A medical facility does MRIs for sports injuries. Occasionally a test yields inconclusive results and must be repeated. Using the following sample data and n = 200, determine the upper and lower
control limits for the fraction of retests using two-sigma limits. Is the process in control?
SAMPLE
1 2 3 4 5 6 7 8 9 10 11 12 13
Number of retests 1 2 2 0 2 1 2 0 2 7 3 2 1

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).

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c is the correct answer.