Unit6Math

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Seneca College *

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MISC

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Industrial Engineering

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Jan 9, 2024

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docx

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6

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Unit 6 6.1: a) Determine the upper and lower control limits and the overall means for x-charts and R- charts. 1. Calculate the overall mean of the sample means (\( \bar{X} \)) and the overall mean of the ranges (\( \bar{R} \)). \[ \bar{X} \] = (sum of all sample means) / number of samples \[ \bar{R} \] = (sum of all ranges) / number of samples 2. Use the appropriate control chart factors for a sample size of 6 (n = 6). Based on standard tables: \( A_2 \) = 0.483 \( D_3 \) = 0 \( D_4 \) = 2.114 3. Calculate the control limits: Upper Control Limit for \( \bar{X} \) (UCLx) = \( \bar{X} \) + \( A_2 \) * \( \bar{R} \) Lower Control Limit for \( \bar{X} \) (LCLx) = \( \bar{X} \) - \( A_2 \) * \( \bar{R} \) Upper Control Limit for R (UCLR) = \( D_4 \) * \( \bar{R} \) Lower Control Limit for R (LCLR) = \( D_3 \) * \( \bar{R} \) Let's start with the calculations for part A. Given the data: Sample Means: 10.012, 10.007, 9.991, 10.006, 9.997, 9.999, 10.001, 10.005, 9.995, 10.001, 10.003, 10.006 Ranges: 0.009, 0.014, 0.007, 0.026, 0.016, 0.012, 0.008, 0.013, 0.015, 0.011, 0.014, 0.009 1. Calculate the overall mean of the sample means (\( \bar{X} \)) and the overall mean of the ranges (\( \bar{R} \)): \[ \bar{X} \] = (sum of all sample means) / 12 \[ \bar{R} \] = (sum of all ranges) / 12
\[ \bar{X} \] = (10.012 + 10.007 + 9.991 + 10.006 + 9.997 + 9.999 + 10.001 + 10.005 + 9.995 + 10.001 + 10.003 + 10.006) / 12 \[ \bar{X} \] = 120.023 / 12 \[ \bar{X} \] = 10.00225 \[ \bar{R} \] = (0.009 + 0.014 + 0.007 + 0.026 + 0.016 + 0.012 + 0.008 + 0.013 + 0.015 + 0.011 + 0.014 + 0.009) / 12 \[ \bar{R} \] = 0.154 / 12 \[ \bar{R} \] = 0.01283 2. Using the control chart factors for n = 6: \( A_2 \) = 0.483 \( D_3 \) = 0 \( D_4 \) = 2.114 3. Calculate the control limits: UCLx = 10.00225 + 0.483 * 0.01283 UCLx = 10.00225 + 0.00620 UCLx = 10.00845 LCLx = 10.00225 - 0.483 * 0.01283 LCLx = 10.00225 - 0.00620 LCLx = 9.99605 UCLR = 2.114 * 0.01283 UCLR = 0.02712 LCLR = 0 * 0.01283 LCLR = 0 So, the control limits are: UCLx = 10.00845 LCLx = 9.99605 UCLR = 0.02712 LCLR = 0 Next, let's plot the values on the x-charts and R-charts. Here are the X-bar and R-charts for the given data:
c) Do the data indicate a process that is in control? To determine if the process is in control, we need to check if any of the sample means or ranges fall outside the control limits. From the charts, we can observe that all the sample means and ranges are within the control limits. d) Why or why not? The process is in control because: 1. None of the sample means fall outside the control limits of the X-bar chart. 2. None of the ranges fall outside the control limits of the R chart. In control chart analysis, if any point falls outside the control limits, it indicates that there may be special causes of variation, and the process might be out of control. Since all our points are within the control limits, we can conclude that the process is in control and only affected by common causes of variation. 6.2: 1. Contribution Margin (CM) for each product: For HR: \[ CM_{HR} = Selling Price_{HR} - Variable Cost_{HR} \] \[ CM_{HR} = $60 - $20 \] \[ CM_{HR} = $40 \] For HA:
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