## Related questions with answers

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

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

Solution

VerifiedA **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 $n=10$.

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

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

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

The centerline is the population mean, which is $\mu = 163$. The process is under control if the sample means are within $155.88$ and $170.12$.

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