An optical company uses a vacuum deposition method to apply a protective coating to certain lenses. The coating is built up one layer at a time. The thickness of a given layer is a random variable...

An optical company uses a vacuum deposition method to apply a protective coating to certain lenses. The coating is built up one layer at a time. The thickness of a given layer is a random variable with mean µ = 0.5 microns and standard deviation σ = 0.2 microns. The thickness of each layer is independent of the others and all layers have the same thickness distribution. In all, 36 layers are applied.

(a) What is the approximate distribution of the coating thickness? Cite the appropriate theorem to justify your answer.

(b) The company has determined that a minimum thickness of 16 microns for the entire coating is necessary to meet all warranties. Consequently, each lens is tested and additional layers are applied if the lens does not have at least a 16-micron-thick coat. What proportion of lenses must have additional layers applied?