Dear, Dr. Gopal

I suggest the following lecture that may help you:

https://www.ics.uci.edu/~jutts/8/Lecture28Compact.pdf

I am not sure if Confidence interval workd best when you are comparing multiple means. I would just use ANOVA followed by Tukey's HSD mean separation, unless there is a pressind reason to using the confidence intervals.

Such check are automated in modern statistical software. However, there is a difference if you want to check paired differences of means, or establish the general absence of mean difference between several groups. You should perform different steps for these two tasks.

you can use post hock test