1. **Problem Statement:** We have 10 samples, each with 4 measurements, and a target process mean of 25. We need to: i) Determine the upper and lower control limits for an X-bar chart. ii) Sketch the process charts and advise if the process is in control. iii) State if the process is properly centered. iv) Complete the table with sample means (X-bar) and ranges. 2. **Formulas and Rules:** - The X-bar chart control limits are given by: $$\text{UCL} = \bar{X} + A_2 \times \bar{R}$$ $$\text{LCL} = \bar{X} - A_2 \times \bar{R}$$ where $\bar{X}$ is the overall average of sample means, $\bar{R}$ is the average range, and $A_2$ is a constant depending on sample size $n=4$. For $n=4$, $A_2=0.729$. - Range $R$ for each sample is $R = \max - \min$ of the sample. 3. **Calculate sample means and ranges:** Sample 1: Mean $= \frac{29.5+23+20.9+24.9}{4} = 24.575$, Range $= 29.5 - 20.9 = 8.6$ Sample 2: Mean $= \frac{22.9+20.1+23.7+26.9}{4} = 23.4$, Range $= 26.9 - 20.1 = 6.8$ Sample 3: Mean $= \frac{25.9+23.6+23+24.3}{4} = 24.2$, Range $= 25.9 - 23 = 2.9$ Sample 4: Mean $= \frac{25.3+23.2+29.8+22.3}{4} = 25.15$, Range $= 29.8 - 22.3 = 7.5$ Sample 5: Mean $= \frac{24.7+25.1+20.6+23}{4} = 23.35$, Range $= 25.1 - 20.6 = 4.5$ Sample 6: Mean $= \frac{28.4+29.7+27.1+23.7}{4} = 27.225$, Range $= 29.7 - 23.7 = 6.0$ Sample 7: Mean $= \frac{21.9+23.5+24.9+24.2}{4} = 23.625$, Range $= 24.9 - 21.9 = 3.0$ Sample 8: Mean $= \frac{26.9+21.1+28.1+20.2}{4} = 24.075$, Range $= 28.1 - 20.2 = 7.9$ Sample 9: Mean $= \frac{25.5+27.6+26.3+27.2}{4} = 26.65$, Range $= 27.6 - 25.5 = 2.1$ Sample 10: Mean $= \frac{23.9+28.6+21.7+20.6}{4} = 23.7$, Range $= 28.6 - 20.6 = 8.0$ 4. **Calculate overall averages:** $$\bar{X} = \frac{24.575 + 23.4 + 24.2 + 25.15 + 23.35 + 27.225 + 23.625 + 24.075 + 26.65 + 23.7}{10} = 24.795$$ $$\bar{R} = \frac{8.6 + 6.8 + 2.9 + 7.5 + 4.5 + 6.0 + 3.0 + 7.9 + 2.1 + 8.0}{10} = 5.33$$ 5. **Calculate control limits:** $$\text{UCL} = 24.795 + 0.729 \times 5.33 = 24.795 + 3.887 = 28.682$$ $$\text{LCL} = 24.795 - 0.729 \times 5.33 = 24.795 - 3.887 = 20.908$$ 6. **Interpretation:** - The process is in control if all sample means lie between $20.908$ and $28.682$. - All sample means calculated are within these limits. - The process mean $\bar{X} = 24.795$ is close to the target 25, so the process is properly centered. 7. **Completed Table 3:** | Sample | X-bar | Range | |--------|-------|-------| | 1 | 24.575| 8.6 | | 2 | 23.4 | 6.8 | | 3 | 24.2 | 2.9 | | 4 | 25.15 | 7.5 | | 5 | 23.35 | 4.5 | | 6 | 27.225| 6.0 | | 7 | 23.625| 3.0 | | 8 | 24.075| 7.9 | | 9 | 26.65 | 2.1 | | 10 | 23.7 | 8.0 | Final answers: - Upper Control Limit (UCL) = $28.682$ - Lower Control Limit (LCL) = $20.908$ - Process is in control and properly centered.