Give step-by-step solution with explanation and final answer:expansion during baking is measured. For this process, the target dimension is 1.5( }
‘microns +/- 0.15. Construct X bar and R chart (n =4). 3
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‘When manufacturing semiconductors, photolithography is an important step. During
this process, a light sensitive photoresist material is applied to the silicon wafer. Next,
the circuit pattern is exposed on the resist using a high-intensity UV light. Once the
resist pattern is in place, the underlying material is removed in the developing
‘process, by cither wet chemical or plasma etching. Following this, a hard-bake
process is performed to increase resist adherence and etch resistance.
An important timing issue in this process is the amount of time the circu pattern on
the resist is exposed to the high-intensity UV light. An important characteristic in this
‘hard bake process is the flow width of the resist. In other words, the amount of resist
Question:
Give step-by-step solution with explanation and final answer:expansion during baking is measured. For this process, the target dimension is 1.5( }
‘microns +/- 0.15. Construct X bar and R chart (n =4). 3
i; 3
:
[Range [verse | ;
[00s | 16 | 1
E Cos [aes]
y ; | 00s | 165 | i
k | 00s | 158 |
i [ooo | as | r
E [006 | 14a] a
et i NNN
Z Joos [Tass | WN AAA
¢ oa [Tas \ Na
Ara TR NNN NAN \
[Foss [Taser] NAA
[or [Tim] NAA
[Loos "us AN NN
no | 1 |
‘When manufacturing semiconductors, photolithography is an important step. During
this process, a light sensitive photoresist material is applied to the silicon wafer. Next,
the circuit pattern is exposed on the resist using a high-intensity UV light. Once the
resist pattern is in place, the underlying material is removed in the developing
‘process, by cither wet chemical or plasma etching. Following this, a hard-bake
process is performed to increase resist adherence and etch resistance.
An important timing issue in this process is the amount of time the circu pattern on
the resist is exposed to the high-intensity UV light. An important characteristic in this
‘hard bake process is the flow width of the resist. In other words, the amount of resist
This is AI generated Response.
Asked by: Umesh
Created at: 2025-08-01 20:25:32
Final Answer:
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Let's solve the problem **step-by-step** as requested.
---
## **Given Data**
- **Target dimension:** 1.50 μm ± 0.15 μm
- **Sample size per subgroup (n):** 4
- **Data:** Provided in the table (Range and Average for each subgroup)
---
## **Step 1: List the Data**
Extract the data from the image:
| Range | Average |
|-------|---------|
| 0.05 | 1.60 |
| 0.10 | 1.45 |
| 0.08 | 1.65 |
| 0.07 | 1.55 |
| 0.04 | 1.58 |
| 0.09 | 1.46 |
| 0.08 | 1.52 |
| 0.06 | 1.44 |
| 0.06 | 1.57 |
| 0.05 | 1.60 |
| 0.07 | 1.48 |
| 0.04 | 1.36 |
| 0.06 | 1.55 |
| 0.09 | 1.53 |
| 0.08 | 1.56 |
| 0.10 | 1.48 |
| 0.07 | 1.41 |
| 0.04 | 1.59 |
There are **18 subgroups**.
---
## **Step 2: Calculate \(\overline{X}\) and \(\overline{R}\)**
\[
\overline{X} = \text{Average of the subgroup averages}
\]
\[
\overline{R} = \text{Average of the subgroup ranges}
\]
### **Calculate \(\overline{X}\):**
Sum of averages:
\[
S_X = 1.60 + 1.45 + 1.65 + 1.55 + 1.58 + 1.46 + 1.52 + 1.44 + 1.57 + 1.60 + 1.48 + 1.36 + 1.55 + 1.53 + 1.56 + 1.48 + 1.41 + 1.59
\]
\[
S_X = 27.18
\]
\[
\overline{X} = \frac{27.18}{18} = 1.51
\]
### **Calculate \(\overline{R}\):**
Sum of ranges:
\[
S_R = 0.05 + 0.10 + 0.08 + 0.07 + 0.04 + 0.09 + 0.08 + 0.06 + 0.06 + 0.05 + 0.07 + 0.04 + 0.06 + 0.09 + 0.08 + 0.10 + 0.07 + 0.04
\]
\[
S_R = 1.13
\]
\[
\overline{R} = \frac{1.13}{18} \approx 0.063
\]
---
## **Step 3: Control Chart Factors for n = 4**
From standard tables:
- \(A_2 = 0.729\)
- \(D_3 = 0\)
- \(D_4 = 2.282\)
---
## **Step 4: Calculate Control Limits**
### **X-bar Chart:**
\[
\text{UCL}_{\overline{X}} = \overline{X} + A_2 \overline{R}
\]
\[
\text{LCL}_{\overline{X}} = \overline{X} - A_2 \overline{R}
\]
Plug in the numbers:
\[
\text{UCL}_{\overline{X}} = 1.51 + (0.729 \times 0.063) = 1.51 + 0.045927 = 1.556
\]
\[
\text{LCL}_{\overline{X}} = 1.51 - 0.045927 = 1.464
\]
---
### **R Chart:**
\[
\text{UCL}_{R} = D_4 \overline{R}
\]
\[
\text{LCL}_{R} = D_3 \overline{R}
\]
\[
\text{UCL}_R = 2.282 \times 0.063 \approx 0.1448
\]
\[
\text{LCL}_R = 0 \times 0.063 = 0
\]
---
## **Step 5: Final Answers**
**X-bar chart:**
- \(\overline{X} = 1.51\)
- **UCL:** 1.56 (rounded)
- **CL:** 1.51
- **LCL:** 1.46 (rounded)
**R chart:**
- \(\overline{R} = 0.063\)
- **UCL:** 0.145 (rounded)
- **CL:** 0.063
- **LCL:** 0
---
## **Summary Table**
| Chart | LCL | CL | UCL |
|---------|-------|-------|-------|
| X-bar | 1.46 | 1.51 | 1.56 |
| R | 0 | 0.063 | 0.145 |
---
**You would plot the subgroup averages and ranges against these limits for your control charts!**
Let me know if you need to see the control chart plot or further explanation!
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expansion during baking is measured. For this process, the target dimension is 1.5( }
‘microns +/- 0.15. Construct X bar and R chart (n =4). 3
i; 3
:
[Range [verse | ;
[00s | 16 | 1
E Cos [aes]
y ; | 00s | 165 | i
k | 00s | 158 |
i [ooo | as | r
E [006 | 14a] a
et i NNN
Z Joos [Tass | WN AAA
¢ oa [Tas \ Na
Ara TR NNN NAN \
[Foss [Taser] NAA
[or [Tim] NAA
[Loos "us AN NN
no | 1 |
‘When manufacturing semiconductors, photolithography is an important step. During
this process, a light sensitive photoresist material is applied to the silicon wafer. Next,
the circuit pattern is exposed on the resist using a high-intensity UV light. Once the
resist pattern is in place, the underlying material is removed in the developing
‘process, by cither wet chemical or plasma etching. Following this, a hard-bake
process is performed to increase resist adherence and etch resistance.
An important timing issue in this process is the amount of time the circu pattern on
the resist is exposed to the high-intensity UV light. An important characteristic in this
‘hard bake process is the flow width of the resist. In other words, the amount of resist