University Hospital has been concerned with the number of errors found in its billing statements to patients. An audit of 100 bills per week over the past twelve weeks revealed the following number of errors:
Week Number of Errors
1 4
2 5
3 6
4 6
5 3
6 2
7 6
8 7
9 3
10 4
11 3
12 4
What type of chart should be used?
p-chart
R-chart
x-bar chart
c-chart
Develop control charts with z = 3: (a) CL, (b) UCL, (c) LCL.
Note. If the lower control limit is negative, round the LCL to zero.
(a) errors (Round your answer to 2 decimal places, the tolerance is +/-0.01)
(b) (Round your answer to 2 decimal places, the tolerance is +/-0.05)
(c) (Round your answer to 2 decimal places, the tolerance is +/-0.05)
Is the process in control?
The process is in control
The process is not in control
P chart
To determine the appropriate type of chart to be used, we need to analyze the nature of the data. In this case, we are measuring the number of errors in billing statements, which is a type of nonconformity data.
For nonconformity data, the appropriate control chart to use is the c-chart.
Next, we will calculate the control limits for the c-chart using the given information.
(a) CL (Center Line): The center line represents the average number of errors per sample and is calculated by summing up the number of errors in each sample and dividing it by the number of samples.
CL = (4 + 5 + 6 + 6 + 3 + 2 + 6 + 7 + 3 + 4 + 3 + 4) / 12 = 4.25 (rounded to two decimal places)
(b) UCL (Upper Control Limit): The upper control limit is calculated by adding three times the square root of the center line value.
UCL = CL + 3 * √CL
UCL = 4.25 + 3 * √4.25 ≈ 9.70 (rounded to two decimal places)
(c) LCL (Lower Control Limit): The lower control limit is calculated by subtracting three times the square root of the center line value. If the result is negative, it should be rounded to zero.
LCL = CL - 3 * √CL
LCL = 4.25 - 3 * √4.25 ≈ 0 (rounded to two decimal places)
The control limits for the c-chart are:
(a) CL = 4.25
(b) UCL ≈ 9.70
(c) LCL ≈ 0
Now, to determine if the process is in control, we need to analyze the data points and check if any points fall outside the control limits on the control chart.
Based on the provided data, we can calculate the number of errors per week and compare it with the control limits.
Week 1: 4 errors (within control limits)
Week 2: 5 errors (within control limits)
Week 3: 6 errors (within control limits)
Week 4: 6 errors (within control limits)
Week 5: 3 errors (within control limits)
Week 6: 2 errors (within control limits)
Week 7: 6 errors (within control limits)
Week 8: 7 errors (within control limits)
Week 9: 3 errors (within control limits)
Week 10: 4 errors (within control limits)
Week 11: 3 errors (within control limits)
Week 12: 4 errors (within control limits)
None of the data points fall outside of the control limits, indicating that the process is in control.
Therefore, the process is in control and no special cause variation is present.
To determine what type of chart should be used, we need to consider the type of data we have and the purpose of the chart. In this case, we have data on the number of errors found in billing statements, which is discrete/count data.
Based on the provided options, the most appropriate chart to use in this scenario is the c-chart. The c-chart is used for monitoring the number of defects or errors when the sample size is constant. It is ideal for situations where the data is discrete and the sample size is large and relatively constant.
Now, let's proceed to develop control charts with z = 3. The control limits can be calculated using the following formulas:
Control Limits:
UCL = CL + 3 * sqrt(CL)
LCL = CL - 3 * sqrt(CL)
(a) To calculate the control limits for errors (CL), we first need to find the average number of errors per week. Sum up the number of errors for each week and divide by the number of weeks (12):
CL = (4 + 5 + 6 + 6 + 3 + 2 + 6 + 7 + 3 + 4 + 3 + 4) / 12
(b) UCL = CL + 3 * sqrt(CL)
(c) LCL = CL - 3 * sqrt(CL)
To determine if the process is in control, we need to check if any data points fall outside the control limits.
Plug in the values obtained above into the formulas mentioned and round them according to the given tolerances, and you will have the answers:
(a) CL = (Sum of errors per week) / (Number of weeks)
(b) UCL = CL + 3 * sqrt(CL)
(c) LCL = max(CL - 3 * sqrt(CL), 0)
Once you have the control limits calculated, you need to plot the data points on a control chart and check if any points fall outside the control limits. If all points are within the control limits, then the process is considered to be in control. If any points fall outside the control limits or show a significant trend or pattern, then the process is not in control.
By analyzing the control chart and checking if the data points fall within the control limits or show any unusual patterns, you can determine if the process is in control or not.