Two mating parts have critical dimensions x1 and x2 as shown

Two mating parts have critical dimensions x1 and x2 as shown below. Assume that x1 and x2 are normally distributed with means 1 and mean2 and standard deviations s1 = 0.400 and s2 = 0.300. If it is desired that the probability of a smaller clearance (i.e., x1 - x2) than 0.09 should be 0.006, what distance between the nominal dimensions of the two parts (i.e., u1 and u2) should be specified by the designer?

Solution

x1-x2 has mean = mu1-mu2

and variance as s1^2+s2^2

= 0.2500

Hence std dev x1-x2 = 0.5

x1-x2 is normal with N(mu1-mu2, 0.5)

P(x1-x2<0.09) = 0.006

i.e.P(Z< x1-x2-(mu1-mu2)/0.5 ) = 0.006

i.e. z <-0.01

or x1-x2/0.5 <-0.001

x1-x2 <-0.005

Hence difference should be less than 0.005