# Confidence interval for the sample mean assuming Johnson SU distribution

I'm validating a machine learning model that outputs a certain sample of N>20000. After fitting the sample to multiple distributions, and then running the typical Anderson-Darling and Kolmogorov-Smirnov tests, the only distribution that performs decently is the Johnson SU. I want to compute the confidence interval for the sample mean, assuming the underlying distribution is the Johnson SU one:

• I know that I can compute confidence interval of certain statistics using bootstrapping too, please don't suggest it
• I'm a student of engineering, with only 3 months of lessons in statistics- most of it i'm sure forgotten by now, so please try to explain it in a way that i can reproduce myself.

*Another question. Can I somehow use the transformation that links the standard normal distribution to this one? Like, i get the x=0.05CDF(N[0,1]) and x=0.95CDF(N[0,1]), transform to the JohnsonSU through the formula and voila (mean = samplemean [+B,-C]@90% confidence)? Feels like im speaking nonsense, damn statistics...