Question

Design specifications for an arrow are that they each weigh 100 ± 10 grams. The process to make the arrows has a standard deviation of four units.

a. What is the process capability index? Assume that the process is centered with respect to specifications. Round your intermediate and final answers to 4 decimal places (e.g., .12345 would be rounded as .1235, not .1234).

Process capability index

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b. Suppose the process average shifts to 92. Calculate the new process capability index. Round your intermediate and final answers to 4 decimal places (e.g., .12345 would be rounded as .1235, not .1234).

New process capability index

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c. What is the probability of defective output after the process shift? Round "z" values to 2 decimal places. Round probabilities to 4 decimal places (so a probability of .12345 would be entered as .1235, not .1234 or 12.345% or something else).

Probability of defective output

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Answer 1

1. UPPER = 110
LOWER = 90
PROCESS MEAN = 100
STANDARD DEVIATION = 4

Cpk = MIN((UPPER - MEAN) / 3 * STANDARD DEVIATION), (MEAN - LOWER ) / 3 * STANDARD DEVIATION)

Cpk = MIN((110 - 100) / 3 * 4), (100 - 90) / 3 * 4)
Cpk = MIN(0.833333, 0.833333)
Cpk = 0.8333

2. UPPER = 110
LOWER = 90
PROCESS MEAN = 92
STANDARD DEVIATION = 4

Cpk = MIN((UPPER - MEAN) / 3 * STANDARD DEVIATION), (MEAN - LOWER ) / 3 * STANDARD DEVIATION)

Cpk = MIN((110 - 92) / 3 * 4), (92 - 90) / 3 * 4)
Cpk = MIN(1.5, 0.166667)
Cpk = 0.1667

3. PROBABILITY OF BEING UNDER 90:
Z-LSL = (LOWER - MEAN) / STANDARD DEVIATION = (90 - 92) / 4 = -0.50 = PROBABILITY = NORMSDIST(Z-LSL) = NORMSDIST(-0.5) = 0.3085

PROBABILITY OF BEING OVER 110:
Z-USL = (UPPER - MEAN) / STANDARD DEVIATION = (110 - 92) / 4 = 4.50 = PROBABILITY = 1 - NORMSDIST(Z-USL) = 1 - NORMSDIST(4.5) = 0

PROBABILITY = PROBABILITY OF BEING UNDER LSL + PROBABILITY OF BEING OVER USL = 0.3085 + 0 = 0.3085