Question: (9 points) The Adventuresports Institute (ASI) at Garrett College is considering the construction of an artificial cave to attract tourists. In order to give everyone a realistic feeling of the cave experience, the entire length of the cave would be chosen such that guests can barely stand upright. Au and Associate PseudoScientific Consulting (AAPC) is asked to determine the appropriate height of the cave so that at least 96% of the population will be accommodated.

AAPC has found out that the mean height of American men is 70 inches with a standard deviation of 2.5 inches. Furthermore, the cave can only accommodate 64 guest at one time. Based on the information given, a height of the cave that can accommodate all 64 guests upright standing position with a 96% probability is determined. What would this height be?

ASI is wondering if AAPC is actually making the right recommendation since it takes millions to construct the artificial cave and any subsequent modifications, especially the height of the cave, can be costly. Now, you are asked to give a second opinion.

Is the AAPC's recommendation appropriate? If not, what should be the proper recommended height to be in compliant with ASI's stipulation? Please explain your answer.