1. **Problem Statement:** We have three separate quality control problems involving x̄ (mean) and R (range) charts for different manufacturing processes. We need to construct control charts, assess statistical control, and evaluate process capability. --- ### Question 1 (a) Set up x̄ and R charts for the hole diameter deviations. 1. Calculate the sample means $\bar{x}_i = \frac{1}{5}\sum_{j=1}^5 x_{ij}$ for each sample $i=1,...,20$. 2. Calculate the sample ranges $R_i = \max(x_{i1},...,x_{i5}) - \min(x_{i1},...,x_{i5})$. 3. Compute the overall mean of means $\bar{\bar{x}} = \frac{1}{20}\sum_{i=1}^{20} \bar{x}_i$. 4. Compute the average range $\bar{R} = \frac{1}{20}\sum_{i=1}^{20} R_i$. 5. Use control chart constants for $n=5$: $A_2=0.577$, $D_3=0$, $D_4=2.114$. 6. Calculate control limits for x̄ chart: $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$ $$LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \bar{R}$$ 7. Calculate control limits for R chart: $$UCL_R = D_4 \bar{R}$$ $$LCL_R = D_3 \bar{R} = 0$$ (b) To check statistical control, verify if all $\bar{x}_i$ and $R_i$ lie within their respective control limits and look for patterns. (c) Given process standard deviation estimate $\hat{\sigma} = 23.3$ (ten-thousandths inch), and specification limits nominal ±100: Calculate process capability ratio: $$C_p = \frac{USL - LSL}{6 \hat{\sigma}} = \frac{200}{6 \times 23.3} \approx 1.43$$ Since $C_p > 1$, the process is capable of meeting specifications. --- ### Question 2 (a) For fill volume data with $n=10$: 1. Calculate sample means $\bar{x}_i$ and ranges $R_i$ for each of the 15 samples. 2. Compute overall mean $\bar{\bar{x}}$ and average range $\bar{R}$. 3. Use constants for $n=10$: $A_2=0.308$, $D_3=0.223$, $D_4=1.777$. 4. Calculate control limits for x̄ and R charts as in Q1. 5. Check if points lie within limits and look for control patterns. 6. If out-of-control points exist, recalculate limits excluding those points. (b) Construct an R chart and compare with the s chart (standard deviation chart) from part (a). The s chart uses $B_3$ and $B_4$ constants for control limits. Compare variability detection sensitivity. --- ### Question 3 (a) For compressive strength data with $n=5$: 1. Use given $\bar{x}_i$ and $R_i$ values. 2. Calculate overall mean $\bar{\bar{x}}$ and average range $\bar{R}$. 3. Use constants for $n=5$: $A_2=0.577$, $D_3=0$, $D_4=2.114$. 4. Calculate control limits for x̄ and R charts. 5. Assess if process is stable by checking points within limits. (b) Process capability analysis: Given specification limits 70 to 95 psi, calculate process standard deviation estimate: $$\hat{\sigma} = \frac{\bar{R}}{d_2}$$ where $d_2=2.326$ for $n=5$. Calculate capability indices: $$C_p = \frac{USL - LSL}{6 \hat{\sigma}}$$ $$C_{pk} = \min\left(\frac{USL - \bar{\bar{x}}}{3 \hat{\sigma}}, \frac{\bar{\bar{x}} - LSL}{3 \hat{\sigma}}\right)$$ Interpret values: $C_p$ and $C_{pk} > 1$ indicate capable and centered process. --- **Summary:** - Calculate sample means and ranges. - Use control chart constants to find control limits. - Check for points outside limits or patterns indicating out-of-control. - Calculate process capability indices to assess if process meets specifications.