A steel manufacturer extrudes scrap for manufacturing blades. Specifications require that the thickness of this scrap has \(\mu=0.020 \mathrm{~mm}\) and \(\sigma=0.005 \mathrm{~mm}\).

(a) Use the specifications to calculate a central line and three-sigma control limits for an \(\bar{x}\) chart with \(n=10\).

(b) Use the specifications to calculate a central line and three-sigma control limits for an \(R\) chart with \(n=10\).

(c) Plot the following means and ranges, obtained in 20 successive random samples of size 10 on charts based on the control-chart constants obtained in parts (a) and (b) and discuss the process.

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Sample R Sample R 1 0.022 0.004 11 0.017 0.003 10 234567890 0.021 0.002 12 0.016 0.008 0.029 0.007 13 0.022 0.006 0.018 0.006 14 0.023 0.004 0.019 0.003 15 0.019 0.002 0.027 0.004 16 0.020 0.005 0.021 0.005 17 0.021 0.003 0.022 0.002 18 0.022 0.002 0.019 0.007 19 0.018 0.008 0.018 0.002 20 0.010 0.006