Please provide steps An optical company uses a Vacuum deposi

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 mu = 0.5 microns and standard deviation sigma = 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 thick ness of 16 microns for the entire coating is necessary to meet all warranties. Consequently, each lens s 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?

Solution

Here we have 36 independent variables,

Let S = X1 + X2 + ... + X36 , where each Xi has mean 0.5 and s.d. = 0.2. Then,

E(S) = 36E(Xi) = 36(0.5) = 18 microns

V(S) = 36V(Xi) = 36(0.2)^2 = 1.44.

Therefore,

Dtandard Deviation SD of S is sqrt 1.44 = 1.2

(2)

Proportion of lenses must have additional layers applied = P(S <16)

P(S <16) = P[z < (16 - 18) / 1.2 ]

= P (z < - 5/3)

= 0.04779 Answer

So approx. 4.8% of lenses must have additional layers applied