# Interpretting the p-value when inverting the null hypothesis

I'm performing an analysis by fitting data points to an exponential distribution and testing the adequacy of this fit using the Kolmogorov-Smirnov test. The standard ks test null hypothesis: the data is indeed derived from an exponential distribution. The ks test statistic is compared with a critical value and the null hypothesis is rejected if it exceeds this critical value. I used this paper to find the critical values. I find the test statistic falls below this threshold, even for a significance level as high as 0.20.

Now here is my question. I've read that a smaller p-value provides stronger evidence supporting an alternative hypothesis, and that one usually looks to 0.05 as a threshold to reject the null hypothesis. However, no significance level is considered to support the null hypothesis. Is there any way to interpret the conventional p-value as supporting the null hypothesis? Is there anything wrong with this interpretation?

• Jun 1, 2018 at 18:54
• It also depends on the specific kind of test. For example the one-way ANOVA's F statistic p-value does have an interpretation as supporting an omnibus test for equivalence (see Wellek, S. (2010). Testing Statistical Hypotheses of Equivalence and Noninferiority. Chapman and Hall/CRC Press, second edition, chapter 7), but the same is definitely not true of t tests which require a different formulation of the null and test statistic. Jun 1, 2018 at 20:19
• You can't use a KS test to test with a fitted parameter -- it's for a fully specified distribution; you need a Lilliefors test (in this case the version for the exponential). Jun 4, 2018 at 14:56