Question

#1#2#3#4#5#6#7 #8 162.2165.8156.4165.3168.6167.0186.8178.3159.8166.2156.4173.3175.8171.4160.4163.0160.3160.6152.2166.4168.2168.4176.8171.3\begin{array}{|c|c|c|c|c|c|c|c|} \hline \# 1 & \# 2 & \# 3 & \# 4 & \# 5 & \# 6 & \# 7 & \text { \#8 } \\ \hline 162.2 & 165.8 & 156.4 & 165.3 & 168.6 & 167.0 & 186.8 & 178.3 \\ \hline 159.8 & 166.2 & 156.4 & 173.3 & 175.8 & 171.4 & 160.4 & 163.0 \\ \hline \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \hline 160.3 & 160.6 & 152.2 & 166.4 & 168.2 & 168.4 & 176.8 & 171.3 \\ \hline \end{array}

Prepare a control chart that specifies a centerline as well as an upper control limit (UCL) and a lower control limit (LCL).

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Answered 1 year ago
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A control chart is a tool for monitoring the behavior of the production process over time. It comprises the centerline and the lower and upper control limits. The centerline indicates the expected value of the variables; meanwhile, the control limits are three standard deviations away from the centerline.

In this case, eight samples were taken with each sample having a size of n=10n=10.

Compute the upper control limit (UCL) and the lower control limit (LCL) for a sample size n=10n=10, population mean μ=163\mu =163, and standard deviation σ=7.5\sigma = 7.5 as shown below

UCL: μ+3σn=163+37.510=170.12\text{UCL: } \mu + 3 \dfrac{\sigma}{\sqrt n} = 163 + 3 \dfrac{7.5}{\sqrt {10}} = \bf 170.12

LCL: μ3σn=16337.510=155.88\text{LCL: } \mu - 3 \dfrac{\sigma}{\sqrt n} = 163 - 3 \dfrac{7.5}{\sqrt {10}} = \bf 155.88

The centerline is the population mean, which is μ=163\mu = 163. The process is under control if the sample means are within 155.88155.88 and 170.12170.12.

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