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?