1. **Problem Statement:** We have three quality control problems involving control charts and process capability. --- ### Question 1 (a) Set up $\bar{x}$ and $R$ charts for the hole diameter deviations. (b) Determine if the process is in statistical control. (c) Calculate process capability ratio $C_p$ given $\hat{\sigma} = 23.3$ and specifications $\pm 100$. --- ### Step 1: Calculate $\bar{x}$ and $R$ for each sample - $\bar{x}_i = \frac{1}{5} \sum_{j=1}^5 x_{ij}$ - $R_i = \max(x_{i1},...,x_{i5}) - \min(x_{i1},...,x_{i5})$ Calculate these for all 20 samples. --- ### Step 2: Calculate overall averages - $\bar{\bar{x}} = \frac{1}{20} \sum_{i=1}^{20} \bar{x}_i$ - $\bar{R} = \frac{1}{20} \sum_{i=1}^{20} R_i$ --- ### Step 3: Control limits for $\bar{x}$ chart - Use constants for $n=5$: $A_2=0.577$ - $UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$ - $LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \bar{R}$ --- ### Step 4: Control limits for $R$ chart - Constants for $n=5$: $D_3=0$, $D_4=2.114$ - $UCL_R = D_4 \bar{R}$ - $LCL_R = D_3 \bar{R} = 0$ --- ### Step 5: Check points against control limits - If all $\bar{x}_i$ and $R_i$ lie within limits, process is in control. --- ### Step 6: Calculate process capability $C_p$ - $C_p = \frac{USL - LSL}{6 \hat{\sigma}} = \frac{200}{6 \times 23.3} \approx 1.43$ Interpretation: $C_p > 1$ means process is capable of meeting specifications. --- ### Question 2 (a) Set up $\bar{x}$ and $R$ charts for fill volume data (15 samples, $n=10$). - Calculate $\bar{x}_i$, $R_i$ for each sample. - Calculate $\bar{\bar{x}}$, $\bar{R}$. - Use constants for $n=10$: $A_2=0.308$, $D_3=0.223$, $D_4=1.777$. - Calculate control limits as in Question 1. - Check for statistical control. - If out of control, revise limits excluding outliers. (b) Set up $R$ chart and compare with $s$ chart. - $s$ chart uses sample standard deviations. - Compare variability measures and control limits. --- ### Question 3 Given partial data for compressive strength with $n=5$ parts per sample. - $\bar{x}$ and $R$ are given. - Use these to check control charts similarly. --- **Summary:** - Calculate sample means and ranges. - Compute overall averages. - Use control chart constants to find limits. - Check if points lie within limits for control. - Calculate $C_p$ for capability. This approach teaches how to analyze process control and capability using $\bar{x}$ and $R$ charts.