Cartons of Plaster of Paris are supposed to weigh exactly $32 \mathrm{oz}$. Inspectors want to develop process control charts. They take ten samples of four boxes each and weigh them. Based on the following data, compute the lower and upper control limits and determine whether the process is in control. \begin{tabular}{|l|l|l|} \hline Sample & Mean & Range \\ \hline 1 & 33.8 & 0.8 \\ \hline 2 & 34.6 & 0.3 \\ \hline 3 & 34.4 & 0.4 \\ \hline 4 & 34.1 & 0.7 \\ \hline 5 & 34.2 & 0.3 \\ \hline 6 & 34.3 & 0.4 \\ \hline 7 & 33.7 & 0.5 \\ \hline 8 & 34.1 & 0.6 \\ \hline 9 & 34.2 & 0.4 \\ \hline 10 & 34.1 & 0.3 \\ \hline \end{tabular} 1. What is the upper control limit for the $\mathrm{x}$-bar chart? 2. What is the lower control limit for $x$-bar chart? 3. What is the upper control limit for R-chart? 4. What is the central line for R-chart?

Step 1 ：Given the data, we first need to calculate the average of the sample means (\(\bar{x}\)) and the average of the sample ranges (\(\bar{R}\)).

Step 2 ：We calculate \(\bar{x}\) by summing up the means of all the samples and dividing by the number of samples, which is 10. Similarly, we calculate \(\bar{R}\) by summing up the ranges of all the samples and dividing by the number of samples.

Step 3 ：Using the given data, we find that \(\bar{x} = 34.15\) and \(\bar{R} = 0.47\).

Step 4 ：The upper control limit (UCL) for an x-bar chart is calculated using the formula UCL = \(\bar{x} + A2\bar{R}\), where A2 is a constant that depends on the sample size. In this case, the sample size is 4, so A2 is approximately 0.729.

Step 5 ：Substituting the values we have found into the formula, we get UCL = \(34.15 + 0.729 \times 0.47\), which simplifies to UCL = 34.49263.

Step 6 ：Rounding to two decimal places, we find that the upper control limit for the x-bar chart is \(\boxed{34.49}\).