The intake valve clearances on new engines of a certain type are normally distributed with mean 200 µm and standard deviation 10 µm.
a. What is the probability that the clearance is greater than 215 µm?
b. What is the probability that the clearance is between 180 and 205 µm?
c. An engine has six intake valves. What is the probability that exactly two of them have clearances greater than 215 µm?
(a) The z-score of 215 is (215 ? 200) / 10 = 1.5.
The area to the right of z = 1.5 is 1 ? 0.9332 = 0.0668.
The probability that the clearance is greater than 215 ???? m is 0.0668.
(b) The z -score of 180 is (180 ? 200) / 10 = ?2.00.
The z -score of 205 is (205 ? 200) / 10 = 0.50.
The area between z = ?2.00 and z = 0.50 is 0.6915 ? 0.0228 = 0.6687.
The probability that the clearance is between 180 and 205 ???? m is 0.6687.
(c) Let X be the number of valves whose clearances are greater than 215 ? m.
From part (a), the probability that a valve has a clearance greater than 215 ? m is 0.0668, so
X ~ Bin(6, 0.0668).
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