Background: I have conducted some testing on a random sample of n=20 parts. The data is variable and I know nothing about the population statistics. I would like to use the data from this sample to make statements about the population (in particular, I would like to say with 95% confidence that the 99th percentile of the population is above some value).
I have read about the importance of checking normality for this type of analysis but have also read good posts about "low power" of a normality test when the sample size is relatively small. Those posts recommend that you check "graphically" and also to transform the data as necessary even if a basic normality check indicates there isn't a compelling reason to reject the null hypothesis that the data is normal.
I transformed the data using a few basic transformations (square root, inverse, etc.) just to see what it would look like. None look drastically different but they do show different "p" values.
My question: Should I choose and use a transformation with the highest "p" value even if a (low powered) normality check of the untransformed data is not below .05?
As I understand it, that transformed data set would have the lowest probability of actually "not being normal". I've attached images of the normality checks I ran on the three sets (untransformed, transformed w/ square root, transformed w/ inversion) using Anderson-Darling.
The only stats tool I have available to me is Minitab.
***EDIT: The reason that I thought normality would be important is that it is typical in my industry to perform these types of reliability calculations using tables of "One-sided and Two-Sided Statistical Tolerance Limit Factors (k)" and the tables are only shown/valid for normal distributions. See, for example, "Tables for One-Sided Tolerance Limits" Industrial Quality Control, vol. XIV, no 10. You do this by calculating X +/- ks where X is sample mean, s is sample std dev, and k is from a table and is a function of desired confidence, reliability, and sample size.