1. **Problem Statement:** We are given sample data from a process with a target of 25. We need to: i) State key indicators of a properly centered process. ii) Determine Upper Control Limit (UCL) and Lower Control Limit (LCL) for an x-bar chart. iii) Sketch the process chart and advise if the process is in control. iv) State if the process is properly centered. 2. **Key Indicators of a Properly Centered Process:** - The process mean (average) is close to the target value. - Data points are symmetrically distributed around the target. - Control limits are set such that most points fall within them. - No patterns or trends indicating shifts or drifts. 3. **Formulas for Control Limits of x-bar Chart:** - $\bar{X} = \text{average of sample means}$ - $R = \text{average of sample ranges}$ - $UCL = \bar{X} + A_2 \times R$ - $LCL = \bar{X} - A_2 \times R$ Where $A_2$ is a constant depending on sample size $n$. For $n=5$, $A_2=0.577$ (standard table value). 4. **Calculate Sample Means and Ranges:** Samples (each of size 5) are: Sample 1: 29.5, 23, 23.7, 23, 23.7 Sample 2: 22.9, 20.1, 23, 29.8, 20.2 Sample 3: 25.9, 23.6, 24.2, 20.6, 27.2 Sample 4: 25.3, 23.2, 20.2, 27.1, 20.6 Sample 5: 24.7, 25.1, 20.2, 24.9, 20.6 Sample 6: 28.4, 29.7, 27.2, 28.1, 27.2 Sample 7: 21.9, 23.5, 20.6, 26.3, 20.6 Sample 8: 26.9, 21.1, 20.6, 21.7, 20.6 Sample 9: 25.5, 27.6, 27.1, 24.9, 26.9 Sample 10: 23.9, 28.6, 24.3, 26.3, 24.3 Calculate mean and range for each sample: - Sample 1 mean: $\frac{29.5+23+23.7+23+23.7}{5} = 24.58$ - Sample 1 range: $29.5 - 23 = 6.5$ - Sample 2 mean: $\frac{22.9+20.1+23+29.8+20.2}{5} = 23.2$ - Sample 2 range: $29.8 - 20.1 = 9.7$ - Sample 3 mean: $\frac{25.9+23.6+24.2+20.6+27.2}{5} = 24.3$ - Sample 3 range: $27.2 - 20.6 = 6.6$ - Sample 4 mean: $\frac{25.3+23.2+20.2+27.1+20.6}{5} = 23.28$ - Sample 4 range: $27.1 - 20.2 = 6.9$ - Sample 5 mean: $\frac{24.7+25.1+20.2+24.9+20.6}{5} = 23.1$ - Sample 5 range: $25.1 - 20.2 = 4.9$ - Sample 6 mean: $\frac{28.4+29.7+27.2+28.1+27.2}{5} = 28.12$ - Sample 6 range: $29.7 - 27.2 = 2.5$ - Sample 7 mean: $\frac{21.9+23.5+20.6+26.3+20.6}{5} = 22.58$ - Sample 7 range: $26.3 - 20.6 = 5.7$ - Sample 8 mean: $\frac{26.9+21.1+20.6+21.7+20.6}{5} = 22.18$ - Sample 8 range: $26.9 - 20.6 = 6.3$ - Sample 9 mean: $\frac{25.5+27.6+27.1+24.9+26.9}{5} = 26.4$ - Sample 9 range: $27.6 - 24.9 = 2.7$ - Sample 10 mean: $\frac{23.9+28.6+24.3+26.3+24.3}{5} = 25.48$ - Sample 10 range: $28.6 - 23.9 = 4.7$ 5. **Calculate Overall Mean and Average Range:** - $\bar{X} = \frac{24.58 + 23.2 + 24.3 + 23.28 + 23.1 + 28.12 + 22.58 + 22.18 + 26.4 + 25.48}{10} = 24.72$ - $R = \frac{6.5 + 9.7 + 6.6 + 6.9 + 4.9 + 2.5 + 5.7 + 6.3 + 2.7 + 4.7}{10} = 5.75$ 6. **Calculate Control Limits:** - $UCL = 24.72 + 0.577 \times 5.75 = 24.72 + 3.32 = 28.04$ - $LCL = 24.72 - 0.577 \times 5.75 = 24.72 - 3.32 = 21.4$ 7. **Interpretation:** - Plot sample means on the control chart with UCL=28.04 and LCL=21.4. - All sample means lie between 21.4 and 28.04, so the process is in control. 8. **Is the Process Properly Centered?** - The target is 25, but the overall mean is 24.72, close to 25. - The process is slightly below target but well within control limits. - Hence, the process is properly centered. **Final Answer:** - $UCL = 28.04$ - $LCL = 21.4$ - Process is in control and properly centered.