A film coating process produces films whose thickness is nor

A film coating process produces films whose thickness is normally distributed with a mean of 110 microns and a standard deviation of 10
microns. For a certain application, the minimum acceptable thickness is 90 microns.
(a) What proportion of films are going to be too thin?
(b) Assuming the standard deviation remains at 10 microns, to what value should the mean be set so that only 1% of the films are too thin?
(c) If the mean remains at 110 microns, what must the standard deviation be so that only 1% of the films are going to be too thin?

Solution

(a) What proportion of lms are going to be too thin?

P(X<90) = P((X-mean)/s <(90-110)/10)

=P(Z<-2) = 0.0228 (from standard normal table)

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(b) Assuming the standard deviation remains at 10 microns, to what value should the mean be set so that only 1% of the lms are too thin?

P(X<90)=0.01

--> P(Z<(90-mean)/10) =0.01

--> (90-mean)/10 = -2.33 (from standard normal table)

So 90-mean= -2.33*10

So mean= 90+2.33*10 =113.3

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(c) If the mean remains at 110 microns, what must the standard deviation be so that only 1% of the lms are going to be too thin?

P(X<90) =0.01

--> P(Z<(90-110)/s) =0.01

--> (90-110)/s = -2.33 (from standard normal table)

So s= (90-110)/-2.33=8.58369