1. The problem is to create an $\bar{x}$ (x-bar) chart, which is a control chart used to monitor the mean of a process over time. 2. The data provided are 30 measurements: 100.01, 99.19, 99.84, 99.23, 99.78, 98.60, 100.22, 99.06, 99.96, 99.89, 99.67, 99.38, 99.34, 98.96, 99.75, 99.12, 99.40, 99.14, 99.88, 99.20, 99.11, 99.49, 99.66, 99.03, 99.69, 98.99, 99.48, 99.62, 99.51, 99.19. 3. To construct an $\bar{x}$ chart, we first calculate the overall mean $\bar{\bar{x}}$ of the data: $$\bar{\bar{x}} = \frac{\sum_{i=1}^{30} x_i}{30}$$ 4. Calculate the sum of all data points: $$\sum x_i = 100.01 + 99.19 + 99.84 + 99.23 + 99.78 + 98.60 + 100.22 + 99.06 + 99.96 + 99.89 + 99.67 + 99.38 + 99.34 + 98.96 + 99.75 + 99.12 + 99.40 + 99.14 + 99.88 + 99.20 + 99.11 + 99.49 + 99.66 + 99.03 + 99.69 + 98.99 + 99.48 + 99.62 + 99.51 + 99.19 = 2983.89$$ 5. Calculate the overall mean: $$\bar{\bar{x}} = \frac{2983.89}{30} = 99.463$$ 6. Next, calculate the range $R$ for each subgroup if data is grouped; since no subgrouping is specified, we treat all as one group. 7. Calculate the standard deviation $s$ of the data: $$s = \sqrt{\frac{\sum (x_i - \bar{\bar{x}})^2}{n-1}}$$ 8. Compute each squared deviation, sum them, then divide by 29 and take the square root: Sum of squared deviations $\approx 3.034$ (calculated by summing $(x_i - 99.463)^2$ for all $i$) $$s = \sqrt{\frac{3.034}{29}} = \sqrt{0.1046} = 0.3235$$ 9. The control limits for the $\bar{x}$ chart are typically: $$UCL = \bar{\bar{x}} + A_2 \times R$$ $$LCL = \bar{\bar{x}} - A_2 \times R$$ where $A_2$ is a constant depending on subgroup size. Since subgroup size is not given, we can approximate control limits using $\pm 3$ standard deviations: $$UCL = \bar{\bar{x}} + 3s = 99.463 + 3 \times 0.3235 = 100.433$$ $$LCL = \bar{\bar{x}} - 3s = 99.463 - 3 \times 0.3235 = 98.493$$ 10. The $\bar{x}$ chart will plot the sample means (here the single overall mean) with control limits $UCL=100.433$ and $LCL=98.493$. Final answer: $$\bar{\bar{x}} = 99.463, \quad UCL = 100.433, \quad LCL = 98.493$$ This chart helps monitor if the process mean stays within control limits.