24. A machine produces large fasteners whose length must be within 0.5 inch of 2

Question

24. A machine produces large fasteners whose length must be within 0.5 inch of 22 inches. The lengths are normally distributed with mean 22.0 inches and standard deviation 0.17 inch. a. Find the probability that a randomly selected fastener produced by the machine will have an acceptable length. b. The machine produces 20 fasteners per hour. The length of each one is inspected. Assuming lengths of fasteners are independent, find the probability that all 20 will have acceptable length. Hint: There is a binomial random variable here, whose value of p comes from part (a).

Explanation / Answer

P(X < A) = P (Z < (A - mean)/standard deviation)

Mean = 22 inch

Standard deviation = 0.17 inch

P(within 0.5 inch of 22 inch) = P(21.5 < X < 22.5)

= P(X < 22.5) - P(X < 21.5)

= P(Z < (22.5 - 20)/0.17) - P(Z < (19.5 - 20)/0.17)

= P(Z < 2.94) - P(Z < - 2.94)

= 0.9984 - 0.0016

= 0.9968

B) P(all 20 will have acceptable length) = 0.996820

= 0.9379

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