Question

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A process quality characteristic that is being monitored is thought to be Normally distributed with a true mean of 90 and a standard deviation of 10. The manufacturer devises a control chart with lower and upper control limits of LCL = 70; UCL = 110, respectively. If the measurement of a manufactured product lies outside these control limits, then the entire process is said to be \out-of-control" and shut down for further investigation.

(a) What is the probability that the first manufactured product indicates out-of-control?

(b) What is the probability that out-of-control is indicated in the 5th product? 10^th product? 100th product? Hint: Consult the additional material on the Geometric probability distribution.

(c) What is the expected number of products manufactured until out-of-control is first indicated? Hint: Consult the additional material on the Geometric probability distribution.

(d) What is the probability that the control chart indicates out-of-control in one of the first three manufactured products?

Answer 1

μ = 90, σ = 10

x1 = 70, x2 = 110

z1 = (70 – 90)/10 = -2 and z2 = (110 – 90)/10 = 2

P(Out-of-control) = Area under the standard normal curve outside z = -2 and z = 2, which is 0.0455

So p = 0.0455

(a) For geometric probability distribution, P(k) = p * [(1 – p)^(k – 1)], where k = 1, 2, 3…

P(1) = p = 0.0455

(b) P(5) = 0.0455 * [(1 – 0.0455)^(5 – 1)] = 0.038

P(10) = 0.0455 * [(1 – 0.0455)^(10 – 1)] = 0.03

P(100) = 0.0455 * [(1 – 0.0455)^(100 – 1)] = 0.0005

(c) Expected value = 1/p = 1/0.0455 = 21.98 (Say 22)

(d) Probability = 0.0455 + 0.0455 + 0.0455 = 0.1365