Answer:
Number of samples (N) = 13
Sample size (n) = 200
Number of retests = (1+2+2+0+2+1+2+0+2+7+3+2+ 1) = 25
(a) Defective rate (P-bar) = Number of retests / Total Number of observations
P-bar = 25 / (13 * 200)
P-bar = 25 / 2600
P-bar = 0.0096
Standard deviation of P-bar (Sp) = √[P-bar x (1 - P-bar)] / n
Sp = √[0.0096 x (1 - 0.0096)] / 200
Sp = 0.0069
Upper Control Limit (UCL) = P-bar + (Z x Sp)
For 2-sigma limits, Z = 2
UCL = 0.0096 + (2 * 0.0069)
UCL = 0.0234
Lower Control Limit LCL = P-bar - (Z x Sp)
LCL = 0.0096 - (2*0.0069)
LCL = -0.0042 or 0 (Number of retests cannot be negative)
LCL = 0
(b) For each of the given retests, defect rate is calculated as,
Number of retests = 1, P-bar = 1/200 = 0.0050
Number of retests = 2, P-bar = 2/200 = 0.0100
Number of retests = 3, P-bar = 3/200 = 0.0150
Number of retests = 7, P-bar = 7/200 = 0.0350 (outside the limits)
No, the process is not in control, since number of retests for sample number 10 (retests = 7) falls outside the calculated values of UCL (0.0234) and LCL (0).
