If the sample standard deviations instead of the sample ranges are used to estimate \(\sigma\), the control limits for the resulting \(\bar{x}\) chart are given by \(\overline{\bar{x}} \pm A_{1} \bar{s}\), where \(\bar{s}\) is the mean of the sample standard deviations obtained from given data, and \(A_{1}\) can be found in Table 8W. Note that in connection with problems of quality control the sample standard deviation is defined using the divisor \(n\) instead of \(n-1\). The corresponding \(R\) chart is replaced by a \(\sigma\) chart, having the central line \(c_{2} \bar{s}\) and the lower and upper control limits \(B_{3} \bar{s}\) and \(B_{4} \bar{s}\), where \(c_{2}, B_{3}\) and \(B_{4}\) can be obtained from Table 8W.

(a) Construct an \(\bar{x}\) chart and a \(\sigma\) chart for these samples of size 3 which had \(\bar{x}\) equal to 25, 28, 26, 29,30,32,22,24,27,26,29,20,25,24,27,28,24,23, 27, 28 and \(s\) equal to 1,2,3,5,4,2,1,3,1,4,4,3,2,1,5,4,1,3,2,2.

(b) Would it be reasonable to use these control limits for subsequent data? Why?

Data From Table 8W

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Hour Coded Sample Values x R 122 1.7 2.2 1.9 1.2 1.75 1.0 0.8 1.5 2.1 0.9 1.32 1.3 3 1.0 1.4 1.0 1.3 1.18 0.4 4 0.4 -0.6 0.7 0.2 0.18 1.3 5 1.4 2.3 2.8 2.7 2.30 1.4. 6 1.8 2.0 1.1 0.1 1.25 1.9 7 1.6 1.0 1.5 2.0 1.52 1.0 8 2.5 1.6 1.8 1.2 1.78 1.3 9 2.9 2.0 0.5 2.2 1.90 2.4 10 1.1 1.1 3.1 1.6 1.72 2.0 11 1.7 3.6 2.5 1.8 2.40 1.9 12 4.6 2.8 3.5 1.9 3.20 2.7 13 2.6 2.8 3.2 1.5 2.52 1.7 14 2.3 2.1 2.1 1.7 2.05 0.6 15 1.9 1.6 1.8 1.4 1.68 0.5 16 1.3 2.0 3.9 0.8 2.00 3.1 17 2.8 1.5 0.6 0.2 1.28 2.6 18 1.7 3.6 0.9 1.5 1.92 2.7 19 1.6 0.6 1.0 0.8 1.00 1.0 20 1.7 1.0 0.5 2.2 1.35 1.7