Can you help me understand the meaning of the area under the curve of a PDF in the interpretation of a Chi-Square test?
This is what I think I know:
If my test statistic (i.e. the chi-square value) increases for a given Degree of Freedom, so does my cumulative distribution function (CDF) value of the Chi-Square CDF(Chi). This value represents the area under the curve of the Chi-Square probability density function (PDF).
CDF(Chi) is also the significance level or "critical region" called $\alpha$ (often used with 0.05).
Is it correct, that if for a given test value the CDF(Chi) > 0.05 my H0 can be rejected when only interested in a one-tailed test - i.e. are two distributions different or not? This approach seems to work if I do a one-tailed test.
This is however currently counterintuitive to me, because as far as I understand, p = 1 - CDF(Chi) and H0 is rejected if p < 0.05. Therefore I would believe that the above statement is not true and I would need to reject H0 if CDF > 0.95. That would also make more sense to me, as it means that H0 is only rejected if there is more than 95% "evidence" that it is not random or less than a 5% chance of wrongfully rejecting H0.
Can someone help clear things up - I assume my last thought makes more sense and in my example, I am getting results that seem to conform to what I want to see but are actually not statically significant.