistic, you compare it to one or two critical values, depending on the alternative hypothesis, to determine whether you should reject the null hypothesis.

Hypothesis testing: Accidents at intersections example

However when I use the union intersection test I reject the null hypothesis

Tests for the supremacy of a multinomial cell probability are developed. The tested null hypothesis states that a particular cell of interest is not more probable than all others. Rejection of this null leads to the conclusion that the cell of interest has a strictly greater probability than all other cells. The null hypothesis constrains the multinomial probability vector to a non-convex region that is a union of closed convex cones. The likelihood ratio test for this problem is derived and shown to be equivalent to an intersection-union test. The least favorable configuration of the multinomial probability vector in the null parameter space is derived, and the limiting null distribution of the test statistic that is stochastically greatest is shown to be a mixture of point mass at zero and a chi-square distribution with a single degree of freedom. Asymptotic and valid finite-sample testing procedures are proposed and examined via a simulation study and the analysis of two data sets. The proposed procedures are extended to test whether the cell with the largest observed frequency is uniquely most probable. An equivalence between a likelihood ratio test for this problem and a union-intersection test is demonstrated.

The critical value or values are used to locate the area under the curve of a distribution that is too extreme to be consistent with the null hypothesis. For a right-tailed test, these are the large positive values, which are collectively known as the right tail of the distribution. The area in the tail equals the level of significance of the hypothesis test.

Intersection accidents: Null and alternative hypotheses in t
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