Apple is interested in using your patented sapphire crystal screen on their upcoming iPhone 8 which is due for release in October. Your patented method sliced the crystal thin enough that the assembly was able to fit within the dimensions of the smartphone. You’re creating 500 sheets per hour on the robotic lasers and as you’re in the development stage, you sample 50 pieces each hour for defects in clarity, thickness and strength. Apple’s specification calls for a failure rate of less than 3%. You are analyzing the following data

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 Sample Defects 1 3 2 2 3 4 4 0 5 3 6 1 7 4 8 0 9 3 10 2 11 4 12 3 13 1 14 1 15 7 16 0 17 0 18 3 19 2 20 3 21 4 22 8 23 2 24 1 25 3

Calculate the p-bar and determine the UCLp and LCLp using 3 standard deviations above and below the mean. Based on your samples, are your results in control and are you meeting the performance standard?

Sample Size, N=50

 Sample Defects (N.p) p = N.p / N 1 3 0.06 2 2 0.04 3 4 0.08 4 0 0.00 5 3 0.06 6 1 0.02 7 4 0.08 8 0 0.00 9 3 0.06 10 2 0.04 11 4 0.08 12 3 0.06 13 1 0.02 14 1 0.02 15 7 0.14 16 0 0.00 17 0 0.00 18 3 0.06 19 2 0.04 20 3 0.06 21 4 0.08 22 8 0.16 23 2 0.04 24 1 0.02 25 3 0.06

p-bar (the central line of the p-chart) = average of the ‘p’ values = average of last column = 0.0512

$UCL_{p}=\bar{p}+3\sqrt{\frac{p.(1-p)}{N}}$

$LCL_{p}=max(0,\bar{p}-3\sqrt{\frac{p.(1-p)}{N}})$

So,

UCLp = 0.0512 + 3*sqrt(0.0512*(1 – 0.0512)/50) = 0.14471
LCLp = max(0, 0.0512 – 3*sqrt(0.0512*(1 – 0.0512)/50)) = 0

The process is not under control as the sample no. 22 is going out of control limit.