A process sampled 20 times with a sample of size 8 resulted in = 28.5 and R = 1.8. Compute the upper and lower control limits for the chart for this process. (Round your answers to two decimal places.) UCL = LCL = Compute the upper and lower control limits for the R chart for this process. (Round your answers to two decimal places.) UCL = LCL =
A process sampled 20 times with a sample of size 8 resulted in = 28.5 and R = 1.8. Compute the upper and lower control limits for the chart for this process. (Round your answers to two decimal places.) UCL = LCL = Compute the upper and lower control limits for the R chart for this process. (Round your answers to two decimal places.) UCL = LCL =
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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![**Control Limits Calculation for the X-bar and R Charts**
**Problem Statement:**
A process sampled 20 times with a sample size of 8 resulted in \(\bar{x} = 28.5\) and \(\bar{R} = 1.8\). Compute the upper and lower control limits for the \(\bar{x}\) chart for this process. (Round your answers to two decimal places.)
* Upper Control Limit (UCL) = ______
* Lower Control Limit (LCL) = ______
Compute the upper and lower control limits for the R chart for this process. (Round your answers to two decimal places.)
* Upper Control Limit (UCL) = ______
* Lower Control Limit (LCL) = ______
**Explanation of Control Limits:**
1. **X-bar Chart (Control Limits for Mean):**
- The X-bar chart monitors the mean values of a sample over time.
- Upper Control Limit (UCL) and Lower Control Limit (LCL) ensure the process remains within these boundaries and exhibits only common cause variation.
2. **R Chart (Control Limits for Range):**
- The R chart monitors the variation or range within the sample over time.
- UCL and LCL for the R chart indicate the acceptable range of variability for the process.
**Formulas:**
For the X-bar chart:
\[ UCL_{\bar{x}} = \bar{x} + A_2 \bar{R} \]
\[ LCL_{\bar{x}} = \bar{x} - A_2 \bar{R} \]
For the R chart:
\[ UCL_{R} = D_4 \bar{R} \]
\[ LCL_{R} = D_3 \bar{R} \]
**Where to Find Constants (\(A_2, D_3, D_4\)):**
These constants depend on the sample size (n). For a sample size of 8:
- \(A_2 = 0.373\)
- \(D_3 = 0.136\)
- \(D_4 = 1.864\)
**Calculation:**
1. For the X-bar chart:
- \(UCL_{\bar{x}} = 28.5 + (0.373 * 1.8) = 28.5 + 0.6714 = 29.17\)
- \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4959d4f7-c515-4b84-8a14-00389cef5562%2F9edf07ab-adbd-47d9-b72f-ff799c87604e%2Fcggq8wv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Control Limits Calculation for the X-bar and R Charts**
**Problem Statement:**
A process sampled 20 times with a sample size of 8 resulted in \(\bar{x} = 28.5\) and \(\bar{R} = 1.8\). Compute the upper and lower control limits for the \(\bar{x}\) chart for this process. (Round your answers to two decimal places.)
* Upper Control Limit (UCL) = ______
* Lower Control Limit (LCL) = ______
Compute the upper and lower control limits for the R chart for this process. (Round your answers to two decimal places.)
* Upper Control Limit (UCL) = ______
* Lower Control Limit (LCL) = ______
**Explanation of Control Limits:**
1. **X-bar Chart (Control Limits for Mean):**
- The X-bar chart monitors the mean values of a sample over time.
- Upper Control Limit (UCL) and Lower Control Limit (LCL) ensure the process remains within these boundaries and exhibits only common cause variation.
2. **R Chart (Control Limits for Range):**
- The R chart monitors the variation or range within the sample over time.
- UCL and LCL for the R chart indicate the acceptable range of variability for the process.
**Formulas:**
For the X-bar chart:
\[ UCL_{\bar{x}} = \bar{x} + A_2 \bar{R} \]
\[ LCL_{\bar{x}} = \bar{x} - A_2 \bar{R} \]
For the R chart:
\[ UCL_{R} = D_4 \bar{R} \]
\[ LCL_{R} = D_3 \bar{R} \]
**Where to Find Constants (\(A_2, D_3, D_4\)):**
These constants depend on the sample size (n). For a sample size of 8:
- \(A_2 = 0.373\)
- \(D_3 = 0.136\)
- \(D_4 = 1.864\)
**Calculation:**
1. For the X-bar chart:
- \(UCL_{\bar{x}} = 28.5 + (0.373 * 1.8) = 28.5 + 0.6714 = 29.17\)
- \(
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