Normality test for x bar charts

c
shewhart control charts are robust and do NOT require a noraml data set.

Best check the instructors credentials.
 
See Wheeler Understanding Statistcal Process Control

This comes down to who you follow: Wheeler or Montgomery et al.?
Personally I an convinced that Wheeler is wrong on the robustness of Individuals with extreme Skewness and/or Kurtosis. Numerous published simulation studies (aside from Wheeler’s books) clearly show that Type 1 & II error rates become unacceptable. 
Therefore a Box-Cox, Johnson transformation or distribution fit is appropriate and necessary.  If you transform and want to plot the raw data, simply apply the inverse transformation to get the control limits.  This is straightforward with software tools.
Of course this debate has gone on for years and will not be resolved here. You are wrong to state that any instructor who does not follow Wheeler is incompetent.  I have read all (yes all) of Wheeler’s books but that doesn’t mean that I have to be his disciple.

Please note that I am not questioning the robustness of the x-bar chart. We are talking about the lack of robustness to non-normality for Individuals charts.
Some references supplied below. Before you go to the library, do the following:
1.     Simulate 100 observations from an Exponential Distribution with scale=1 and threshold = 0.
2.     Create an IMR chart with all tests for special causes. Count the number of alarms (these are all type I error).
3.     Run a Box-Cox Transformation on the simulated data (save the transformed data)
4.     Repeat Step 2.
SPC References:
Spedding TA and Rawlings PL, “Non-normality in Statistical Process Control Measurements” International Journal of Quality and Reliability Management, Vol 11 No 6, 1994, pp 27-37.
Montgomery DC, Introduction to Statistical Quality Control – 5th Ed.,  pp237-238.
Borror CM, Montgomery DC, Runger GC, “Robustness of the EWMA Control Chart to Non-normality”, Journal of Quality Technology, Vol 31, No 3, 1999.  (Shewhart Charts are included in the simulation study)
Process Capability References:
Somerville SE and Montgomery DC, “Process Capability Indices and Non-Normal Distributions, Quality Engineering, 9(2), 305-316.
Tang LC and Than SE, “Computing Process Capability Indices for Non-Normal Data: A Review and Comparative Study” Quality and Reliability Engineering International, 15: 339-353.
English JR and Taylor GD, “Process Capability Analysis – A Robustness Study” International Journal or Production Research, 1993, Vol 31, No 7, 1621 – 1635. 
 

Repeat after me.
I will obey the teachings of Dr. Wheeler for He is Omniscient.
I will obey the teachings of Dr. Wheeler for He is Omniscient.
I will obey the teachings of Dr. Wheeler for He is Omniscient.

You must be a GE BB  they all hang on normality. Even when it doesn;t matter.
 

I would look at the raw data on a control chart, to see if the process appears stable. Process instability can make data normally distributed data look non-normal.
It also is OK to check for normality, once you know whether your process is stable. It’s OK to use Anderson-Darling and Box-Cox, but also remember to generate probability plots – a graph is worth a thousand hypotheses.
If the data truly are non-normal, you need to align your decision with the current use for the data. If it’s a look back to establish process capability, you simply want an appropriate model to make a sound estimate. If you are moving forward to control the process, you need to determine how big a penalty you pay for a decision error. If the penalties are moderate, and if your control limits lie well within spec limits, the hassle of a transform may not be warranted.
Whatever else you do, watch out for one-size-fits-all answers, whether they come from the Wheeler or the Montgomery camps.

I’ll be there (I live in greater PHX). My email is [email protected].