A fraction nonconforming control chart with center line 0.10, UCL = 0.19, and LCL = 0.01 is used to control a process . 1. If three-sigma limits are used, find the sample size for the control chart....

Solution 1:
We have p-bar = 0.07, k = 3 and n = 400
UCL = p-bar + 3 √ p (1 – p)/n
= 0.07 + 3 √0.07 (1 – 0.07)/400
= 0.07 + 3 (0.0128)
= 0.07 + 0.038
= 0.108
LCL = p-bar - 3 √ p (1 – p)/n
= 0.07 - 3 √0.07 (1 – 0.07)/400
= 0.07 - 3 (0.0128)
= 0.07 - 0.038
= 0.032
b) np = 400 (0.10) ≥ 40. Since it is greater than 5, we use normal approximation to binomial.
P (detect on 1st sample) = 1 – P (not detect on 1st sample)
= 1 – β
= 1 – [P (p-bar < UCL|p) – P (p-bar < LCL|p)]
= 1 – P [(UCL – p)/√ p (1 – p)/n + (LCL – p)/√ p (1 –p)/n]
= 1 - P [(0.108 – 0.1)/√ 0.1 (1 – 0.1)/400 + (0.032 – 0.1)/√ 0.1 (1 –0.1)/400]
= 1 – [P (Z <...