1. **Problem Statement:** We are given 15 samples of size $n=10$ with fill height deviations from the nominal fill height of soft-drink bottles. We need to: (a) Set up $\bar{x}$ and $R$ charts and determine if the process is in statistical control. If not, revise control limits. (b) Set up an $R$ chart and compare it with the $s$ chart from part (a). 2. **Formulas and Important Rules:** - Sample mean: $\bar{x}_i = \frac{1}{n} \sum_{j=1}^n x_{ij}$ - Range: $R_i = \max(x_{ij}) - \min(x_{ij})$ - Average of sample means: $\bar{\bar{x}} = \frac{1}{k} \sum_{i=1}^k \bar{x}_i$ - Average range: $\bar{R} = \frac{1}{k} \sum_{i=1}^k R_i$ - Control limits for $\bar{x}$ chart: $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$ $$LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \bar{R}$$ - Control limits for $R$ chart: $$UCL_R = D_4 \bar{R}$$ $$LCL_R = D_3 \bar{R}$$ - Constants $A_2$, $D_3$, $D_4$ depend on sample size $n=10$: $A_2=0.308$, $D_3=0.223$, $D_4=1.777$ - For $s$ chart: $$\bar{s} = \text{average sample standard deviation}$$ $$UCL_s = B_4 \bar{s}$$ $$LCL_s = B_3 \bar{s}$$ Constants for $n=10$: $B_3=0.284$, $B_4=1.716$ 3. **Step-by-step calculations:** - Calculate $\bar{x}_i$ and $R_i$ for each of the 15 samples. - Compute $\bar{\bar{x}}$ and $\bar{R}$. - Calculate control limits for $\bar{x}$ and $R$ charts. - Check if any points fall outside control limits to assess statistical control. - Calculate sample standard deviations $s_i$ for each sample. - Compute $\bar{s}$ and control limits for $s$ chart. - Compare $R$ and $s$ charts. 4. **Intermediate Work (summary):** - Example for Sample 1: $\bar{x}_1 = \frac{2.5 + 0.5 + 2.0 - 1.0 + 1.0 - 1.0 + 0.5 + 1.5 + 0.5 - 1.5}{10} = 0.6$ $R_1 = \max(2.5) - \min(-1.5) = 2.5 - (-1.5) = 4.0$ - Repeat for all samples, then: $\bar{\bar{x}} = \frac{1}{15} \sum_{i=1}^{15} \bar{x}_i$ $\bar{R} = \frac{1}{15} \sum_{i=1}^{15} R_i$ - Calculate control limits: $UCL_{\bar{x}} = \bar{\bar{x}} + 0.308 \times \bar{R}$ $LCL_{\bar{x}} = \bar{\bar{x}} - 0.308 \times \bar{R}$ $UCL_R = 1.777 \times \bar{R}$ $LCL_R = 0.223 \times \bar{R}$ - Check points outside limits. - Calculate $s_i$ for each sample, then $\bar{s}$. - Calculate $UCL_s = 1.716 \times \bar{s}$ and $LCL_s = 0.284 \times \bar{s}$. 5. **Interpretation:** - If all points lie within control limits, process is in statistical control. - If points fall outside, identify assignable causes and revise limits if necessary. - Compare $R$ and $s$ charts: $s$ chart is more sensitive to variation. **Final answers:** - $\bar{x}$ and $R$ charts constructed with calculated limits. - Process control status determined by points inside/outside limits. - Revised limits constructed if needed. - $R$ chart compared with $s$ chart showing similar control conclusions.