Question

12. An IBM subcontractor was hired to make ceramic substrates which are used to distribute power and signals to and from computer silicon chips. Specifications require resistance between 1.500 ohms and 2.500 ohms, but the population has normally distributed resistances with a mean of 1.978 ohms and a standard deviation of 0.172 ohm.

a. What percentage of ceramic substrates will not meet the manufacturer specifications? Does this manufacturing process appear to be working well?

b. If the required specifications are to be modified so that 3% of the devices are rejected because their resistances are too low and another 3% are rejected because their resistances are too high, find the cutoff values for the acceptable devices.

Answer 1

a)
Here, μ = 1.978, σ = 0.172, x1 = 1.5 and x2 = 2.5. We need to compute P(1.5<= X <= 2.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (1.5 - 1.978)/0.172 = -2.78
z2 = (2.5 - 1.978)/0.172 = 3.03

Therefore, we get
P(1.5 <= X <= 2.5) = P((2.5 - 1.978)/0.172) <= z <= (2.5 - 1.978)/0.172)
= P(-2.78 <= z <= 3.03) = P(z <= 3.03) - P(z <= -2.78)
= 0.9988 - 0.0027
= 0.9961

i.e. 99.61%
Yes manufacturing process appears to be working well.

b)
z-value = 1.88

lower cut off = 1.978 - 1.88*0.172 = 1.655
upper cut off = 1.978 + 1.88*0.172 = 2.301