Temperature is used to measure the output of a production process. When the process is in control, the mean of the process is = 127.5 and the standard deviation is a 0.4. (a) Compute the upper and lower control limits if samples of size 6 are to be used. (Round your answers to two decimal places.) UCL = LCL =
Temperature is used to measure the output of a production process. When the process is in control, the mean of the process is = 127.5 and the standard deviation is a 0.4. (a) Compute the upper and lower control limits if samples of size 6 are to be used. (Round your answers to two decimal places.) UCL = LCL =
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ISBN:9781337406659
Author:WINSTON, Wayne L.
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![## Control Limits in Production Process Monitoring
**Topic: Computing Control Limits for Monitoring Production Processes Using Temperature**
Temperature is often employed to measure the output of a production process to ensure quality control. When evaluating the process, if it is deemed to be in control, it typically follows defined statistical parameters.
### Given Parameters:
- **Mean of the process (μ)**: 127.5
- **Standard deviation (σ)**: 0.4
### Problem:
Compute the upper and lower control limits if samples of size 6 are to be used. Ensure to round your answers to two decimal places.
### Solution Approach:
1. **Control Limits** are determined based on the desired confidence level, typically using the formula for control limits in a standard normal distribution.
2. **Formulas for Control Limits**:
- **Upper Control Limit (UCL)**:
\[ UCL = \mu + Z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]
- **Lower Control Limit (LCL)**:
\[ LCL = \mu - Z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \mu \) = Mean of the process
- \( \sigma \) = Standard deviation
- \( Z \) = Z-value (typically 3 for 99.73% confidence)
- \( n \) = Sample size
### Solution:
- **Sample Size (n)**: 6
- **Z-value for 99.73% Confidence** (commonly used in control charts): 3
- **Calculation**:
\[ UCL = 127.5 + 3 \times \left(\frac{0.4}{\sqrt{6}}\right) \]
\[ LCL = 127.5 - 3 \times \left(\frac{0.4}{\sqrt{6}}\right) \]
### Results:
- **UCL =** [Input box for users to compute]
- **LCL =** [Input box for users to compute]
This computation allows process engineers to monitor and control the production process efficiently, ensuring that deviations remain within acceptable limits for quality assurance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4959d4f7-c515-4b84-8a14-00389cef5562%2F0e11cfb0-96bd-4812-a46c-84f055b3490f%2Ffrbd9pm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Control Limits in Production Process Monitoring
**Topic: Computing Control Limits for Monitoring Production Processes Using Temperature**
Temperature is often employed to measure the output of a production process to ensure quality control. When evaluating the process, if it is deemed to be in control, it typically follows defined statistical parameters.
### Given Parameters:
- **Mean of the process (μ)**: 127.5
- **Standard deviation (σ)**: 0.4
### Problem:
Compute the upper and lower control limits if samples of size 6 are to be used. Ensure to round your answers to two decimal places.
### Solution Approach:
1. **Control Limits** are determined based on the desired confidence level, typically using the formula for control limits in a standard normal distribution.
2. **Formulas for Control Limits**:
- **Upper Control Limit (UCL)**:
\[ UCL = \mu + Z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]
- **Lower Control Limit (LCL)**:
\[ LCL = \mu - Z \times \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \mu \) = Mean of the process
- \( \sigma \) = Standard deviation
- \( Z \) = Z-value (typically 3 for 99.73% confidence)
- \( n \) = Sample size
### Solution:
- **Sample Size (n)**: 6
- **Z-value for 99.73% Confidence** (commonly used in control charts): 3
- **Calculation**:
\[ UCL = 127.5 + 3 \times \left(\frac{0.4}{\sqrt{6}}\right) \]
\[ LCL = 127.5 - 3 \times \left(\frac{0.4}{\sqrt{6}}\right) \]
### Results:
- **UCL =** [Input box for users to compute]
- **LCL =** [Input box for users to compute]
This computation allows process engineers to monitor and control the production process efficiently, ensuring that deviations remain within acceptable limits for quality assurance.
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