1. **State the problem:** We have tensile test data for a plastic cord with subgroup size $n=4$ and $N=10$ samples. Each sample has a mean ($\bar{X}$) and range ($R$). We need to construct mean and range control charts to check if the process is in control. 2. **Given data:** Sample No. | Mean ($\bar{X}$) | Range ($R$) 1 | 476 | 32 2 | 466 | 24 3 | 484 | 32 4 | 466 | 26 5 | 470 | 24 6 | 498 | 25 7 | 464 | 24 8 | 484 | 24 9 | 482 | 22 10 | 506 | 23 3. **Calculate overall mean of means ($\bar{\bar{X}}$):** $$\bar{\bar{X}} = \frac{476 + 466 + 484 + 466 + 470 + 498 + 464 + 484 + 482 + 506}{10} = \frac{4796}{10} = 479.6$$ 4. **Calculate overall mean of ranges ($\bar{R}$):** $$\bar{R} = \frac{32 + 24 + 32 + 26 + 24 + 25 + 24 + 24 + 22 + 23}{10} = \frac{256}{10} = 25.6$$ 5. **Determine control chart constants for subgroup size $n=4$:** From standard tables: $$A_2 = 0.729, \quad D_3 = 0, \quad D_4 = 2.282$$ 6. **Calculate control limits for mean chart:** $$UCL_{\bar{X}} = \bar{\bar{X}} + A_2 \times \bar{R} = 479.6 + 0.729 \times 25.6 = 479.6 + 18.6624 = 498.2624$$ $$LCL_{\bar{X}} = \bar{\bar{X}} - A_2 \times \bar{R} = 479.6 - 18.6624 = 460.9376$$ 7. **Calculate control limits for range chart:** $$UCL_R = D_4 \times \bar{R} = 2.282 \times 25.6 = 58.4192$$ $$LCL_R = D_3 \times \bar{R} = 0 \times 25.6 = 0$$ 8. **Check if any points are out of control:** - For mean chart, check if any $\bar{X}$ is outside $[460.94, 498.26]$. - For range chart, check if any $R$ is outside $[0, 58.42]$. From data: - Mean values: 506 (sample 10) is above UCL (498.26), so out of control. - Range values: all are within limits. 9. **Conclusion:** The process is not in control due to sample 10 mean exceeding the upper control limit. This indicates an assignable cause. 10. **Revise central line and control limits excluding out-of-control sample 10:** - New $\bar{\bar{X}}$ excluding sample 10: $$\frac{4796 - 506}{9} = \frac{4290}{9} = 476.67$$ - New $\bar{R}$ excluding sample 10: $$\frac{256 - 23}{9} = \frac{233}{9} = 25.89$$ 11. **Recalculate control limits:** $$UCL_{\bar{X}} = 476.67 + 0.729 \times 25.89 = 476.67 + 18.87 = 495.54$$ $$LCL_{\bar{X}} = 476.67 - 18.87 = 457.80$$ $$UCL_R = 2.282 \times 25.89 = 59.06$$ $$LCL_R = 0$$ 12. **Revised conclusion:** With revised limits, all samples except sample 10 are within control limits. The process is now in control after removing the assignable cause. **Final answer:** The process was initially out of control due to sample 10. After removing it and recalculating control limits, the process is in control.