answer all parts plz A statistical process control chart example. Samples of 25

Question

answer all parts plz A statistical process control chart example. Samples of 25 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 25 that require rework. A process problem is suspected if x exceeds its mean by more than three standard deviations. Round your answers to four decimal places. (a) If the percentage of parts that require rework remains at 1%, what is the probability that X exceeds its mean by more than three standard deviations? Probability = (b) If the rework percentage increases to 3%, what is the probability that X exceeds 1? Probability = (c) If the rework percentage increases to 3%, what is the probability that x exceeds 1 in at least one of the next six hours of samples? Probability =

Explanation / Answer

(a)

mean + 3*sigma = 0.25 + 3*0.4975 = 1.7425

P(x > 1.7425) = P(z > (1.7425 - 0.25)/0.4975) = P(z > 3) = 0.0014

(b)

P(x > 1) = P(z > (1 - 0.75)/0.8529) = P(z > 0.2931) = 0.3847

(c)

Probability that x exceeds 1, p = 0.3847 (calculated above)

Consider an event where x does not exceed 1 in next six hours, and its probability = (1-p)^6 = (1 - 0.3847)^6 = 0.0543

Required probability that atleast one of the next hours of samples x exceeds 1 = 1 - 0.0543 = 0.9457

n 25 p 0.01 q = 1 - p 0.99 mean np 0.25 std. dev. sqrt(npq) 0.4975

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