an optical company uses a vacuum deposition method to apply a protective coating

Question

an optical company uses a vacuum deposition method to apply a protective coating on certain lenses. the method produces a lens thickness that is normally distributed with a mean of 0.5 microns and a population standard deviation of 0.2 microns.

A) what percentage of lenses are between 0.3 and 0.7 microns?

B) 15% of all lenses are thicker than how many microns?

C) suppose a random sample of 35 lenses is taken. what is the probability the average thickness for all 35 lenses is less than 0.46 microns?

Explanation / Answer

mean is 0.5 and s is 0.2

z is given as (x-mean)/s

a) P(0.3<x<0.7)=P((0.3-0.5)/0.2<z<(0.7-0.5)/0.2)=P(-1<z<1)=2*P(z<1)-1, from normal table we get 2*0.8413-1=0.6826

b) we need to find the z for 85% which is 1.04 from normal table thus mean+z*s=0.5+0.2*1.04=0.708

c) for sample size of 35, the standard error SE is s/sqrt(N)=0.2/sqrt(35)=0.03381

thus P(x<0.46)=P(z<(0.46-0.5)/0.03381)=P(z<-1.18) or 1-P(z<1.18)=1-0.8810=0.119

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