VIPSolutionsGive step-by-step solution with explanation and final answer:Question 1 [16 marks] 1.1 Distinguish between random variation and nonrandom variation in a process. (6) 1.2 Discuss the different nonrandom signals that can occur when using a process control chart to analyse a process. (5) 1.3 Explain the various types of attributes. (5) Question 2 [10 marks] 2.1 Acertain hotel would like to analyse the time (measured in minutes) it takes for the guests’ luggage to be delivered to their rooms. The following data was recorded over a 5-day period. Con [men] 67 117 97 [ 2 [76 114 90 [5 [110 99 113 Calculate the control limits for an ¥ chart. with the range as measure of dispersion. (5)
Question:
Give step-by-step solution with explanation and final answer:
Question 1 [16 marks]
1.1 Distinguish between random variation and nonrandom variation in a process. (6)
1.2 Discuss the different nonrandom signals that can occur when using a process control
chart to analyse a process. (5)
1.3 Explain the various types of attributes. (5)
Question 2 [10 marks]
2.1 Acertain hotel would like to analyse the time (measured in minutes) it takes for the guests’
luggage to be delivered to their rooms. The following data was recorded over a 5-day
period.
Con [men]
67 117 97
[ 2 [76 114 90
[5 [110 99 113
Calculate the control limits for an ¥ chart. with the range as measure of dispersion. (5)
This is AI generated Response.
Asked by: Veera manikanta
Created at: 2025-09-23 23:43:00
Asked by: Veera manikanta
Created at: 2025-09-23 23:43:00
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## Question 2.1: X-bar Chart Control Limits Calculation
Let's solve the question step by step:
### **Step 1: Calculate the Mean (\(\bar{x}\)) for Each Day**
| Day | Time (min) | \(\bar{x}_i\) |
|-----|--------------------|---------------|
| 1 | 6.7, 11.7, 9.7 | (6.7+11.7+9.7)/3 = **9.37** |
| 2 | 7.6, 11.4, 9.0 | (7.6+11.4+9.0)/3 = **9.33** |
| 3 | 9.5, 8.9, 9.9 | (9.5+8.9+9.9)/3 = **9.43** |
| 4 | 9.8, 13.2, 6.9 | (9.8+13.2+6.9)/3 = **9.97** |
| 5 | 11.0, 9.9, 11.3 | (11.0+9.9+11.3)/3 = **10.73** |
---
### **Step 2: Calculate the Range (\(R\)) for Each Day**
| Day | Max | Min | Range (\(R_i\)) |
|-----|------|------|------------------|
| 1 | 11.7 | 6.7 | 11.7 - 6.7 = **5.0** |
| 2 | 11.4 | 7.6 | 11.4 - 7.6 = **3.8** |
| 3 | 9.9 | 8.9 | 9.9 - 8.9 = **1.0** |
| 4 | 13.2 | 6.9 | 13.2 - 6.9 = **6.3** |
| 5 | 11.3 | 9.9 | 11.3 - 9.9 = **1.4** |
---
### **Step 3: Calculate the Grand Mean (\(\overline{\bar{x}}\)) and Average Range (\(\bar{R}\))**
\[
\overline{\bar{x}} = \frac{9.37 + 9.33 + 9.43 + 9.97 + 10.73}{5} = \frac{48.83}{5} = \textbf{9.77}
\]
\[
\bar{R} = \frac{5.0 + 3.8 + 1.0 + 6.3 + 1.4}{5} = \frac{17.5}{5} = \textbf{3.50}
\]
---
### **Step 4: Find Control Chart Constants**
For a sample size \(n = 3\):
- \(A_2 = 1.023\) (from standard statistical tables for X-bar charts)
---
### **Step 5: Calculate Control Limits**
\[
\text{UCL} = \overline{\bar{x}} + A_2 \bar{R}
\]
\[
\text{LCL} = \overline{\bar{x}} - A_2 \bar{R}
\]
\[
\text{UCL} = 9.77 + (1.023 \times 3.50) = 9.77 + 3.58 = \textbf{13.35}
\]
\[
\text{LCL} = 9.77 - (1.023 \times 3.50) = 9.77 - 3.58 = \textbf{6.19}
\]
---
## **Final Answers**
- **Grand Mean (\(\overline{\bar{x}}\))**: **9.77 min**
- **Average Range (\(\bar{R}\))**: **3.50 min**
- **Upper Control Limit (UCL)**: **13.35 min**
- **Lower Control Limit (LCL)**: **6.19 min**
---
### **Summary Table**
| Statistic | Value |
|-------------------------------|---------|
| Grand Mean (\(\overline{\bar{x}}\)) | 9.77 |
| Average Range (\(\bar{R}\)) | 3.50 |
| UCL | 13.35 |
| LCL | 6.19 |
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